By Andy May
You can read this post in German here, courtesy of Christian Freuer.
Here we go again, writing on the proper use of statistics in climate science. Traditionally, the most serious errors in statistical analysis are made in the social sciences, with medical papers coming in a close second. Climate science is biting at their heels.
In this case we are dealing with a dispute between Nicola Scafetta, a Professor of Atmospheric Physics at the University of Naples and Gavin Schmidt, a blogger at RealClimate.org, a climate modeler, and director at NASA’s Goddard Institute for Space Studies (GISS).
Scafetta’s original 2022 paper in Geophysical Research Letters is the origin of the dispute (downloading a pdf is free). The essence of the paper is that CMIP6 global climate models (GCMs) that produce an ECS (Equilibrium Climate Sensitivity) higher than 3°C/2xCO2 (“°C/2xCO2” means °C per doubling of CO2) are statistically significantly different (they run too hot) from observations since 1980. This result is not surprising and is in line with the recent findings by McKitrick and Christy (2020). The fact that the AR6/CMIP6 climate models run too hot and that it appears to be a function of too-high ECS is acknowledged in AR6:
“The AR5 assessed with low confidence that most, though not all, CMIP3 and CMIP5 models overestimated the observed warming trend in the tropical troposphere during the satellite period 1979-2012, and that a third to a half of this difference was due to an overestimate of the SST [sea surface temperature] trend during this period. Since the AR5, additional studies based on CMIP5 and CMIP6 models show that this warming bias in tropospheric temperatures remains.”
(AR6, p. 443)
And:
“Several studies using CMIP6 models suggest that differences in climate sensitivity may be an important factor contributing to the discrepancy between the simulated and observed tropospheric temperature trends (McKitrick and Christy, 2020; Po-Chedley et al., 2021)”
(AR6, p. 443)
The AR6 authors tried to soften the admission with clever wording, but McKitrick and Christy showed that the AR5/CMIP5 models are too warm in the tropical troposphere and fail to match observations at a statistically significant level. Yet, regardless of the evidence that AR5 was already too hot, AR6 is hotter, as admitted in AR6 on page 321:
“The AR5 assessed estimate for historical warming between 1850–1900 and 1986–2005 is 0.61 [0.55 to 0.67] °C. The equivalent in AR6 is 0.69 [0.54 to 0.79] °C, and the 0.08 [-0.01 to 0.12] °C difference is an estimate of the contribution of changes in observational understanding alone (Cross-Chapter Box 2.3, Table 1).”
(AR6, p. 321).
So, we see that the AR6 assessment that the AR6 and AR5 climate sensitivity to CO2 may be too high and that AR6 is worse than AR5 supports the work that Scafetta, McKitrick, and Christy have done in recent years.
Now let’s look at the dispute on how to compute the statistical error of the mean warming from 1980-1990 to 2011-2021 between Scafetta and Schmidt. Schmidt (2022)’s objections to Scafetta’s error analysis are posted on his blog here. Scafetta’s original Geophysical Research Letters paper was later followed by a more extended paper in Climate Dynamics (Scafetta N., 2022b) where the issue is discussed in detail in the first and second appendix.
Scafetta (2022a)’s analysis of climate model ECS
The essence of Scafetta’s argument is illustrated in figure 1.
In figure 1 we see that when ECS is greater than 3°C/2xCO2 the models run hot. The righthand plots show a comparison of the mean difference between the observations and models between the 11-year periods of 1980-1990 and 2011-2021. Scafetta’s 2022a full analysis is contained in his Table 1 where 107 CMIP6 GCM average simulations for the historical + SSP2-4.5, SSP3-7.0, and SSP5-8.5 IPCC greenhouse emissions scenarios provided by Climate Explorer are analyzed. The ERA5-T2m[1] mean global surface warming from 1980-1990 to 2011-2021 was estimated to be 0.578°C from the ERA5 worldwide grid. The IPCC/CMIP6 climate model mean warming is significantly higher for all the models plotted when ECS is greater than 3°C/2xCO2.
Schmidt’s analysis
The plots shown on the right in figure 1 are the essence of the debate between Scafetta and Schmidt. The data plotted by Schmidt (shown in our figure 2) is slightly different but shows the same thing.
In figure 2 we see that the only model ECS ensemble mean estimates (green dots) that equal or fall around the ERA5 weather reanalysis mean difference between 1980-1990 and 2011-2021 are ECS estimates of 3°C/2xCO2 or less. All ensemble ECS estimates above 3°C/2xCO2 run too hot. Thus, on the basic data Schmidt agrees with Scafetta, which is helpful.
The Dispute
The essence of the dispute is how to compute the 95% uncertainty (the error estimate) of the 2011-2021 ERA5 weather reanalysis mean relative to the 1980-1990 period. This error estimate is used to decide whether a particular model result is within the margin of error of the observations (ERA5) or not. Scafetta computes a very small ERA5 error range of 0.01°C (Scafetta N. , 2022b, Appendix) from similar products (HadCRUT5, for example) because ECMWF (European Centre for Medium-Range Weather) provides no uncertainty estimate with their weather reanalysis product (ERA5), so it must be estimated. Schmidt computes a very large ERA5 margin of error of 0.1°C using the ERA5 standard deviation for the period. It is shown with the pink band in figure 2. This is the critical value in deciding which differences between the climate model results and the observations are statistically significant.
If we assume that Scafetta’s estimate correct, figures 1 and 2 show that all climate model simulations (the green dots in figure 2) for the 21 climate models with ECS >3°C and the great majority of their simulation members (the black dots) are obviously too warm at a statistically significant level. Whereas, assuming Schmidt’s estimate correct, figure 2 suggests that three climate models with ECS>3°C partially fall within the ERA5 margin of error while the other 18 climate models run too hot.
Although Schmidt’s result does not appear to significantly change the conclusion of Scafetta (2022a, 2022b) that only the climate models with ECS<3.01°C appear to best hindcast the warming from 1980-1990 to 2011-2021, it is important to discuss the error issue. I will refer to the standard stochastic methods for the evaluation of the error of the mean discussed in the classical textbook on error analysis by Taylor (1997).
In the following I repeat the calculation made by Schmidt and comment on them using the HadCRUT5.0.1.0 annual mean global surface temperature record instead of the ERA5-T2m because it is easier to get, it is nearly equivalent to ERA5-T2m, and especially because it also reports the relative stochastic uncertainties for each year, which, as already explained, is a crucial component to evaluating the statistical significance of any differences between reality and the climate models.
Schmidt’s estimate of the error of the mean (the pink bar in Figure 2) is ± 0.1°C (95% confidence). He obtained this value by assuming that the interannual variability in the ERA5-T2m from 2011 to 2021 from the decadal mean is random noise. Practically, he calculated the average warming (0.58°C) from 2011 to 2021 using the ERA5-T2m temperature anomalies relative to the 1980-1990 mean. That is, he “baselined” the values to the 1980-1990 mean. Then he estimated the error of the mean by computing the standard deviation of the baselined values from 2011 to 2021, he then divided this standard deviation by the root of 11 (because there are N=11 years) and, finally, he multiplied the result by 1.96 to get the 95% confidence. Download a spreadsheet performing Schmidt’s and Scafetta’s calculations here.
Figure 3 shows Schmidt’s equation for the error of the mean. When this value is multiplied by 1.96, to get the 95% confidence, it gives an error of ± 0.1°C.
The equations used by Schmidt are those reported in Taylor (1997, pages 100-102). The main concern with Schmidt’s approach is that Taylor clearly explains that the equation in figure 3 for the error of the mean only works if the N yearly temperature values (Ti) are random “measurements of the same quantity x.” For example, Taylor (page 102-103) uses the above equation to estimate the error of the mean for the elastic constant k of “one” spring by using repeated measurements with the same instrument. Since the true elastic constant is only one value, the variability of the repeated measurements can be interpreted as random noise around a mean value whose standard deviation is the Standard Deviation of the Mean (SDOM).
In using the SDOM, Schmidt et al. implicitly assume that each annual mean temperature datum is a measurement of a single true decadal value and that the statistical error for each datum is given by its deviation from that decadal mean. In effect, they assume that the “true” global surface temperature does not vary between 1980 and 1990 or 2011-2021 and all deviations from the mean (or true) value are random variability.
However, the interannual variability of the global surface temperature record over these two decades is not random noise around a decadal mean. The N yearly mean temperature measurements from 2011 to 2021 are not independent “measurements of the same quantity x” but each year is a different physical state of the climate system. This is easily seen in the plot of both decades in this spreadsheet. The x-axis is labeled 2010-2022, but for the orange line, it is actually 1979-1991, I did it this way to show the differences in the two decades. Thus, according to Taylor (1997), SDOM is not the correct equation to be adopted in this specific case.
As Scafetta (2022b) explains, the global surface temperature record is highly autocorrelated because it contains the dynamical interannual evolution of the climate system produced by ENSO oscillations and other natural phenomena. These oscillations and trends are a physical signal, not noise. Scafetta (2022b) explains that given a generic time series (yt) affected by Gaussian (randomly) distributed uncertainties ξ with standard deviation σξ, the mean and the error of the mean are given by the equation in figure 4.
The equation in figure 4 gives an error of 0.01°C (at the 95% confidence level, see the spreadsheet here for the computational details). If the standard deviation of the errors are not strictly constant for each datum, the standard error to be used in the above equation is the square root of the mean of the squared uncertainties for each datum.
Scafetta’s equation derives directly from the general formula for the error propagation discussed by (Taylor, 1997, p. 60 and 75). Taylor explains that the equations on pages 60 and 75 must be adopted for estimating the error of a function of “several” independent variables each affected with an individual stochastic error, corresponding to different physical states, such as the average of a global surface temperature record of N “different” years. The uncertainty of the function (e.g., the mean on N different quantities) only depends on the statistical error of each quantity, not on the variability of the various quantities from their mean.
We can use an error propagation calculator tool available on the internet to check our calculations. I uploaded the annual mean ERA5 temperature data and the respective HadCRUT5 uncertainties and had the calculator evaluate the mean with its relative error. The result is shown in Figure 5.
Schmidt’s calculation of the standard deviation of the mean (SDOM) is based on the erroneous premise that he is making multiple measurements of the same thing, using the same method, and that, therefore, the interannual variability from the decadal mean is some kind of random noise that can be considered stochastic uncertainty. None of these conditions are true in this case. The global yearly average surface temperature anomaly is always changing for natural reasons, although its annual estimates are also affected by a small stochastic error such as those incorporated into Scafetta’s calculation. According to Taylor, it is only the errors of measure of the yearly temperature means that can determine the error of the 11-year mean from 2011 to 2021.
As Scafetta writes in the appendix to Scafetta 2022b, HadCRUT5’s global surface temperature record includes its 95% confidence interval estimate and, from 2011 to 2021, the uncertainties for the monthly and annual averages are monthly ≈ 0.05°C and annual ≈ 0.03°C. Berkeley Earth land/ocean temperature record uncertainty estimates are 0.042°C (monthly), 0.028°C (annual), and 0.022°C (decadal). The longer the time period, the lower the error of the mean becomes.
Each of the above values, year-by-year, must averaged and divided by the square-root of the number of years (in this case 11) to determine the error of the mean. In our case, the HadCRUT5 error of the mean for 2011-2021 is 0.01°C. Scafetta’s method allows for the “true” value to vary in each year, Schmidt’s method does not.
The observations used for the ERA5 weather reanalysis are very nearly the same as those used in the HadCRUT5 dataset (Lenssen et al., 2019; Morice et al., 2021; Rohde et al., 2020). As Morice et al. note, the MET Office Hadley Centre uses ERA5 for quality control.
Lenssen et al., which includes Gavin Schmidt as a co-author, does an extensive review of uncertainty in several global average temperature datasets, including ERA5. Craigmile and Guttorp provide the plot in figure 6 of the estimated yearly standard error in several global surface temperature records: GISTEMP, HadCRUT5, NOAA, GISS, JMA and Berkeley Earth.
Figure 6 shows that from 1980 to 2021, at the annual scale and at 95% confidence, the standard error of the uncertainties is much less than Schmidt’s error of the mean of 0.10°C, which, furthermore, is calculated on a time scale of 11 years. The uncertainties reported in Figure 6 are not given by the interannual temperature variability around a decadal mean. This result clearly indicates that Schmidt’s calculation is erroneous because at the 11-year time scale the error of the mean must be significantly smaller (by the root of 11 = 3.3) than the annual value.
Scafetta (2022b) argues that the errors for the annual mean of the ERA5-T2m should be of the same order of magnitude as those of other temperature reconstructions, like the closely related HadCRUT5 dataset. Thus, the error at the decadal scale must be negligible, about ±0.01°C, and this result is also confirmed by the online calculator tools for estimating the error of given functions of independent variables as shown in figure 5.
The differences between Scafetta and Schmidt are caused by the different estimates of ERA5 error. I find Scafetta’s much more realistic.
Patrick Frank helped me with this post, but any errors are mine alone.
Download the bibliography here.
ERA-T2m is the European Centre for Medium-Range Weather (ECMWF) Reanalysis 2-meter air temperature variable.
Irrespective of trivial quibbling about the surface, even Schmidt concedes the mid troposphere is way off. So there can be no dispute the physics has been misrepresented. Schmidt likes to zero in on surface quibbling because some models kinda-sorta get it right. But those same kinda-sorta-right models get other layers totally wrong. what gives?
Any model that shows any ocean surface waters sustaining more than 30C are wrong.
The INM model does not exceed the 30C but it lowers the present by 4C to get its warming trend. So WRONG now rather than by the end of the century.
They run too hot on top of inflated temperature readings. As they say, a twofer.
or maybe a toofer.
From a Climate Twoofer? a.k.a. Truther
”on top of inflated temperature readings”
Yes, although I do notice poor old 1998 a been emasculated.
Hansen dethroned 1934 as the warmest year recorded in the United States, and then he did the same for 1998, because both years did not fit the “hotter and hotter and hotter” narrative of the climate change alarmists.
If it’s just as warm in the recent past as it is today, yet there is much more CO2 in the atmosphere today, then logic would tell you that increased amounts of CO2 have not resulted in higher temperatures.
So to stifle that argument, the Temperature Data Mannipulators like Hansen went into their computers and changed the temperature profile from a benign temperature profile (not too cold and not too hot), into a “hotter and hotter and hotter” temperature profile.
It’s all a BIG LIE and our radical Leftwing politicians are using this BIG LIE to destroy Western Democracy.
That’s a pretty big lie you perpetrated, James. Proud of yourself? You shouldn’t be.
Never mind that a global mean is a meaningless concept with regards to Earth’s environment which is characterized by wild swings locally and regionally. Extreme variation is the norm.
Hacks like Gavin need to pretend there was stability unbalanced by human emissions to push their phony narrative littered with errors.
If you look at the atmosphere, there are clearly at least TWO indicators of the energy content present, one being temperature the other being wind.
The global average published only includes one of these energy systems. And hence can never be a measure of the energy total in the system.
We all know that a difference in temperature can cause wind and most would assume that this can also work in reverse. So the energy in the system is capable of being in either of two modes. You could consider the wind to be the kinetic energy form and the temperature to be the potential energy.
Imagine trying to define the energy of a park full of swings, all with kids on board, some big kids, some light kids some moving quickly some moving slowly and yet you only define the energy present by the height of the swings at any one instant.
I’d argue that any sense of an average is meaningless without reference to the inclusion of the kinetic energy at the same instant.
So when will we see a plot of the total energy in the atmosphere and not just some measure of the temperature, which is otherwise missing large portions, if not most of the data.
“which is otherwise missing large portions, if not most of the data.”
Yes, there are thousands of variables in continuous flux regulating Earth’s temperature. Because we lack the grid cell resolution to capture most of those variables accurately they must be modeled. The error bars in model assumptions are greater than the effect they are trying to isolate. Pretending that model output has any statistical significance for real world conditions is a primary error.
Averages tell you nothing about the magnitude of individual fluxes which can be determinant for future conditions.
Actually, in measuring atmospheric energy content, you also have to take into account moisture content (latent energy), pressure/altitude (potential energy), and not only temperature (dry-static energy), and wind (kinetic energy).
It makes no sense to compute temperature anomalies in the winter Arctic, where the atmosphere is extremely dry, together with temperature anomalies in the rest of the world. The result is that Arctic warming is driving most of the warming seen in the GSAT anomaly when the energy change involved is very small. That’s how they got HadCRUT 5 to show a lot more warming and absence of pause than HadCRUT 3. Expanding Arctic coverage, as the Pause was accompanied by an increase in Arctic warming.
The real measure of atmospheric energetics would require the use of enthalpy, but it is a lot more difficult to compute than temperature, and it is a concept that most people don’t handle.
Would it be difficult to calculate enthalpy? I think the relative humidity and air pressure are recorded as well as temperatures.
It would be interesting to just weight the HadCRUT grid cells by average humidity in calculating the HadCRUT global mean time series. They already weight the grid cells in proportion to area to do the calculation.
Nice suggestion! Wonder why the so-called “climate scientists” have never done this? It shouldn’t be hard to do!
Perhaps they have done it but didn’t like the results?
It has been done many times. Song et al. 2022 is a more recent example. They found that equivalent potential temperature (theta-e) increased by 1.48 C despite the dry bulb temperature only increased by 0.79 C.
The paper is BS. They claim hurricane frequency and strength has increased.
The deal killer- “Here we examine the Thetae_sfc changes under the Representative Concentration Pathways 8.5″
Can you post a link to another study that shows values of temperature and equivalent potential temperature for the period 1980 to 2019 that are significantly different from 0.79 C and 1.48 C respectively?
“Can you post a link to another study that shows values of temperature and equivalent potential temperature for the period 1980 to 2019 that are significantly different from 0.79 C and 1.48 C respectively?”
Can you understand that using RCP 8.5 invalidates their alarmist claptrap? The paper oozes with bias.
I had the opinion you had better chops than this.
RCP8.5 (or any RCP for that matter) was not used to provide the 0.79 C and 1.48 C temperature and theta-e values respectively. Those come from HadCRUT.
They designed a study to show the greatest amount of future warming. This is activism not science.
Surface equivalent potential temperature (Thetae_sfc) doesn’t match surface air temperature observations.
“The magnitude of the Thetae_sfc trends is significantly different from that of SAT. Thetae_sfc has much larger temporal variations than SAT, and the linear trend (1.48 °C) is roughly double that of SAT (0.79 °C) in the observations.”
Yep. It would be very unusual if they did match.
Yep. That’s what happens when there is a positive trend in temperature and humidity.
Potential temperature isn’t real. It would be seen in observations if it was. Since observations didn’t give them the warming rates desired they had to make up something that did.
If the objective is to be as obtuse as possible then why not indict all thermodynamic metrics as fake? Anyway, theta-e is included in observations. You see it in most upper-air station observations. Furthermore theta-e is not used to assess the warming rate. However, it is used to assess the enthalpy rate. That’s partly what this subthread is about. The other part is kinetic energy. And as I’ve said already ERA already provides a total integrated energy product already that combines enthalpy and kinetic energy.
It would be even easier to just use the total vertically integrated energy product provided by ERA already.
“Extreme variation is the norm.”
Yes, we need to keep reminding people of this, as the climate change alarmists claim every extreme weather event is abnormal. No, they are not. Extreme weather should be expected.
And there has never been any connection made between any extreme weather event and CO2, although you wouldn’t know that listening to climate change alarmists who see CO2 in everything.
Extreme weather is normal weather.
The extreme level of propaganda people are exposed to is the problem. Because blatant lies are presented as unquestionable fact people can’t even conceive that it could be wrong. They cling to their “trusted news source’s” lies over empirical evidence because they have heard the lies hundreds of times and can’t accept new information because the belief is so ingrained.
It’s an interesting article that should encourage some good conversation.
It looks like Scaffetta did a type B evaluation while Schmidt did a type A evaluation. The Schmidt type A evaluation includes the component of uncertainty caused by natural variation whereas the Scaffetta type B evaluation only includes the component of uncertainty caused by measurement variation.
Both evaluations are correct for their respective intents. The question is…which one is best suited for comparison with CMIP6? I think there are valid arguments both ways. However, given that the comparisons are sensitive to the timing of natural variation within the decade I think it makes sense to include the natural variation component. Case in point…we are currently in a Monckton Pause lasting 8.75 years primarily because the early part of the period was marked by a very strong El Nina while the later part of the period is marked by a triple dip La Nina. If in the future Schmidt compared 2023-2033 with 1980-1990 he might get a very different result due to 2023 starting off as an El Nino (assuming one does indeed form later this year) even though long term forcings might remain unchanged.
BTW…your correct statements about the uncertainty of the average being assessed with the 1/sqrt(N) rule is going to absolutely trigger some of the familiar posters here causing hundreds of posts. As always…it will be interesting.
BTW #2…ERA5 actually does include uncertainty. It’s just really hard to work with because you have to download the grids and type B it spatially and temporally up to global values and then daily, monthly, and annual values.
Great Scott! A hypocrisis to the nth degree. Tangential yawnfest.
The earth’s climate has been warming for three hundred years or more. The odds are that it will continue to warm for another hundred years or more, regardless of what anyone tries to do to stop it.
For those who are more interested in the complexities of public policy analysis than we are in the complexities of IPCC climate modeling, the energy marketplace and the impacts of public policy decision making on that energy marketplace are much more interesting and much more rewarding as topics to be spending serious personal time in discussing.
Not sure who the proverbial “we” is there, but I can guarantee that 100 comments re-litigating the 1/sqrt(N) rule just isn’t going to get you very far.
JCM, I’m presuming that I’m not the only one who posts comments on WUWT who is more interested in public policy decision making than in the low level details of climate-related temperature measurement and prediction.
Hence the use of ‘we’ assuming there is one more person who has the same outlook as I do.
However, as the number of comments on this article continues its steep rise, just as Mr. Bdgwx said would happen, the tempation to ask Nick Stokes a question or two concerning natural variability in GMT versus anthropogenic effects on GMT is becoming almost too powerful to resist.
The intent of both Schmidt and Scafetta was to evaluate at what point ECS became too large. That is, when is greenhouse gas warming statistically higher than observed warming between the two decades.
How do we treat natural oscillations inside the two decades being compared? Are they part of the random noise and included in the uncertainty? Or not.
Since the IPCC has declared in AR6, and previous reports, that natural forcings are zero and greenhouse gases have caused 100% of modern warming, I don’t think it is justified to include natural forces in the uncertainty. It is only appropriate to include measurement error and biases.
If we include it in the uncertainty here, it must also be included in the calculation of ECS, which would lower the ECS, since ECS is determined by subtracting natural “forcings” from the computed greenhouse gas warming. You can’t have it both ways and both ways, if done consistently, lower ECS.
“Since the IPCC has declared in AR6, and previous reports, that natural forcings are zero and greenhouse gases have caused 100% of modern warming, I don’t think it is justified to include natural forces in the uncertainty.”
This is the primary error which is the source of all the false attributions and the reason models run hot. Ignoring what actually drives climate is the only way they can elevate CO2 to a dominant position. Natural forces are the uncertainty being swept under the rug.
Always appreciate Gavin being shown up but arguing about error bars from fictional models feels like validation that they represent Earth’s actual climate system.
”Always appreciate Gavin being shown up but arguing about error bars from fictional models feels like validation that they represent Earth’s actual climate system.”
Sweeter words have not been written. It’s like arguing that fairies would beat leprechauns at poker.
Don’t be silly. How else do you think the leprechauns got their pots of gold?
old cocky: “Don’t be silly. How else do you think the leprechauns got their pots of gold?”
I was told by my Irish ancestors, usually on or about the 17th of March of every year, that it was alchemy.
“Ignoring what actually drives climate is the only way they can elevate CO2 to a dominant position. Natural forces are the uncertainty being swept under the rug. ”
Absolutely right.
It’s kind of ridiculous, isn’t it. Natural forces (Mother Nature) drive the climate until the 1980’s and then, all of a sudden, CO2 takes control, and Mother Nature goes away.
We should keep in mind that today’s CO2 levels are at about 415ppm, and climate change alarmists say this is too much and will cause a runaway greenhouse effect that will overheat the Earth with dire consequences for all life.
But CO2 levels have been much higher in the past, 7,000ppm or more, yet no runaway greenhouse occurred then, so why should we expect one to occur now?
CO2 Scaremongering is the problem. A little cooling from Mother Nature might fix that problem.
Andy
“since ECS is determined by subtracting natural “forcings” from the computed greenhouse gas warming”
Absolutely untrue. It makes no sense. ECS for models is usually determined by direct experiment – ie double CO2 and see what happens.
First you need to know what happens before you double co2.
“Absolutely untrue. It makes no sense. ECS for models is usually determined by direct experiment – ie double CO2 and see what happens.”
In practice this IS true. The GCMs cant produce natural forcings beyond 17 years (by Santer) and then average out over the timescales applicable to ECS. By using GCMs to investigate ECS, they’re effectively ruling out natural forcings.
However natural forcings produced a habitable greenland for example. The GCMs wont do that.
“The GCMs cant produce natural forcings beyond 17 years (by Santer)”
Santer said nothing like that.
The relevant paper is
https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2011JD016263
where a key point is named as
And they use control runs (no forcings) to establish that the model doesn’t show climate change beyond 17 years and therefore any observed climate change longer than 17 years must be forced (by CO2). Namely
“The GCMs cant produce natural forcings beyond 17 years (by Santer)”
The paper you linked says nothing about GCM’s at all. It is about surely identifying a trend from data. It doesn’t say that a model, or anything else, can’t reproduce natural variation. It says you need that length of time to say that trend was probably due to CO2. Then you can distinguish it from natural variation; that does mean there wasn’t any.
That hilarious. I recommend you read it.
And when you do, remember that GCM control runs by definition dont produce climate change. Beyond 17 years apparently. They’re comparing control runs to observed warming trends.
Yes, my apologies, my memory of the paper was faulty. Santer does use GCM’s to calculate an error structure for determining the significance of trends in measured tropospheric temperature.
But the claim
“The GCMs cant produce natural forcings beyond 17 years (by Santer)”
is still nonsense. What he says is that in models, pauses (periods under forcing with zero or less trend) of greater than 17 years are very infrequent (there are important extra bits about removing known sources of variation). That is just a frequency observation. It doesn’t say they can’t do anything. And it has nothing to do with “produce natural forcings“. I don’t even know what that could mean.
I gave you the relevant quote.
“[30] On timescales longer than 17 years, the average trends in RSS and UAH near-global TLT data consistently exceed 95% of the unforced trends in the CMIP-3 control runs (Figure 6d), clearly indicating that the observed multi-decadal warming of the lower troposphere is too large to be explained by model estimates of natural internal variability.”
The unforced trends are control runs where GCMs produce no climate change. It’s by definition. Its how they’rebuilt. And the implication is clearly GCMs don’t model natural forcings resulting in climate change like most climate change over most history. They’re just not capable.
So ECS is essentially as a result of the forcing as modelled by a GCM.
There is no issue of capability. That is a frequency observation. 95% of control periods do not reach the current trend level over 17 years. Therefore it is unlikely that the trend we see happened without forcing. It doesn’t say it couldn’t happen.
What it says is that after 17 years the trend reverses. And over time it averages to nothing. Its by definition. So from an ECS perspective, GCMs describe ECS purely from a forcing point of view. There is no natural variability built in over the timescales needed for ECS in GCMs.
Unlike the real world where climate change persists longer than 17 years and results in a habitable Greenland or frozen Thames.
Nick the first line says he used models:
Nick,
ECS is not a scientific value since it can never be measured in the real world. An instantaneous doubling of CO2 can never happen outside the model world. And, even if it did, everything else would not stay the same while the world came into equilibrium with the doubled CO2. It is a model fantasy number. And ECS, in the model world, is determined by subtracting natural only models from all forcings models. The problem is they assume natural only is zero, and it clearly isn’t.
See figure 3 here:
https://andymaypetrophysicist.com/2020/11/10/facts-and-theories-updated/
TCR is a value that can probably be tested and falsified. A 1% rise per year, until doubled, is a decent hypothesis. And we are on track to show TCR is at the very low end of the IPCC range.
While STILL assuming all of the warming is caused by CO2 – which it is not.
You need no model for that. There is sufficient reverse correlation between temperature and atmospheric CO2 in the Earth’s climate history to confirm ECS as zero.
I believe I understand what the point of the analysis is. My point is that the observed temperature is subject to both natural variation and forced change. If the forced change is +0.6 C natural variation could cause us to observe +0.5 C to +0.7 C assuming the natural variation is ± 0.1 C (2σ). In other words, an observation of 0.58 C is consistent with a forced change of 0.48 to 0.68 C. If would be 0.48 C if we happen to be in an extreme cool phase of the natural oscillation and 0.68 C if we happen to be in an extreme warm phase of the natural oscillation.
The question…do we conclude that a model under/over estimated if the observation was only 0-0.1 below/above the prediction. I don’t think so because the observation could be high/low because ENSO (or whatever) happened to be high/low.
I think you mean “the observed temperature is subject to the natural variation, forced change and adjustments in search of fitting it into the required result.”
If you assume the temperature trend line is entirely due to a constant monotonic increase in forcing, the natural variation is the excursions from that trend line. From the graphs, those excursions appear to be along the lines of +/- rand(0.5)
Yeah. Exactly. I sometimes use the example y = x + sin(x). The x term is like the forced change while the sin(x) term is like the natural variation.
BTW…using the NIST uncertainty machine we see that if u(x) = 1 then when Y = x it is the case that u(Y) = 1 as well. But if Y = x + sin(x) then u(Y) = 1.63. The point…adding variation adds uncertainty.
I see where you’re coming from as far as noise adding to the uncertainty. It was more a case that the trend and noise in your example were reversed.
It can take quite a difference in trend to allow noisy time series to be distinguished at a sensible level of significance.
“If we include it in the uncertainty here, it must also be included in the calculation of ECS, which would lower the ECS, since ECS is determined by subtracting natural “forcings” from the computed greenhouse gas warming..
No. Even if this were true (https://wattsupwiththat.com/2023/04/13/the-error-of-the-mean-a-dispute-between-gavin-schmidt-and-nicola-scafetta/#comment-3708016), since they both have expected values (like ’em or not),and approximately normally distributed standard errors*, that are either independent (worst case) or correlated (which would tend to increase the difference), then, assuming worst case – independence – the expected value of the subtraction would be taken from the 2 expected values. Now, the standard error of it would then be (u”natural”^2+u”GHG”^2)^0.5, which is certainly greater.
*per modern CLT.
Dear bdgwx,
Heck, I thought the science was settled and that is why here in Australia they are blowing-up power stations, destroying electricity generation networks, flushing perfectly good fresh water down the rivers to irrigate the ocean, bringing in carbon taxes (run by ‘the market’) and spending zillions of dollars they don’t have in order to make everybody poorer and more dependent, in a grovelling sort of way, on China.
The absurdity of the claim that computer games can be used to predict anything is beyond scientific belief. Which particular scientists at the University of NSW, or ANU or anywhere else can be held to account for the wreckage created out of thin-air by their green-dreams? They forget the atmosphere is still 99.96% CO2-free.
Against this background, I don’t see the point of arguing the toss about uncertainty, when in fact all the data being used is modeled. All of it is modeled, and each model within models suffer from the same problem – they take specks of data from weather balloons, ships, planes, temperatures measured in Stevenson screens or not, rub them together and come up with a trend. Then they squabble about which trend is ‘real’ and which models are best, meanwhile there is a war going-on, most nations are broke, many have been furiously back-pedalling, the rest will eventually once all this is revealed to be nonsense.
Two years ago we presented a study on WUWT, about sea surface temperature along the Great Barrier Reef (https://wattsupwiththat.com/2021/08/26/great-barrier-reef-sea-surface-temperature-no-change-in-150-years/). A detailed report downloadable at https://www.bomwatch.com.au/wp-content/uploads/2021/10/GBR_SST-study_Aug05.pdf outlined our methods, results and discussed the findings. I was clear subsequently, that a journal version of the report submitted to Frontiers in Marine Science Coral Reef Research would never be published, probably because the apple-cart was too-full of rotten apples, the same ones running the peer review process, and upsetting the cart would not be ‘collegiate’ as they say.
The paper presented a substantial amount of data and raised valid questions about biases embedded in HadISST that caused trend, which was not measurable in Australian Institute of Marine Science temperature timeseries. There is also no long-term trend in maximum temperature observed at Cairns, Townsville and Rockhampton that is not explained by site moves and changes. Confirmed by aerial photographs and archived documents, some site changes were fudged by Bureau of Meteorology (BoM) scientists, most recently Blair Trewin, to imply warming in the climate, which actually does not exist.
So here in Australia, the BoM massages the data, which is used to underpin CSIRO State of the Climate Reports which are then used by the government to lecture and eventually screw the populace into believing in something that has been manufactured for the last 30-years.
My question is if there is no warming in surface temperature, and no warming in SST, irrespective of uncertainties amongst them, how can any of the models be correct? And if the data (HadISST and homogenised maxima) have been fudged to support the models, how can such data be used to verify the models?
Science in Australia has been totally corrupted, top-down by governments, intent on ‘proving’ warming to drive their absurd WEF-UN-agendas. The same seems true in Canada, New Zealand and the USA.
Yours sincerely,
Dr Bill Johnston
http://www.bomwatch.com.au
They can’t
Those who claim problems with surface temperature measurements can’t dispute that the world has been warming, unless they say UAH v6 TLT is also wrong by showing warming.
Dear donklipstein,
I am not claiming anything, I am explaining what I’ve found (there is a difference). I have also only published a small proportion of all the datasets I have examined over the last decade. The “can’t dispute that the world has been warming” thing, reflects success of a marketing strategy, not unambiguous scientific study.
The original thesis that temperature is increasing, goes back to before the satellite era. For Australia, to before 1990 when the government started on about “greenhouse”. From that point, in order to support the thesis they (the Bureau of Meteorology and CSIRO) had to find warming in surface temperature data and they did that using various forms of homogenization, including arbitrary ‘adjustments’.
Their main trick is to adjust changes in data that made no difference, and/or, not mention/not adjust changes that did. I have used objective statistical methods and aerial photographs and documents/pictures from the National Archives to track such changes and verify station metadata.
Check out my Charleville report for example, where for 13 years they made the data-up (the full report is here: https://www.bomwatch.com.au/wp-content/uploads/2021/02/BOM-Charleville-Paper-FINAL.pdf). Or Townsville, where they forgot they moved the Stevenson screen to a mound on the western side of the runway in 1970 (https://www.bomwatch.com.au/wp-content/uploads/2021/02/BOM-Charleville-Paper-FINAL.pdf). There are also many examples of where they did not know where the original sites were or that they had moved (https://www.bomwatch.com.au/wp-content/uploads/2020/08/Are-AWS-any-good_Part1_FINAL-22August_prt.pdf).
Your faith is misplaced. While you say “Those who claim problems with surface temperature measurements can’t dispute that the world has been warming”, it is no joke that scientists within the Bureau of Meteorology have cheated and changed data in order to homogenise-in the ‘trend’.
Most of the individual site reports I have published on http://www.bomwatch.com.au are accompanied by the datasets used in the study, so anyone or any of the multitude of fake fact-checkers is welcome to undertake replicate analysis.
Regardless of what UAH has to say (and they don’t estimate surface T), no medium to long-term datasets in Australia are capable of detecting small trends that could be attributed to the ‘climate’.
The outdoor ‘laboratory’ is not controlloed enough, instruments are too ‘blunt’ in their response, and observers too careless, for data to be useful, and no amount of patching and data wrangling can change that.
Data-shopping using Excel, as practiced by some and the use of naive timeseries methods – technical for “add trendline” to an excel plot, has confused themselves and many others into ‘believing’ in trend that in reality is due to site changes, not the climate.
Yours sincerely,
Dr. Bill Johnston
scientist@bomwatch.com.au
‘It’s an interesting article that should encourage some good conversation.’
Agreed. I thought it was interesting that no matter which method is used to test for significance, it’s very clear that a hefty proportion of the models are unfit for the purpose. Do you have any insight as to when Gavin will be advising the Administration to perhaps defund some of the worse performers and/or let up on the alarmism?
That should be “El Niño,” or at least “El Nino.”
Doh. That is embarrassing. You are, of course, correct.
It is just the new name for a transgender El Niño.
Wouldn’t that be Niñx?
Not. It is Ellos Ninos
The issue is *not* using 1/sqrt(N). The issue is what that tells you. It is *NOT* a measure of the accuracy of anything. It is only a measure of the interval in which the population average can lie. That does *NOT* tell you that the average you calculate is accurate at all.
Remember, the examples mentioned from Taylor are from Chapter 4.
Taylor specifically states in the intro to Chapter 4:
“We have seen that one of the best ways to asses the reliability of a measurement is to repeat it several times and examine the different values obtained. In this chapter and Chapter 5, I describe statistical methods for analyzing measurements in this way.
As noted before, not all types of experimental uncertainty can be assessed by statistical analysis based on repeated measurments. For this reason, uncertainties are classified into two groups: The random uncertainties, which can be treated statistically, and the systematic uncertainties, which cannot. This distinction is described in Section 4.1. Most of the remainder of this chapter is devoted to random uncertainties. ”
In Section 4.1 Taylor states: The treatment of random erros is different from that of systematic errors. The statistical methods described in the following sections give a reliable estimate of random errors, and, as we shall see, provide a reliable procedure for reducing them. ….. The experienced scientist has to learn to anticipate the possible sources of systematic erro and to make sure that all systematic errors are much less than the required precision. Doing so will involve, for example, checking meters against standards and correcting them our buying better ones if necessary. Unfortunately, in the first-year physics laboratory, such checks are rarely possible, so treatment of systematic errors is often awkward.”
It seems to be an endemic meme in climate science, and that includes you bdgwx, that all measurement uncertainty is random, Gaussian, and cancels leaving the stated values as 100% accurate. Therefore the variation in the stated values can be assumed to be the uncertainty associated with the data. The problem is that when it comes to temperature records climate science is in the same boat as the first-year physics class Taylor mentions. Systematic error in the climate record *is* awkward to handle – but that is *NOT* a reason to ignore it.
Taylor discusses this in detail in Section 4.6. (when systematic bias exists)
“As a result, the standard deviation of the mean σ_kavg can be regarded as the random component ẟk_ran of the uncertainty ẟk but is certainly not the total uncertainty ẟk.”
If you have an estimate of the systematic bias component then you can add them either directly or in quadrature – let experience be your guide.
About adding in quadrature Taylor says: “The expression (4.26) for ẟk cannot really be rigorously justified. Nor is the significance of the answer clear; for example we probably cannot claim 68% confidence that the true answer lies in the range of kavg +/- ẟk. Nonetheless, the expression does at least provide a reasonable estimate of our total uncertainty, give that our apparatus has systematic uncertainties we could not eliminate.”
YOU keep on wanting to use the standard deviation of the mean calculation as the measure of uncertainty in the record. The typical climate science assumption that all error is random, is Gaussian, and cancels out.
Again, the issue is what 1/sqrt(N) tells you. And it does *NOT* tell you the accuracy of the mean you have calculated since systematic bias obviously exists in the temperature measurement devices and, according to all stated values for acceptable uncertainty of the devices, can range from +/- 0.3C for the newest measurement devices up to +/- 1.0C (or more) for older devices.
No where in any climate science study I have seen, and that includes those referenced here, provides for propagating measurement uncertainty onto the averages calculated from the stated values of the measurements. Doing so would mean that trying to identify differences in the hundredths digit is impossible due to the uncertainties inherent in the measurements.
I am just dismayed beyond belief when I see statisticians that know nothing of the real world trying to justify the standard deviation of the mean as the actual uncertainty in temperature measurement averages by assuming that all measurement uncertainty is random, Gaussian, and cancels. Even Possolo in TN1900, Example 2, had to assume that all measurement uncertainty cancelled or was insignificant in order to use 1/sqrt(N).
No one is ever going to convince me that all temperature measuring devices in use today have 100% accurate stated values for their measurement. Not even you bdgwx.
Example Tim.
Systematic error could be a maximum thermometer with a bubble or break; a scale with a loaded pan; or a measuring jug with sludge in the bottom.
Because they occur across sequential measurements they have the property of being non-random, and also that they are additive to the thing being measured. The simple solution is to detect and deduct their effect in situ using unbiased methods, is it not?
All the best
Bill Johnston
This is the mantra of the data mannipulators, who assume it is possible to determine and remove all “biases” from historic air temperature data.
It is not possible.
Bias errors are typically unknown and can also change over time.
They cannot be reduced by subtraction or by averaging.
They are best handled by estimating their bounds and constructing a corresponding uncertainty limit.
“Systematic error could be a maximum thermometer with a bubble or break; a scale with a loaded pan; or a measuring jug with sludge in the bottom.”
How do you detect these effects remotely? As Taylor states they are not amenable to identification using statistical means. Bevington and Possolo agree.
If you mean they should be detected *at the site* then how often are those stie inspections done, especially considering the global dispersion of the measurement devices?
Consider a station whose air intake is clogged by ice in late December. That ice may stay there for two months or more till air temps will be high enough to melt it. If the site inspection is done in the summer time, you’ll find nothing yet the winter temps for that period will contain a systematic bias.
I could give you numerous examples. Ant infestations, decaying organic matter, brown grass in winter vs green grass in summer, a nearby lake that is frozen in winter but not in summer, a nearby soybean/corn field whose evapotranspiration changes with season, and on and on.
These are all systematic biases that would be undetectable using statistical analysis – but they *are* real. And they can’t be eliminated by “homogenization” with stations up to 1200km away.
It’s why uncertainty intervals are important for the temperature measurements. I know al lot of climate scientists just assume that systematic biases will cancel just like random errors but that just isn’t a good assumption for an old mechanic/machinist/carpenter/physical-scientist like me. Similar electronics tend to have the same temperature coefficient (TC), e.g. many resistors, even SMD types, have the same TC, usually positive. That means that you will see a consistent systematic bias introduced – no cancellation across different stations of the same type. Heck, some resistors, esp ones using nichrome, have a non-linear TC.
It just seems that in climate science the overriding assumption is that measurement uncertainty *always* cancels leaving the stated values as 100% accurate. It’s why variance (or sd) represented by measurement uncertainty is *NEVER* propagated throughout the statistical analyses. First off, it is difficult to do and second, it would obviate the ability to identify differences in the hundredths digit! Instead, uncertainty is just assumed to be the standard deviation of the sample means even though that tells you nothing about the accuracy of the population mean you are trying to identify.
After giving yet another dose of measurement reality, all the trendologists can do is push the downvote button.
They well and truly can’t handle the truth.
Tim Gormon and karlomonte (below),
In the context of climate timeseries, Taylor is incorrect. For a start, if you can’t measure systematic bias in data, or don’t look for it, you would not know it was there. Further, if you could measure it you can also explain it, allow for it or correct for it. I don’t want to get bogged-down in semantics or foggy concepts. If you have some clear examples, preferably something you have published then why not use those to support your argument.
To be clear Tim and karlomonte, with colleagues I undertook week-about, 9AM standard weather observations at a collaborative site for about a decade. So I know the routine and I know many of of the pitfalls. As most people don’t know much about how data arrives at a data portal such as BoM’s climate data online, in situ in the context I am meaning is “as you find the data”.
Exploratory data analysis is about quality assurance of data, and before embarking on any analysis it is an essential first step. An effective QA protocol includes descriptive statistics as well as analysis of inhomogeneties that could indicate systematic bias.
Contrary to your claims, tests do exist and I invite you to read the downloadable reports available on http://www.bomwatch.com.au, where I have given example after example of an objective method for undertaking such analyses. The methodology is all there in methods case studies, backed-up by hundreds of individual analyses.
There are methods case studies here: https://www.bomwatch.com.au/data-quality/part-1-methods-case-study-parafield-south-australia-2/; here: https://www.bomwatch.com.au/data-quality/part-1-methods-case-study-parafield-south-australia-2/; and here: https://www.bomwatch.com.au/data-quality/part-1-methods-case-study-parafield-south-australia-2/.
Examples of detailed studies are here: https://www.bomwatch.com.au/bureau-of-meterology/part-6-halls-creek-western-australia/; and here: https://www.bomwatch.com.au/bureau-of-meteorology/charleville-queensland/.
If you think the methods are wrong, or want to contest what I am saying, I invite you to grab some daily data and analyse it and thereby demonstrate the things you talk about.
Finally, as I think we have discussed before, measurement uncertainty (strictly 1/2 the interval scale) is constant and it does not propagate. For two values having known uncertainty, to be different, their difference must exceed the sum of their uncertainties. Thus for a meteorological thermometer, where instrument uncertainty = 0.25 degC (rounds to 0.3), the difference between two values must be greater than 0.6 degC.
Tim, your argument goes around and round when you talk a bout 1/100ths of a digit. Nothing is measured to 1/100ths of a unit. I think you are also confused about uncertainty of a mean (typically SD, but also 1.96*SE) and accuracy which relates to an observation as discussed above.
Yours sincerely,
Bill Johnston
I’ve looked at some of your papers at bomwatch, Bill. They do not mention field calibrations against an accurate temperature standard, such as a PRT in an aspirated screen.
Field calibrations against an accurate measurement sensor are the only way to detect the systematic error that attends naturally ventilated shields due to the environmental variables of irradiance and insufficient wind speed.
Published work from the19th century shows that meteorologists of that time were well-aware of this problem, but latterly (post-WWII) attention to this critical detail seems to have evaporated.
No statistical inter-comparison of regional sensors will detect weather-related systematic error in an air temperature record.
I have published several papers on the neglect of systematic error in constructing the global air temperature record, most recently here.
The field measurement uncertainty width of any naturally ventilated meteorological air temperature sensor is not better than 1σ = ±0.3 C, which means the anomaly record is not better than 1σ = ±0.42 C.
The 95% uncertainty in the anomaly record is then on order of ±0.8 C. There’s no getting away from it.
That’s strange because 1) type A evaluations of the anomaly record show significantly lower values and 2) if it were on the order of ±0.8 C then we wouldn’t be able to see natural variation like ENSO cycles in the record yet we clearly do and 3) we wouldn’t be able to predict it with any better skill than an RMSD of 0.4 C yet even a complete novice like me can do so with far better skill than that.
Thanks Pat,
Australian screens are not aspirated. PRT probes are lab calibrated. They are also field checked at least once/year and the Bureau has a range of cross checks they apply to assess inter-site bias and drift. They also employ a local error-trapping routine. Details are contained in a range of publications on their website. Although I have worked with PRT probes, I have not focused on probes per se on BomWatch.
Measurement uncertainty (the uncertainty of an observation) is 1/2 the index interval, which for a Celsius met-thermometer is be 0.25 degC, rounds up to 0.3 degC. I don’t know how that translates in anomaly units, neither do I see how 1σ applies to an individual datapoint.
There is a need I think to write a note about metrology that sorts out and clarifies issues surrounding measurements.
All the best,
Bill Johnston
Bill,
I’m not sure who you are trying to snow. I looked at the first case study you listed. No where in there was there a study done to actually identify any systematic bias due to measurement devices or microclimate. If the rest of them are the same then there is no use even looking at them.
Hubbard and Lin showed in 2002 that measurement stations corrections *must* be done on a station-by-station basis. Regional adjustments, like those based on homogenization, only serve to spread systematic biases around to other stations.
“Tim, your argument goes around and round when you talk a bout 1/100ths of a digit. Nothing is measured to 1/100ths of a unit. I think you are also confused about uncertainty of a mean (typically SD, but also 1.96*SE) and accuracy which relates to an observation as discussed above.”
Unfreakingbelievable. What do you think I have been trying to get across in all these long threads? How close you are to the population mean is *NOT* the accuracy of that mean. The uncertainties of the measuring stations *must* be propagated onto the population mean. That propagated uncertainty is *NOT* the variation of the stated value of the measurements either.
Systematic uncertainty is FAR more than the resolution of the measuring device. Calibration drift is a fact of life and no amount of statistical analysis can identify it. Learn it, love it. live it.
Or if you are only concerned with the change in temperature you don’t need to detect and deduct it since that happens via algebra. That, of course, applies only if the bias is truly systematic and not random and remains unchanged between measurements.
“not random and remains unchanged between measurements” — the main point about bias error that you refuse to acknowledge is that there is no way for you to know if this is true or not.
Systematic bias is always there in measurement devices. Period. Anomalies do not get rid of it. You do not know what it is and no amount of statistical analysis can tell you what it is.
Anomalies carry the systematic biases with them. If you use them with measurements from other stations, or even with measurements from the same station, you simply do not know if your anomalies are accurate or not.
Systematic bias CAN change between measurements, even using the same device. As a statistician I’m sure you are totally unaware of the impacts hysteresis can have in the real world. A device can read differently when the temperature is falling than it does when it is rising. It’s just a plain fact of real world devices. Do you even have a clue as to what snap-type micrometers are for?
Someday you really should join us here in the real world and leave your blackboard world behind.
The whole landscape experiences hysteresis, it is measurable and therefore can be quantified using data. However, hysteresis is not a property of data per se, but a property of the ‘thing’ being measured.
b.
This is nonsense.
It is also the property of the device doing the measuring!
You can *only* measure hysteresis through calibration procedures.
You say: Or if you are only concerned with the change in temperature you don’t need to detect and deduct it since that happens via algebra.
Tell us what algebra. Systematic bias is NOT factors affecting the background ambience of a site, which for a good unchanging site should be constant.
Cheers,
Bill
Let T = Mi + Ei where T is the measurand, Mi is a measurement of T, and Ei is a random variable effecting all measurements Mi that cause it to be different from T. If we then add a systematic bias B to all measurements we have T = Mi + Ei + B. Then if we want to know the difference between two measurands Ta and Tb we have ΔTab = Ta – Tb = (Mai + Eai + B) – (Mbi + Ebi + B). Simplifying this equation via standard algebra we have ΔTab = (Mai – Mbi) + (Eai – Ebi). Notice that the systematic bias B cancels out. This leaves us with ΔTab = Ma – Mb ± √(Ea^2 + Eb^2).
I know. Or at least I wasn’t assuming that was the case here. I assumed you were talking about a bias in the measurement only. Though, in reality, it doesn’t really matter the source of the systematic bias. If it can be expressed as B then it cancels regardless of how it arises. The more interesting scenario is when there is a systematic bias that changes over time. But that’s a more complex scenario.
“The more interesting scenario is when there is a systematic bias that changes over time. But that’s a more complex scenario.”
Only if you can credibly claim that such a bias trend has a significant impact on the trends under discussion. We have yet to see any such claim, with reality based documentation,
Are you aware of the Temperature Coefficient of electronic devices?
The TC can be either negative or positive depending on the exact components used and how they are used.
If *every* identical temperature measurement device has a similar systematic bias due to calibration drift then how can that *NOT* affect the trend? The trend will be made of of the *real* change in the measurand PLUS the change in the calibration of the measuring device.
Did you *really* think about this before posting?
Please show me where your claim results in the referenced “significant impact on the trends under discussion”. I.e., global temperature and sea level trends over physically/statistically significant time periods w.r.t. ACC. This is what you continuously fail to demonstrate.
Please read completely, slowly, for comprehension…
From the main post:
“Now let’s look at the dispute on how to compute the statistical error of the mean warming from 1980-1990 to 2011-2021 between Scafetta and Schmidt.”
These methods are useless for dealing with bias, Type B errors. Climate astrology assumes that subtracting a baseline (the anomaly calculation) removes all bias error so they can safely ignore it.
But this assumption is quite false.
Another in the never ending series of posts that claims – with no proof – that there’s a Big Foot trail of systemic measurement/evaluation errors that qualitatively influence global average temperature and sea level trends over physically/statistically significant periods w.r.t. ACC.
Will you remain Dan Kahan System 2 hard wired to the fallacy that the world is upside down, and that you have (internally inconsistent) alt.statistical theories that superterranea denies? Because, after all, they are all part of a Dr. Evil conspiracy against you!!
That you can’t grasp what I wrote is yet another indication that you have zero real metrology experience, so you resort to fanning from the hip, hoping to land a round on those who expose your abject ignorance.
So, once again, no proof or even evidence. Therefore, you deflect to a fact free authority appeal to your supposed – but undemonstrated – “real metrology experience”.
I really doubt there is any “proof” that would convince blob that he is completely wrong.
And I can push the downvote button too, blob! It’s fun!
Ah, the predictable, convenient excuse for failing to provide any such proof. And a snowflake, pearl clutch whine about down votes in the bargain….
What exactly are you asking for? A list of all possible bias errors?
As Rumsfield said, there are known knowns, known unknowns, and unknown unknowns (or something like that).
BoB wants us to list out all the unknown unknowns.
It’s why there is an *interval* associated with uncertainty. You try to include all the possible unknown unknowns. BoB and climate science wants to just assume they are all ZERO’S.
“BoB wants us to list out all the unknown unknowns.”
No, I acknowledge both systematic and random sources of error. I merely remind you that the chance of systematic errors both sufficiently large and lining up just right over statistically/physically significant time periods to significantly change any of the most likely trends is TSTM.
Again, you’re counting on your chimps to accidentally type the Encyclopedia Britannica. Might happen soon. Probably won’t….
Ok blob, here’s your chance to shine:
Using the manufacturer’s error specification, construct a standard uncertainty interval for the Fluke 8808A Digital Multimeter.
Wayback a few weeks, and ponder. We’ve already done that. You conveniently forgot where it landed…
No evidence, blob? hahahahahahah
Hypocrite.
That you have not the first clue about how to proceed is noted.
And who are “we”?
“that there’s a Big Foot trail of systemic measurement/evaluation errors that qualitatively influence global average temperature and sea level trends over physically/statistically significant periods w.r.t. ACC.”
What do you think the measurement uncertainty of the ARGO floats being +/- 0.5C consists of? Purely random error?
No, which is why it has been accounted for.
“– with no proof –“
Says the guy who clearly has never done the research.
From only the most recent of field calibrations: Yamamoto, et al (2017) “Machine Learning-Based Calibration of Low-Cost Air Temperature Sensors Using Environmental Data”
Abstract, first two sentences: “The measurement of air temperature is strongly influenced by environmental factors such as solar radiation, humidity, wind speed and rainfall. This is problematic in low-cost air temperature sensors, which lack a radiation shield or a forced aspiration system, exposing them to direct sunlight and condensation.” (my bold)
The negative impact of environmental variables on the accuracy naturally ventilated air temperature sensors has been known since the field calibrations of the 19th century.
No proof indeed. Your dismissive insouciance disrespects past scientists and the hard careful work they did.
First off, per your linked paper. all errors that both tend to be random, and can be reduced. Second, you are still defecting from demonstrating any such steadily changing sources of systemic error that woulda coulda result in significant change in either temperature or sea level trends over physically/statistically significant time periods.
You are essentially making the Chimps Who When Left Alone With Typewriters Will Eventually Type The Encyclopedia Brtiannica argument. Yes, the chances are greater than zero. Just not much….
Another fine word salad, blob, totally sans any meaning.
You get a green!
What kind of malarky is this?
Do you think that all the water impoundments the Corp of Engineers have built over the years didn’t have an impact on the temperature measurements at locations surrounding them? Over physically/statistically significant time periods?
Jeesssh, you *really* need to think things through before spouting nonsense.
Evidently dyslexia is a critical part of your armamentarium, bob.
When you add (subtract) random variables the variances add they don’t cancel! Why is this so hard to understand?
Anyone that has ever built a staircase should have a basic understanding of this!
I can’t grok where they are, at all. Makes no sense to me.
“We have yet to see any such claim”
Aitken, 1884.
Lin, 2005
Two among many, since the 19th century.
The first link errored out. The second link is paywalled, but the abstract ends up in predictable coulda’ woulda.
“The debiasing model could be used for the integration of
the historical temperature data in the MMTS era and the current US CRN temperature data and it also could be useful for achieving a future homogeneous climate time series.”
Maybe if you tried harder. After all, this is just “Two among many…
Apologies for the bad link, Cambridge apparently changed it. Try this.
From Aitken page 681: “It is only within the last few days that I have been in possession of a Stevenson screen, and been able to make comparative trials with it. During these tests its action was compared with the fan apparatus, and readings were taken at the same time of the thermometers in the draught tube screens. These latter, as already stated, had an average error of about 0.6° too high. The smallest error recorded in more than thirty readings of the thermometer in the Stevenson screen was l.3°, and it only fell to that on two occasions. The excess error was generally more than 2°, and was as high as 2.8 on two occasions.”
For Lin, 2005, you ignored this part of the Abstract: “The results indicate that the MMTS shield bias can be seriously elevated by the snow surface and the daytime MMTS shield bias can additively increase by about 1 °C when the surface is snow covered compared with a non-snow-covered surface.”
Naturally ventilated MMTS sensors dominated at US stations after 1990.
The debiasing method is real-time filtering. It provides the only way to remove most of the systematic error produced by unventilated screens, and is a technique in use at no meteorological stations.
Once again, you carelessly dismiss.
You’ve got no clue that you’ve got no clue, bob.
Bob and bdgwx are both the same. They think systematic error ALWAYS CANCELS.
It seems to be some kind of a plague among those in climate scientists and statisticians.
All uncertainty cancels, always. So it can always be ignored.
Is that why none of the statistics textbooks I own have any examples using “stated value +/- uncertainty” to teach the students how to handle uncertainty?
I don’t think that.
The way I figure it, Tim, is that if standard scientific rigor was applied, none of them would have anything to talk about.
Hence the refractory ignorance.
Jeez guy, just how do you conclude that the temperatures are subtracted? What if they are added like for calculating an average?
You need to show a reference that “B” cancels out when averaging measurements. That isn’t even logical regardless what your made up algebra example proves.
If I use a micrometer with a bent frame, each and every measurement will carry that error and you can not recognize that error by using algebra or statistics. Calibration is the only way observe systematic error and correct it.
“Let T = Mi + Ei “
ROFL!!
It is T = Mi ± Ei
Please note carefully the ±
It then progresses to T = Mi ± (Ei ± B)
What this results in is that the uncertainty must be evaluated on its own, not as part of Mi. The possible range of T is *expanded* as you add elements, they do *NOT* cancel!
First, that model comes from NIST TN 1900. Second, ± isn’t a mathematical operation so what you wrote is nonsensical.
So + is not a mathematical operator? So – is not a mathematical operator?
The use of ± is just a short hand for not having to write it out:
T = Mi – Ei to T = Mi + Ei
I am not surprised that you can’t make sense of this. in TN1900 Ei can be either negative OR positive, Possolo just doesn’t state that.
You can have Mi + (-Ei) or Mi + (+Ei)
Why you would think that all error terms are positive is just beyond me. It goes hand in hand with your lack of knowledge of metrology!
+ and – are mathematical operators.
ALGEBRA MISTAKE #24: ± is not a mathematical operator.
Like NIST 1900 Ei is the random variable representing the difference between the measurement M and the true value T. It already takes on both positive and negative negative numbers. There’s no need for two equation.
That’s what I said!
I don’t think that.
ALGEBRA MISTAKE #25: A plus (+) operator in an equation does not imply that the rhs operand is positive.
Clown.
KM says you are a clown. He’s right!
“ALGEBRA MISTAKE #24: ± is not a mathematical operator.”
I told you: ± is short hand for X + e and X-e. You just write it as X ± e. Shorter, sweeter, and just takes up less space in the text!
“Like NIST 1900 Ei is the random variable representing the difference between the measurement M and the true value T. It already takes on both positive and negative negative numbers. There’s no need for two equation.”
If that difference can be either positive or negative then why do you always write it as positive and claim it always cancels?
You *should* write your equation as T + (+e, -e) to make clear what is going on. Then you wouldn’t make the mistake of thinking e always cancels when subtracting two T values!
If T1 is X1-e and T2 is X2 + e and you subtract T1 from T2 you get:
X1 -e1 -X2 -e2. ==>< X1 – X2 – 2e. NO CANCELLATION!
It appears that *YOU* are the one that can’t do simple algebra.
“ALGEBRA MISTAKE #25: A plus (+) operator in an equation does not imply that the rhs operand is positive.”
That’s the only possible conclusion when you claim the errors cancel! You *have* to be considering that both errors are either positive or negative. If one can be positive and one negative then you don’t get cancellation!
You are hoist on your own petard!
You can defend and rationalize your position until you are blue in the face. ± still isn’t a mathematical operator and the rhs operand of the plus (+) operator is not assumed to always be positive. That’s not debatable. And no, I’m not going to start using some made up symbology to write algebra equations. The established standard developed over the last hundred years or so is perfectly fine for the job as-is.
Here is what yer fav reference has to say (you know, one of the bits you skipped over):
7.2.4 When the measure of uncertainty is U, it is preferable, for maximum clarity, to state the numerical result of the measurement as in the following example. […]
“m_S = (100,021 47 ± 0,000 79) g, where the number following the symbol ± is the numerical value of (an expanded uncertainty) U = ku_c […].”
Looks like you need to contact the BIPM PDQ and tell them they don’t know algebra.
Get cracking…
I have no problem with the BIPM or the statement of uncertainty using the ± symbol. I do it all of the time myself.
What I have problem with is someone 1) using ± as an operator in an algebra equation and 2) assuming that the rhs operand in a + operation is always positive and 3) inventing a new paradigm for the expression of algebra.
Nutter.
When you assume that “e” cancels in an anomaly you *ARE* assuming that the operand is always positive ( or negative).
(A + e) – (B + e) = (A-B) + (e – e).
This is YOUR math. And (e-e) only cancels if they are equal and of of the same sign.
You can whine about the fact you didn’t specify if “e” is only positive (or negative) but that is the only way your math works.
I’m still waiting for an explanation as to how an error that is sometimes positive and sometimes negative can also be systematic.
Duh!
For one, because you don’t how it changes over time!
Why is this so hard to grasp?
Because if an error changes over time it is not a systematic error
Uncontrolled environmental variables. Systematic, deterministic, may vary through either sign, not random.
Because it is an INTERVAL containing a value that is unknown and therefore uncertain!
Look at a normal distribution. A standard deviation tells you that 68% of the values of that distribution lay within that interval, some plus, some minus. An normal distribution’s standard deviation has values of μ + ε and μ – ε, and ±ε defines the interval.
Uncertainty can define that interval that surrounds the measurement, but you do not know, AND CAN NOT KNOW, where within that interval the actual value truly is.
You are searching for a method of showing that you can eliminate uncertainty in measurements. Eliminating uncertainty can not be done. Even computing via RSS has a basic assumption that all measurements have a normal distribution.
What if they aren’t normal but one is skewed to the minus side and the other to the positive side? Do they cancel as they would using RSS? You end up with results that are incorrect and you can’t even identify why!
You want to convince everyone that uncertainty is reduced beyond an RSS value thru dividing by “n”! Show a general mathematic proof for that.
Let’s do a standard quality assurance problem. My assembly line creates 1000 widgets a day. Each widget is designed to weigh 5 units and is made up of two materials of equal weight with an uncertainty in each of 0.5 units. What should I quote the uncertainty of the average weight to be for any two widgets?
“Because it is an INTERVAL containing a value that is unknown and therefore uncertain!”
That describes random uncertainty.
“Uncertainty can define that interval that surrounds the measurement, but you do not know, AND CAN NOT KNOW, where within that interval the actual value truly is.”
Again, what’s the difference between random and systematic uncertainty? If you are defining a systematic error in terms of a probability distribution (that is probability in terms of a personal belief) the error is assumed to be the same for each measurement, as opposed to a random uncertainty where the value will be different each time.
“You are searching for a method of showing that you can eliminate uncertainty in measurements.”
I absolutely am not. The only certainty is there will always be uncertainties.
Can’t see what the resto of your comment has to do with systematic errors.
Because you have no clues about the subject.
Are you joking 😭! You just sealed the fact that you know nothing of physical measurements.
What experience do you have using sophisticated measuring equipment?
Why do you think there are calibration intervals required for using devices?
You would know devices drift both up and down over time. Some do go the same direction over time but you NEVER know.
Read E.3 in the GUM again and try to understand there is no difference between the treatment of Type A and Type B. The intervals are all based on variance and standard deviations.
^^^
+100
“You would know devices drift both up and down over time.”
Yes, though I doubt that’s the issue here. What I think Frank wants to say is that there are systematic errors that will change over time due to environmental factors. I don’t dispute that, and I think looking at systematic biases that change over time is a useful thing to asses.
But it doesn’t have anything to do with claiming the uncertainty of the global anomaly as very large due to all the errors for all instruments being treated as systematic, and then turning round and saying the error can change randomly from year to year.
Just quite while you are behind, this is not working for you.
Not quite yet. I’m quite enjoying it.
Oh look, a spelling lame. How lame.
No Tim. That is not my math. That is your math. You and you alone wrote.
Here is my math using the notation Pat Frank preferred from Vasquez.
M_k = T + β + ε_k
M_k is a measurement. T is the true value of the measurand. β is the bias or systematic error which is “fixed” and “a constant background for all the observable events” according to Vasquez. ε_k is the random error which is different for each of the k measurements. Note that this is similar to the model Possolo used in NIST TN 1900 except here we have added a systematic bias term. Now consider two measurements: a and b. The difference M_b – M_a is then M_b – M_a = (T + β + ε_a) – (T + β + ε_b) = ε_a – ε_b. Notice that β is eliminated from the equation via trivial algebra.
And like told Pat this can actually be proven with the law of propagation of uncertainty using GUM (16) and the model Vasquez suggested in (1) which is u_k = β + ε_k. Because the correlation term in GUM (16) is signed via the product of the partial derivatives it is necessarily the case that it is negative for a function involving a subtraction. And because β is the same for each measurement then the correlation matrix r(xi, xj) > 0 with a magnitude proportional to the effect β has on the measurements thus leading to the cancellation of the β component of uncertainty. I encourage you to prove this out for yourself using the NIST uncertainty machine.
Garbage-In-Garbage-Out…
From TN 1900.
“””””The equation, tᵢ = τ + εᵢ, that links the data to the measurand, together with the assumptions made about the quantities that figure in it, is the observation equation. The measurand τ is a parameter (the mean in this case) of the probability distribution being entertained for the observations. “””””
In TN 1900
“tᵢ” is not a true value to start with. A true value comes from establishing a normal distribution by measuring THE SAME THING, MULTIPLE TIMES, WITH THE SAME DEVICE.
“tᵢ” is a mean of measurements of different things under repeatable conditions. It is an expanded EXPERIMENTAL determination of a mean and standard deviation of what values may be expected at that location, with that device, and that time of year.
εᵢ is not a measurement error term. εᵢ is an interval describing the spread of the experimental observed data over time.
TN 1900 specifically says:
“””””Assuming that the calibration uncertainty is negligible by comparison with the other uncertainty components, and that no other significant sources of uncertainty are in play,”””””
That is, measurement ERROR plays no part in the analysis.
You have really gone off the tracks and overturned!
The use of “±” is to designate an interval surrounding a mean, mode, or median in a distribution. It is convenient because it defines a given interval of values in a normal distribution and because the square root of variance is a ± value!
There are two unique numbers defined by each. They must remain connected to define those numbers.
A ± does signify a symmetrical interval of (μ – ε) (μ + ε). It is not designed to be an algebraic operator.
You may subtract the two defined values, just as you do to find the distance between two points on a number line.
Where (μ + ε) > (μ – ε)
D = |(μ + ε) – (μ – ε)| = 2ε
The total size of the interval!
I know.
I’m glad you and I agree. Now can you convince TG of that for me? He won’t listen to me.
Here is what TG said.
“””””X1 -e1 -X2 -e2. ==>< X1 – X2 – 2e. NO CANCELLATION"""""
It is exactly correct! There is no cancelation. The intervals size is always 2ε.
“I told you: ± is short hand for X + e and X-e. You just write it as X ± e. ”
Is this were Karlo gets all his ± is illegal nonsense from?
That isn’t what ± means. At least not in this context. It’s simply expressing an interval.
https://wattsupwiththat.com/2022/12/09/plus-or-minus-isnt-a-question/
It does mean that in, say the quadratic formula, where the squarevroot can be both positive and negative.
“If that difference can be either positive or negative then why do you always write it as positive and claim it always cancels?”
When writing an error term, the error can be either positive or negative so you only need to write +. if the error is positive you are adding a positive number, if it’s negative you are adding a negative number.
It’s the fact it can be positive or negative that results in cancellation.
“If one can be positive and one negative then you don’t get cancellation!”
If you are talking about systematic errors, then they both have to have the same sign or they wouldn’t be systematic.
“a systematic bias that changes over time.”
It’s called weather.
It could be depending on what you want to include in your uncertainty budget. I was thinking more of the biases caused changing the time-of-observation, changing the station location, changing the station instrumentation, instrument aging, etc.
Yep. The grass turns brown in the winter when the weather gets cold. The grass turns green in the spring when it warms up. The grass turns brown again during the summer heat. The grass turns back green in the fall when the temperature goes down.
All of that is systematic bias in the temperature readings because of changes in the microclimate at the measurement site.
I could go on – like trees loosing leaves changing the wind impacting the stations, etc. None of it is random. The trees don’t just drop their leaves on a whim!
“which for a good unchanging site should be constant”
This is the nonsense — you have no basis on which to make this assumption. Bias errors are unknown, and change with time. All you can do is estimate them.
I simply can’t tell you how dismayed I am to see people in climate science saying that calibration drift doesn’t impact the trend identified by the measurement device and that they think a field measurement device can be “unchanging”.
Calibration drift is one of major reasons to point to DMM uncertainties, if you study the error specs it is clear that it is a very real factor. And at the core of all modern temperature measurements is digital voltmeter.
The GUM makes an interesting statement about Type B uncertainties to 4.3.7:
But this really doesn’t you much at all. What it is really saying is that it has to be studied and evaluated on a case-by-case basis, there is no one-size-fits-all.
I see to recall another statement in the GUM that I can’t find ATM which goes on to say that if a standard uncertainty is dominated by Type B factors, you need to do more work to get rid of them.
—————————————————————–
From the International Vocabulary of Metrology:
calibration:
operation performed on a measuring instrument or a measuring system that, under specified conditions 1. establishes a relation between the values with measurement uncertainties provided by measurement standards and corresponding indications with associated measurement uncertainties and 2. uses this information to establish a relation for obtaining a measurement result from an indication (bolding mine, tpg)
————————————————————-
Even a lab calibration should provide a measurement uncertainty statement.
——————————————————————–
From SOP 1 Recommended Standard Operating Procedure for Calibration Certificate Preparation:
2.10 A statement of the measurement uncertainty, and corresponding measurement unit, coverage factor, and estimated confidence interval shall accompany the measurement result.
———————————————————–
Even lab calibration winds up with an uncertainty factor so that measurements taken immediately after calibration will have an uncertainty interval. Calibration problems just proceed from that point!
Site ambience is exactly as variable as the weather.
“since that happens via algebra”
The religious myth that all systematic error is a constant offset; widespread in climate so-called science and fiercely there embraced.
I didn’t say that all systematic error is constant. What I said is that when systematic error is constant it cancels via standard algebriac steps when converting to anomalies. I’ve also said repeatedly that it is the time-varying systematic error is among the most troublesome problems in the quantification of the global average temperature change. I’d love to discuss it. The problem is that not many people can make it past the trivial idealized case of a constant systematic bias B. And as I always say…if you can’t understand the trivial idealized case then you won’t have any better luck with the vastly more complex real world.
It does *NOT* cancel. I showed you that it doesn’t cancel. It *is* simple algebra.
If T1 = X1 +/- e and T2 = X2 +/- e then the difference can range from
T1 – T2 = X1 – X2 + 2e or X1 – X2 – 2e.
NO CANCELLATION. Your algebra assumes all systematic bias is constant in the same amount and in the same direction!
After all this blather and whining about plus/minus signs, he forgot they are used about a hundred places in the GUM!
That’s not my algebra. Don’t expect me to defend it.
It *is* your algebra when you make the assertion that systematic bias always cancels in an anomaly!
The only way for that to happen is if the “e” term for both are the same and in the same direction. You have no justification for assuming that.
No Tim. This..
If T1 = X1 +/- e and T2 = X2 +/- e then the difference can range from
T1 – T2 = X1 – X2 + 2e or X1 – X2 – 2e.
…is not my algebra. It is your algebra. You and you alone wrote it.
I never used “e” in my algebra.
Just freaking great!
You used “Ei” instead of “e”.
ROFL!!!! Your algebra still sinks.
Are you still asserting systematic uncertainty *ALWAYS* cancels in an anomaly?
He keeping looking for loopholes to justify his tiny numbers, but doesn’t grasp Lesson One about the subject. There is a reason it is called Uncertainty, it deals with what isn’t known. But he thinks he does know it!
The distinction is important because the i signifies that it is a random variable. And just like Possolo did in NIST TN 1900 it is the random effect. It’s not even the bias I was speaking of. You literally switched the entire meaning “e” and created a completely different scenario to build an absurd strawman that you and you alone created.
It’s not my algebra. It’s the established standard. Specifically ± is not a valid operator. And the valid operators + and – make no assumptions about the sign of the right hand side operator. You can challenge basic algebra and make up as many absurd strawman as you want and ± still won’t be a valid operator and the valid + still doesn’t make assumptions about the sign of the rhs operand.
No. I’m not. And I never have. What I’m saying is that when the systematic uncertainty B is fixed and constant for both components of the anomaly then it cancels. Notice what I didn’t say. I didn’t say systematic uncertainty always cancels independent of the measurement model y. I didn’t even say it always cancels for the measurement model y = f(a, b) = a – b. It obviously won’t if it is different such that there is a Ba and Bb terms.
You forgot the link to the NIST Uncertainty Machine.
HTH
And of course, you are the world’s foremost expert on measurement uncertainty, surpassing even the Great Nitpick Nick Stokes.
“when systematic error is constant ”
Yes, and when pigs fly they won’t need to be trucked to market.
Pat Frank said: Yes, and when pigs fly they won’t need to be trucked to market.
And yet this is what you are effectively doing with your calculations. Your calculations are accidentally equivalent to an assumption of correlation of r(xi, xj) = 1 using GUM 16. I say accidentally here because the use of Bevington 4.22 at all is incorrect. It just happens to produce the same result as Bevington 3.13 with full correlation. A correlation of 1 between inputs is statement that their errors are exactly the same (ie the bias is constant and the same for all stations).
A farm animal reference!
You really aren’t just a blackboard scientist, are you?
Now you are just posting lies and propaganda.
My guess is that they have not a clue as to what temperature coefficients of electronic parts is.
Of course not, which are typically given as ±X ppm!
Yeah. That’s a good one. I’ve used it before. The NIST uncertainty machine is really good too. One advantage of the NIST tool is that it works with non-linear measurement models since it does a monte carlo simulation in addition to the GUM method.
That is a good reference. JCGM 100:2008 is also good and more comprehensive. There are other equally good references as well. They are all rooted in the law of propagation of uncertainty which uses the partial derivative technique.
Since I am reading some older papers right now, I thought this quote from Ramanthan et al 1987 might show where “the science” comes from..
Although they talk about equilibrium(!) surface warming, we are pretty close to 2030, but not to 1.5K. I think it is funny.
“For the 180-year period from 1850 to 2030”
Written in 1987as if 2030 were known … and people seem okay with it.
When I first saw fig2, looking at the pink vertical bar I said to myself “Schmidt has assumed a constant mean”
Which directly contradicts the entire notion of ECS. And this contradiction proves Schmidt’s method is wrong.
You cannot have constant mean if you have increasing CO2, unless ECS=0.
That contradiction occurred to me also, his whole approach makes absolutely no sense.
It makes sense, when you understand Schmidt’s purpose to be ‘pimping CO2 as the bringer of climate doom,’ no matter how many times reality shows that notion to be nonsense.
If you are dealing with data that are not stationary, the mean and standard deviation will drift over time. That means it will exhibit a trend. One can calculate a mean for a data set that has a trend, but then the question should be asked, “Of what utility is it, and what does the standard deviation tell us?” To get a sense of the ‘natural variation’ one should probably de-trend the data.
Agreed, or use the residuals of lag 1, meaning subtract the previous values from all values and compute the residuals to remove some or all the autocorrelation.
It would be difficult to learn with a million people checking your work. Then again, he volunteered.
“When I first saw fig2, looking at the pink vertical bar I said to myself “Schmidt has assumed a constant mean””
Well, you didn’t understand the figure. Fig 2 plots the difference between averages for two different decades. ERA5 returns just one figure for that, with the uncertainty shown. There is no progression in time.
Sorry Nick, the method Schmidt uses only applies to repeated measurements of the same quantity with the same instrument, see the Taylor reference in the post. The average annual global surface temperature changes each year and so does the measurement error. The entire climate state changes each year.
This makes no sense. Firstly, that isn’t the method Schmidt used. Secondly, his diagram described a single fixed number, the difference between the 1980-1990 mean and the 2011-2021 mean, per ERA5. It doesn’t vary over years.
But the real absurdity is that you keep saying that Schmidt underestimated the error, at 0.1°C, but applaud Scafetta’s estimate of 0.01°C, for all years.
I didn’t say that. I said that Schmidt used the standard deviation of ERA5 over the decade in question to compute the error (0.1). This will include natural variability (both El Ninos and La Ninas) to compare ECS with observations. Wrong! ECS is computed assuming that natural variability is zero. It is only valid to include measurement error, not natural variability.
“ECS is computed assuming that natural variability is zero.”
That’s just not true. But it’s also irrelevant to the task of measuring a decadal average.
What is happening is that you have GCM trajectories which diverge from ERA5. The statistical test requirement is to rule out any possibility that this is due to something other than wrong model. So you need to include all the ways that ERA5 might be legitimately different from what is quoted. That includes measurement error, sampling error, and weather variations, eg ENSO. If none of those can account for the difference, the difference is significant and the model is probably wrong.
Of course, it is true, see attached, from page 961, AR6.
That image is not showing or describing neither the ECS nor the natural variation. What it is showing the contribution from various forcing agents to the global average temperature. You might be able to make some inferences about ECS from this graph, but I’m not sure how reliable they will be.
Look closer at the graph. The bottom panel is labeled “natural” and it is zero.
That is natural forcing. We’re not talking about forcing here. We’re talking about variation. Remember variation is caused by things that effect the ebb and flow of energy as it is moved around the climate system. Forcing is caused by things that perturb the energy balance of the climate system.
These things are unnatural?? This word salad is total nonsense.
Well said. I think these two sentences explain the intent of Schmidt better than anything I’ve said so far.
Speaking of Gavin Schmidt, GISTEMP’s Land Ocean Temperature Index (LOTI) for March just came out. A comparison to February’s LOTI, all of the changes since 1974 were positive and most prior to 1974 were negative. All totaled there were 345 changes made to the 1718 monthly entries since 1880. This goes on month after month, year after year it’s a steady drone.
There could have been 1718 changes this month. In fact, there likely was. I think you’re only seeing a subset because that file is limited to two decimal places. If you change the gio.py and then run the GISTEMP code yourself you can output the same file with more digits and you’ll likely see more changes. Remember, all it takes it for a single observation to get uploaded to the GHCN or ERSST repositories between 1951-1980 for all 1718 monthly values to change.
Meaningless round-off errors? What does the precision of the input temperatures say about the valid number of significant figures that should be retained in the final answer?
Typically yes. That’s why the fewer digits shown the less likely you are to notice a change.
Steve,
You just need to get with the program and understand that the number of coincidences in mainstream climate science is nearly infinite. Data tampering to obliterate unseemly cooling trends is one example. Incorrectly calculating error bars so that at least a few of the crummy models appear plausible is another.
I’ll start off with this.
https://www.ncei.noaa.gov/pub/data/uscrn/documentation/program/NDST_Rounding_Advice.pdf
“”””Multi-month averages, such as seasonal, annual, or long term averages, should also avoid rounding intermediate values before completing a final average calculation:
Seasonal MeanT = (Seasonal MaxT.xxx… + Seasonal MinT.xxx…)/2 = XX.x
Annual MeanT = (Annual MaxT.xxx… + Annual MinT.xxx…)/2 = XX.x For final form, also round Seasonal or Annual MaxT and MinT to one place: XX.x”””””
The adage I learned and it in almost all college lab manuals was the averages could not exceed the precision of actual measurement. IOW, you could not add decimal places through mathematical calculations.
NOAA lists CRN accuracy as ±3°C. NOAA shows one decimal as the final form. Somehow claiming 0.01 uncertainty just looks like mathburtation.
There are other issues we can address. Variance is one. Both monthly averages and baseline averages are random variables with a distribution with variances. When subtracting random variables to obtain an anomaly the variances add. If those variances exceed the “measurement uncertainty” then again, the measurement uncertainty is meaningless.
Last for now, the SDOM/SEM must be calculated from a sample mean that is normal. Have any of the involved data distributions been checked for this? Skewed distributions can really hose stuff up.
All the series are heavily autocorrelated, they are not normally distributed. This is probably the most important reason why Schmidt’s technique is invalid. Using the standard deviation as a stand in for error only works with a very small variance that is due to measurement errors. When nature is involved, as it is here, it is an inappropriate way to measure error/uncertainty. You need to back out the natural part. This can be done by detrending the data or simply lagging it by one sample and working with the residuals. What Schmidt did was break all the rules.
This post can supply more details:
https://andymaypetrophysicist.com/2021/11/13/autocorrelation-in-co2-and-temperature-time-series/
“All the series are heavily autocorrelated, they are not normally distributed. This is probably the most important reason why Schmidt’s technique is invalid.”
I’m not sure I see the logic there. Usually you expect auto-correlation to increase uncertainty.
As I said below, it seems this is simply a difference between only looking at measurement uncertainty verses looking at year to year variance.
“I’m not sure I see the logic there. Usually you expect auto-correlation to increase uncertainty.”
Indeed there is no logic. Andy is reeling off all the stuff usually put to claim uncertainty is underestimated, and then saying that Schmidt has overestimated.
Bellman,
Quite the opposite. Autocorrelation decreases the uncertainty because the previous value in the time series determines most of the next value. It does inflate the R^2 and the other statistical metrics and they have to be corrected for the degree of autocorrelation. But the next value has more certainty than if all the y values were independent of one another.
Schmidt’s error value requires that each measurement be independent, but of the same quantity and with the same tool. It cannot be a constantly changing quantity like the global average surface temperature.
“Schmidt’s error value requires that each measurement be independent”
You still don’t seem to have any idea where Gavin’s error value came from. It’s nothing like that. He simply reasoned that ERA5, being derived from basically the same data, should have the same uncertainty of the global mean as GISS and other measures.
And that is not derived rom measurement uncertainty, which is indeed small. Scafetta’s figure 0.01C may even be about right for that (do you still commend it?). But the main source of error in the global average is spatial sampling. You have measures at a finite number of places; what if they had been somewhere else?
“Autocorrelation decreases the uncertainty “
Autocorrelation increases the uncertainty of the mean. It does so because, to the extent that each reading is predictable, it offers less new information.
“And that is not derived rom measurement uncertainty, which is indeed small. “
Upon what foundation do you make this claim? Even the Argo floats were assessed at having an uncertainty interval of +/- 0.5C.
The measurement uncertainty is *NOT* the standard deviation of the sample means. Measurement uncertainty is not the same as trying to calculate how close you are to the population mean. How close you are to the population mean is irrelevant if that population mean is inaccurate.
“Upon what foundation do you make this claim?”
Well, Andy is telling us that it is 0.01C.
“Autocorrelation decreases the uncertainty because the previous value in the time series determines most of the next value.”
That makes no sense. If you are using the standard deviation of annual values to estimate the SEM, then the fact that the previous value determines most of the next value, means you have less variation in annual values. That in tern means your calculated uncertainty is smaller, but spurious. You need to correct for the autocorrelation by increasing your calculated uncertainty.
That’s not consistent with JCGM 100:2008 equations (13) and (16). You can also test this out with the NIST uncertainty machine which allows you to enter the correlation matrix.
bdgwx,
Both of those equations are “valid only if the input quantities Xi are independent or uncorrelated.”
They do not apply to autocorrelated time series, like ERA5 obviously.
That’s not correct. Equations (13) and (16) are generalized to handle both correlated and uncorrelated (independent) input quantities.
BTW…the quote in its entirety is as follows.
It is possible that you read this paragraph and missed the verbiage talking about equations (10), (11a), and (12), which are special cases that apply only to uncorrelated inputs, and conflated them with equations (13) and (16), which apply to both uncorrelated and correlated inputs.
“the correlations must be taken into account.”
And where is this taken into account exactly when it comes to climate science?
It isn’t, and certainly not by Schmidt. Regarding eq. 13 and 16, u is the variance, representing uncertainty. These equations are suspiciously similar to those used by Scafetta. Except they are for 2 inputs.
Scafetta assumed the years were independent/uncorrelated.
Being pedantic…u is the standard uncertainty and u^2 is the variance.
Scafetta’s equation can be derived from (13) or (16) if you assume uncorrelated inputs such that u(x_i, x_j) = 0 for (13) or more conveniently r(x_i, x_j) = 0 for (16). Notice that in both case (13) and (16) reduce to equation (10). And since the partial derivative of ∂f/∂x_i = 1/N when f = Σ[x_i, 1, N]/N then equation (10) reduces to σ_ς / √N when σ_ς = u(x_i) for all x_i.
Equations (10) (13) and (16) are all for an arbitrary number of inputs. The inputs don’t have to be of the same thing or even of the same type. They may even have different units. For example, f(V, I) = V*I where X_1 = V is voltage and X_2 = I is the current.
The correlations will only increase the uncertainty. The equations are literally just the equation for data with no correlation plus another term based on the correlation.
“These equations are suspiciously similar to those used by Scafetta. Except they are for 2 inputs.”
They are for as many inputs as required. And there’s nothing suspicious about the equations being similar – they are all based on the same concepts.
I’m not following that argument. If there is autocorrelation that will increase the uncertainty in this case.
Using the NIST uncertainty machine with r(x_i, x_j) = 0 we get 0.578 ± 0.010 C. But adding some correlation of say r(x_i, x_j) = 1 – ((j – i) / 10) we get 0.578 ± 0.027 C
It might be helpful to point out that I chose r(x_i, x_j) = 1 – ((j – i) / 10) such that a value is 90% affected by the value 1 year ago, 80% for 2 years ago, and so forth. You can certainly plug in any correlation matrix you want. I just thought this one was reasonable illustration of the point.
bdgwx, Again, autocorrelated time series are not “independent nor are they uncorrelated”
I know.
Andy perhaps a review of sampling theory is worthwhile.
1) Sampling is done when you are unable to measure the entire population. Sampling can use the Law of Large Numbers (LLN) and the Central Limit Theory (CTL) to ESTIMATE or infer the properties of the population.
2) Sampling requires one to determine two things, the size of each sample and the number of samples to obtain.
3) Both the LLN and the CTL require two things:
a. the samples must be independent, and,
b. the samples must have a distribution like the population, i.e., an identical distribution.
c) This is known as IID.
4) If done properly the mean of each sample taken together will form a normal distribution called the sample means distribution. This distribution is important since it determines the accuracy of statistics obtained from it.
5) If the sample means distribution is normal,
a. The sample means average “x_bar” will estimate the population mean μ.
b. The sample means distribution variance, s², and the standard deviation of the sample means distribution is √s² = s
c. The population Standard Deviation, σ, can be inferred by the formula s • √n = σ. Where “n” is the sample size and NOT the number of samples.
6) The common term Standard Error of the sample Means (SEM) IS the standard deviation of the sample means distribution, “s”.
What does all this mean?
First, if the stations are considered samples, then all the above applies. Importantly, do the means of the samples form a normal frequency distribution? If not, then all bets are off. The sampling was not done properly. None of the inferences about the population statistical parameters made from the sample means distribution statistics are valid.
Second, a very important point about the SEM, which may affect the perceived error of the mean! See item 6) above. THE STANDARD DEVIATION OF THE SAMPLE MEANS DISTRIBUTION IS THE SEM.
YOU DO NOT DIVIDE THE SEM BY ANYTHING TO OBTAIN A SMALLER NUMBER. You certainly don’t divide by the number of samples (stations)!
The following documents discusses this issue
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1255808/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2959222/#
I have never seen in any paper or blog where the “global” data used has been checked to see if there is a normal distribution. This makes the inferences suspect of both suspect.
I’ll address your references from Dr. Taylor’s book in another post.
And the number of samples for just a single time series is always exactly one.
“Sampling requires one to determine two things, the size of each sample and the number of samples to obtain.”
How many more times. There is usually only one sample. There is no point in taking multiple samples from the same population because if you do you can merge them into one bigger sample.
Holy crap guy, you just denigrated every polster, medical researcher, marketing manager and hundreds of math professors. Have you ever searched the Internet for sampling theory?
That is beside the fact that no climate scientists does what you say. Why do we track and calculate by station? Just dump every temperature reading into a big bucket! Sounds good to me actually.
“Holy crap guy, you just denigrated every polster, medical researcher, marketing manager and hundreds of math professors.”
No, I was denigrating you.
“That is beside the fact that no climate scientists does what you say.”
How on earth do you take multiple samples of the same planet over the same time period? As you keep saying, you can only measure the temperature once.
The issue here isn’t combining different averages, it’s your claim that in order to determine the SEM you repeatedly take samples.
It’s this misunderstanding that means that most of the rest of what you say is either self evident or wrong.
e.g.
“If done properly the mean of each sample taken together will form a normal distribution called the sample means distribution. This distribution is important since it determines the accuracy of statistics obtained from it.”
How many different samples do you want to take to determine the sampling distribution of the mean in this way?
You can do it in a simulation, maybe you do it in your workshop, but if say you you are measuring the heights of trees in a forest, and you take a sample of size 30, why on earth would you then take another 100 samples of the same size, just to determine the sampling distribution?
As you say, the point of sampling is that it’s too expensive or impossible to measure every tree in the forest, so you only take a sample of sufficient size to give you a good enough estimate of the population size. If you had to expand your sample of 30 into a sample of 3000, how is that practicable. And if you could measure 3000 random trees, why not just combine them into a single same of 3000, and get a more accurate estimate?
Maybe you are thinking of bootstrapping or Monte Carlo techniques that do simulate taking multiple samples, but that doesn’t seem to be what you are saying.
“If the sample means distribution is normal,
a. The sample means average “x_bar” will estimate the population mean μ.”
The mean of a sample will estimate the population mean regardless of the shape of the sampling distribution. Though with a sufficiently large sample size it should be normal.
“b. The sample means distribution variance, s², and the standard deviation of the sample means distribution is √s² = s”
Gosh, the square root of the square of something is equal to the original value.
But this is where you start to confuse yourself. s is usually used for the standard deviation of the sample. But here you are trying to use it for the standard deviation of the sampling distribution – that is SEM. And why would you be interested in the variance of the sampling distribution?
” c. The population Standard Deviation, σ, can be inferred by the formula s • √n = σ. Where “n” is the sample size and NOT the number of samples.”
You keep saying this as if it was a useful result. Usually you have one sample, and you infer the population standard deviation from that sample standard deviation. If you have taken many different samples, you could infer the population standard deviation by looking at the standard deviation of the pooled results. There’s zero point in taking the sampling distribution of the multiple samples, and multiplying by root n.
“First, if the stations are considered samples, then all the above applies.”
Samples of what? You keep making these vague statements without ever specifying exactly what you are attempting to do. If you want to know, say, the global average anomaly for one specific month, how are you considering stations as samples? Do you mean treat the 30 daily values for any one station as one sample, or do you mean take each monthly average from an individual station as one value from the sample that is all stations?
“YOU DO NOT DIVIDE THE SEM BY ANYTHING TO OBTAIN A SMALLER NUMBER.”
Who has ever suggested you do? And why did this trivial point need to be written in capitals?
It seems obvious that thsi simply illustrates that you don’t know what the SEM is or how it is calculated.
“You certainly don’t divide by the number of samples (stations)!”
If you want to treat each individual station as being a sample unto itself over a particular period, fine. You have 1 single sample per instrument, and you could calculate it’s SEM just as in Exercise 2 of TN1900. But if you then average all of the instrument averages over the globe to get a global average, you are not producing a sampling distribution. Each station is a completely different sample. If you want to use multiple samples to estimate the sampling distribution they all have to come from the same population (IID remember).
So if what you are trying to say, is you can take the standard deviation of all the individual station means, and treat it as the SEM, you really haven’t understood anything. The global average is a single sample of all the station means. The standard deviation of all the station means is the standard deviation of that sample, and the SEM is that standard deviation divided by root N, where N is the number of stations.
And to be clear, none of this is how you would calculate the uncertainty in an actual global anomaly. Stations are not random observations from the surface. There are multiple ways of creating global averages from them, and all will require specific estimates of the uncertainty involved in each procedure.
Diminishing returns can bite in a big way with increasing sample sizes. It’ often useful to compare statistics (including variance/sd) for multiple samples to see whether they could come from the same population. That gives more information than pooling them.
Comparing samples seems to be what this whole topic is about 🙂
Your comment makes no sense. Multiple samples is why there are so many stations.
Once again, individual stations are not sampling the population. If you want a global average the population is the entire globe. One individual station can give you a sample from one point on the globe.
A sampling distribution requires multiple samples, each taken from the entire population. Unless your sample from an individual station is somehow sampling random points around the globe for each value it records, it cannot be considered a random sample from the global population.
Maybe it would be a fun exercise to simulate that. Take all the daily station data, and create simulated stations where every daily value is taken from a random station. Then each of these synthetic stations could be treated as a random sample of the global average for that month, and the distribution of all such stations would give you the standard error of the mean. But it would only be the error of the mean of a sample of size 30, so not much certainty.
“NOAA lists CRN accuracy as ±3°C.”
I think that should be ±0.3°C.
I laughed when I saw the output line of that on-line error propagation tool in figure 5 — ten digits!
Meaningless digits! I was teaching when hand-held calculators first came out in the ’70s. I’d give a problem on a quiz with all the inputs having 3 significant figures, and I’d get answers with all the digits displayed on their calculator. It was surprisingly difficult to get the students to understand that I expected them to be smarter than their calculators.
Looking at the details of the input values in the figure, there is enough information to get a handle on the real number of digits, but they throw it all away.
A whole lot of statistics relies on a constant mean and deviation. You cannot apply these formulas and methods to climate because.
1. Climate does not have a constant mean because ECS > 0.
2. Climate does not have a constant deviation because extreme weather is becoming more common (reportedly).
Climate data is probably a fractal distribution that has been sampled with bias simply because the weather stations are not randomly distributed. Assumptions that rely on the Central Limit Theorem are probably wrong.
Would I be correct in thinking that the difference between Figures 3 and 4 is that 4 is using the standard deviation of de-trended residuals? Otherwise the two equations are both the same. Just the standard deviation divided by the square root of sample size.
Equation 4 uses an outside estimate of measurement error. This is used because some of the standard deviation is not actual error, it is natural variation. The IPCC ECS estimate assumes natural variability is zero and that ECS accounts for all the warming. So, throwing the natural variability into the mix and saying that the resulting large “error bars” mean a higher ECS is valid is disingenuous in the extreme.
The ECS is only large in the first place because they assumed natural variability was zero.
Ah got it – I think. The Scafetta error is based on the assumed measurement uncertainty of a round ±0.03°C for each annual value, whereas the Schmidt one is based on the standrd deviation of the annual values.
Schmidt is using the same technique used in Example 2 of TN1900, treating each year as a random variable about a true average, whereas Scafetta is treating the decade as an exact average with thew only uncertainty coming from the measurements.
Yes. Obviously assuming one valid decadal mean is not correct, so Schmidt’s technique is invalid.
But it is Scafetta who is assuming the valid decadal mean.
Of course he isn’t Nick. His error varies every single year. Schmidt is the one with the constant error because he simply took the standard deviation. He explicitly assumed that the mean of both decades was invariant.
Originally Scafetta assumed the decadal values had no uncertainty. That was the basis for Schmidt’s objection. Schmidt estimated an error of 0.1C. The mean is invariant; it is just one number that we are uncertain of. Scafetta then conceded some uncertainty, although you say it was much less, about 0.01C. But I think you are wrong there. Anyway, schmidt never assumed a valid decadal mean (in the sense of no uncertainty).
I see Pat Frank helped you with this. My understanding of his error propagation paper is that each model is rendered useless by its uncertainty in cloud analysis. If that is right the analysis in this article is discussing how to average meaningless model results. That thought is supported by Howard Hayden’s work showing a warming of 3C would require a forcing of about 17 W/M2- about 5 times the IPCC estimate of the forcing from doubling CO2.
I’m intrigued that Pat Frank contributed to this, given he claims that the annual uncertainty in HadCRUT is ±0.5°C. If that figure was used the uncertainty would be 0.5 / sqrt(11) = 0.15.
Don’t confuse two types of error. Pat may be correct that the total annual uncertainty in HadCRUT5 is 0.15C. But we are not talking about total uncertainty in this post. This is simply evaluating what the measurement uncertainty is in the ERA5 record. The total uncertainty includes natural variability, especially the uncertainty in cloud feedback and ocean oscillations, ENSO, AMO, PDO, etc. This latter component is unknown and much larger than the measurement uncertainty. Plus, there seems to be a persistent upward trend in the 1980s that must be dealt with.
If the natural forces were known (they aren’t) and properly modeled, then the IPCC assumption that they are zero is not correct. Nature is involved in recent warming, this means the greenhouse effect is smaller, and ECS is smaller as a result.
On the other hand, if we include the natural variability in the total error estimate, it causes the error bars to be inflated and larger ECS values are within the margin of error. See the problem? You can’t say nature is zero and compute a large ECS, then turn around and say the measurements have a lot of error because the standard deviation is large when it is actually large because of natural forces.
Probably both Pat and Nicola are correct. The total uncertainty in our estimates of global average temperature are large and the measurement error is small, and the natural contribution to warming is large.
“But we are not talking about total uncertainty in this post.”
Well, you should be. You have a number from ERA5, and one from models. Whether they are significantly different depends on your total uncertainty about each.
No, we are showing that the ECS computed in the climate models is not meaningful above 3. All we are concerned with is the measurement error.
The model computation of ECS, the quantity we are checking, assumes that natural variability is zero. If we include natural variability in our error estimate it invalidates the test.
It is clear, at least to me, that natural variability is not zero. But the models assume it is, so to check model-ECS we must do that also. Schmidt, on the other hand, claims it is zero in one calculation and then claims it isn’t when he is checking the calculation. Sorry, Schmidt, an obvious flaw of reasoning.
“It is clear, at least to me, that natural variability is not zero. But the models assume it is…”
Completely untrue.
“Schmidt, on the other hand, claims it is zero in one calculation”
No. Schmidt never made such a claim.
” All we are concerned with is the measurement error.”
Of ERA5? Why?
Scafetta is testing the significance of the difference between ERA5 and models. To do that, all sources of uncertainty should be included. For example, the main uncertainty in ERA5 is spatial sampling error. Models do not have such an error. It is obviously a source of uncertainty in the difference.
Models don’t produce natural variability except as noise. This is because they can’t. Natural variability was unknown when models were first built in the 1960s. The AMO was discovered in the 1980s and the PDO in 1997. It is unknown how they are produced. We don’t have a working theory for ENSO, so they cannot be put into models in a realistic way. The NAO is red noise in models and they cannot reproduce its long positive or negative trends seen in observations.
The result of such ignorance is this:
This figure is adapted from IPCC AR5 (a & b) and the 4th National Climate Assessment of 2017 (c). Models believe natural climate change has played no role since 1750 because nothing has been put into them that allows them to believe otherwise. And climate scientists accept this result unquestioningly instead of saying models are unfit for purpose due to our lack of knowledge.
It is such madness that it will haunt climate science for decades to come and it will tarnish all science for looking away instead of demanding scientific standards be upheld.
Why do you think I, a molecular biologist, had to study and write about climate science when I have no dog in this fight? I couldn’t care less if it is anthropogenic CO2 or some natural factor. I have no political inclinations. Climate scientists are not doing their job, and are dragging science into a dark alley to be abused by people with an agenda. It is the duty of every scientist to defend science from such abuse.
Love your response. Everything I see from climate science (not other fields who claim it as an excuse) is geared to proving that temperatures can be measured more and more accurately through statistical manipulation. They ignore the fact that temperatures have never been measured to a precision that can justify that, let alone that temps vary from pole to pole and sea level to tops of mountains by 10’s of a degree..
“This is because they can’t. Natural variability was unknown when models were first built in the 1960s.”
Of course natural variability has always been known. It is weather. Models evolved from weather forecasting programs. It was found that, although the weather they calculate is unreliable as a forecast after a few days, it is still weather that reasonably could have happened, and it responds in the same way to climate forcings. Then it becomes a GCM. It started with natural variation.
You have listed some particular modes of natural variation. They are not used in any way by the GCMs, which work from the fluid dynamics fundamentals. Those modes are an outcome, and would have appeared whether the modelers knew about them or not. GCM’s do a very realistic ENSO, even though no specific information of ENSO was supplied.
Nope. Climate is the average of weather and includes weather variability. Climate variability is not weather variability.
If we define climate as the 30-yr average of weather (WMO definition), then climate variability is the variability that takes place from a 30-yr period to the next.
Weather variability is generally considered chaotic, climate variability can be deterministic (volcanic, solar, orbital, oceanic oscillations).
Nick, you clearly don’t understand natural climate change. I recommend you read my book, it is very inexpensive.
Models don’t understand climate variability because nobody does, and models can’t understand what is not in their programming.
A common but false myth. Manabe and col. built their atmospheric model in the 1960s from scratch.
That’s just opinion. Many models still produce a double intertropical convergence zone in the Pacific.
Samanta, D., Karnauskas, K.B. and Goodkin, N.F., 2019. Tropical Pacific SST and ITCZ biases in climate models: Double trouble for future rainfall projections?. Geophysical Research Letters, 46(4), pp.2242-2252.
How could you possibly believe that models do a very realistic ENSO with a double ITCZ? Gullibility? Ignorance? Wishful thinking?
“If we define climate as the 30-yr average of weather (WMO definition)”
A common misconception. WMO does not make that definition, nor AFAIK does anyone else. The WMO recommends a 30 year baseline for the purpose of calculating temperature anomalies. That is long enough to smooth interannual noise, and short enough to be reasonably achieved with the records we have. It is not a definition of climate.
“then climate variability”
Moving the goal post here. You were talking about natural variability. That is what GCMs are founded on. “volcanic, solar, orbital” are forcings (natural) and are input as data.
The splitting of the ITCZ is a fine point which is being solved. But here is GFDL from 8 years ago
https://www.youtube.com/shorts/KoiChXtYxOY?watch
Nick Stokes April 14, 2023 5:50 pm
Sorry, wrong.
NASA: “Some scientists define climate as the average weather for a particular region and time period, usually taken over 30-years.”
Canadian Met Office: “When it comes to making decisions which incorporate future climate change, or determining how the climate has changed at a specific location, the advice is to use at least 30 years of data.”
National Geographic Society: “A region’s weather patterns, usually tracked for at least 30 years, are considered its climate.”
WMO Itself: “Climate, sometimes understood as the “average weather,” is defined as the measurement of the mean and variability of relevant quantities of certain variables (such as temperature, precipitation or wind) over a period of time, ranging from months to thousands or millions of years. The classical period is 30 years, as defined by the World Meteorological Organization (WMO).”
NASA: “Climate, on the other hand, refers to the long-term (usually at least 30 years) regional or even global average of temperature.”
American Chemical Society: “Climate, derived from the Greek klima meaning “inclination,” is the weather averaged over a long period of time, typically 30 years.”
There are lots more.
w.
You’re right, Willis.
I ran across a paper describing the origin of the 30-year averaging period defining climate.
It’s in a 1948 meteorological review about the changing climate: Hans Wilhelmsson Ahlmann (1948) “The Present Climatic Fluctuation” The Geographical Journal, 112(4/6) 165-193.
The “Fluctuation” of the title is the early 20th century warming trend. It was a surprise to them.
The relevant text is page 165, note 3: the 30-year climate period was accepted by the Directors of the International Meteorological Organization (the WMO predecessor) at a 1935 Meteorological Conference in Warsaw.
They also decided the normal period should be 1901-1930 — a period that has since been changed, evidently according to convenience.
The 1935 meeting of the full IMO membership was in August in Danzig. The 30-year normal was considered there by the attending meteorologists.
Page 343, beginning “For many years it has been recognized that there should be some method for expeditious international exchange of current climatological data” discusses the issue of the 30-year normal period raised in Danzig.
The Warsaw meeting of the IMO Directors ratified the recommendation of the Danzig meeting.
So that’s it. The 30-year normal was decided by the IMO membership at the August meeting in Danzig and ratified in September by the Directors meeting in Warsaw.
I’ve done a careful translation of the in-French Conference statement, using Google Translate. Here is the original followed by the translation:
Note 3: “Earlier definitions of climate took no account of time, because it was supposed that climate was constant except for short periodic variations, and that the characteristic means of the climatic elements gradually became more certain as longer series of years became available for their calculation. The occurrence of the present climate fluctuation as a secular variation has caused the “normal value” of climatology to lose its significance. At the Meteorological Conference in Warsaw in 1935 the following definitions were agreed upon: “Pour des raisons pratiques, on définiera comme climat, les conditions météorologiques moyennes pour le mois et l’année, calculées sur une de période de 30 annees. On définiera comme fluctuations du climat les différences entre deux moyennes calculées sur 30 années. On ne peut parler d’une variation du climat que dans le cas oú cette différence dépasse une certaine valeur qui dépend de la dispersion des observations individuelles. La Conference est d’avis que les discussions concernant les fluctuations des moyennes climatiques devront être reportées à une période universelle et synchrone allant de 1901 à 1930. Elle recommande d’utiliser de préférence cette période lorsqu’on dresse des cartes climatographiques.”
Google Translated, with French accents corrected in:
“Earlier definitions of climate took no account of time, because it was supposed that climate was constant except for short periodic variations, and that the characteristic means of the climatic elements gradually became more certain as longer series of years became available for their calculation. The occurrence of the present climate fluctuation as a secular variation has caused the “normal value” of climatology to lose its significance. At the Meteorological Conference in Warsaw in 1935 the following definitions were agreed upon: “For practical reasons, we will define as climate, the average weather conditions for the month and the year, calculated over a period of 30 years. Climate fluctuations will be defined as the differences between two averages calculated over 30 years. We cannot speak of a variation of the climate only in the case where this difference exceeds a certain value which depends on the dispersion of the individual observations. The Conference is of the opinion that the discussions concerning the fluctuations of the climatic means should be postponed to a universal and synchronous period extending from 1901 to 1930. It recommends that this period be preferably used when making climatographic maps.”
So there it is. Ahlmann also noted (p. 172) that it was not until recently (1941) that meteorologists had to finally dispense with the belief that the climate was generally stable.
I also ran across papers of Robert Griggs who, in 1937, introduced the idea that shifting timberlines could be used to deduce long-term climate change.
Griggs quoted Asa Gray, mid-19th century, examining at Darwin’s request to investigate the problem of islands of the same species separated by, e.g., glacial barriers. Did God make special creations in every mountain valley? Gray apparently cautiously cast doubt on the idea. Phylogenetic origin was a real problem.
Griggs 1937 wrote further, “Only recently have we become aware of the fact that in some countries there have been vast changes in climate within the period of civilization, or even within the last thousand years. On the contrary, mankind has generally assumed that such climatic changes as obviously must have occurred in periods of glaciation came on too slowly to have any human importance.”
Trusting his references, Griggs’ “only recently” means the early 1920s.
The history of thought is fascinating. It’s a revelation how recently our modern views originated.
The idea of a spontaneously changing climate is recent to modern times. And GCMs haven’t kept up.
The ocean model in AOGCMs does not converge. They cannot simulate physically valid ENSOs or anything else.
One only has to read about how terribly models handle both clouds and oceans in Mototaka Nakamura’s book “Confessions of a climate scientist” to understand why they never work properly!
Nick Stokes April 14, 2023 1:13 pm
Oh, please. The GCMs only “work from the fluid dynamics fundamentals” in the same way that Hollywood movies are “based on a true story” …
Here’s how they actually work, through a complex bunch of guardrails, kludges, and arbitrary tunings and adjustments.
w.
Nick is right on that point. There is no “make a cold front here”, “make a hurricane there”, “make an El Nino”, “make a positive AMO”, “make a negative PDO”, etc. heuristics in GCMs. Those variations are consequences primarily of the ebb and flow of energy and moisture constrained by the primitive equations. Those variations just…happen…organically…for lack of a better way to explain it.
Now, for heuristic/statistical models natural variation is handled explicitly. For example, the CPC-Markov model is a heuristic/statistical model for predicting ENSO. But, the discussion was not about heuristic/statistical models. It was about global circulation models.
“It is the duty of every scientist to defend science from such abuse.”
That’s exactly why I got into it, too, Javier.
I was compelled to rescue science itself and as the source of every worthwhile value we have.
Nick, spatial uncertainty is one of the categories estimated in HadCRUT5, where Scafetta got his year-to-year error estimates.
Right. That’s the issue. Because HadCRUT has spatial uncertainty (and other kinds of uncertainty) there is going to be expected discrepancies between it and a modeled value. The question is…is the observation with the uncertainty envelope included inconsistent with the prediction. For example, if the observation is 0.58 ± 0.1 C and the prediction is 0.65 then we cannot say the observation is inconsistent with the prediction. However, if the observation is 0.58 ± 0.01 C then we can say it is inconsistent. But the problem is that ± 0.01 C may not be including all sources of uncertainty that are essential for this specific comparison.
Nick, Schmidt made exactly that claim according to Zeke Hausfather:
See here:
Analysis: Why scientists think 100% of global warming is due to humans – Carbon Brief
Your quote doesn’t say it was zero.
But there is endless confusion here between calculating something, and finding that it comes out to be zero (or near), one the one hand, and on the other assuming that it is zero.
Again, I don’t understand this argument. The modeled ECS does not contain a component of natural variation. Or put another way the ECS plays out over 100+ years so any natural variation reduces to near zero because it is cyclic and the time period is so long.
The observed temperature, however, does contain natural variation. And the shorter the time period the more the natural variation dominates the temperature changes. It is certainly true that at 10 years the natural variation starts washing out, but it is there a bit.
So when you compare one value without the variation to another value with the variation you have to account with the potential difference caused by the variation.
The original paper was an examination of ECS, how big can it be? ECS cannot be measured, instantaneous doubling of CO2 is not possible in nature, it is a totally model-based value. Insulated from proper scientific testing.
If we compare it to nature with all its complexities, are we doing a fair comparison? I think not. It isn’t a real value that can be measured.
Besides the elephant in the room is natural variability. If natural variability is significant on the decadal scale, the ECS is wrong anyway.
Nah. If say ECS = 3 C and variation has an SD of ±0.05 C then once the ECS plays out the temperature will randomly walk between 2.9 and 3.1 C (2σ). Over long periods of time the temperature will average out to 3 C. But if you randomly pick a decade to sample you may get 2.9 or 3.1 C. Observing 2.9 or 3.1 C does not invalidate the ECS of 3 C.
Not according to John Taylor or me. The SD estimate of error is only valid when doing repeated measurements of the same quantity with the same instrument. If the quantity is “randomly walking” as you say, you must have a measurement error for each measurement (or year in our case).
Nobody said statistics was easy.
NIST TN 1900 E2 is the perfect example of how to handle this particular scenario if you are interested in reading through it.
No. That’s not true. The random walk here is caused by natural variation (ie ENSO cycles). Assuming hypothetically you had a perfect measurement of temperature with no uncertainty the temperature would still exhibit a random walk due to natural variation. In fact, you would easily see the natural variation in a series of measurements. They are even easily discernible today even though our measurements do have uncertainty.
For sure.
No one said the science of measurement, metrology, was easy either.
This fact is confirmed every time a statistician claims that the standard deviation of the sample means is the uncertainty of the mean. You can calculate the population mean down to the millionth digit and it can still be inaccurate because of systematic error in the measurements the population mean is calculated from.
Funny you mention variation. Especially washing out! The baseline variance never “washes out”. It is there year to year and decade to decade. It affects the uncertainty of what an average DISPERSION truly is.
“Or put another way the ECS plays out over 100+ years so any natural variation reduces to near zero because it is cyclic and the time period is so long.”
Really? So the interglacial temperature variation over time reduces to near zero?
“Pat may be correct that the total annual uncertainty in HadCRUT5 is 0.15C.”
Checking he actually says the uncertainty is ±0.46°C, as a lower limit, and, it may be at least twice that.
“The total uncertainty includes natural variability, especially the uncertainty in cloud feedback and ocean oscillations, ENSO, AMO, PDO, etc.”
I don’t see any mention of that in the paper. Here’s the abstract
https://www.science-climat-energie.be/wp-content/uploads/2019/07/Frank_uncertainty_global_avg.pdf
Andy,.
Let me post some initial stuff from the GUM.
2.2.3 The formal definition of the term “uncertainty of measurement” developed for use in this Guide and in the VIM [6] (VIM:1993, definition 3.9) is as follows:
uncertainty (of measurement)
parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand
NOTE 1 The parameter may be, for example, a standard deviation (or a given multiple of it), or the half-width of an interval having a stated level of confidence.
NOTE 2 Uncertainty of measurement comprises, in general, many components. Some of these components may be evaluated from the statistical distribution of the results of series of measurements and can be characterized by experimental standard deviations. The other components, which also can be characterized by standard deviations, are evaluated from assumed probability distributions based on experience or other information.
NOTE 3 It is understood that the result of the measurement is the best estimate of the value of the measurand, and that all components of uncertainty, including those arising from systematic effects, such as components associated with corrections and reference standards, contribute to the dispersion.
2.2.4 The definition of uncertainty of measurement given in 2.2.3 is an operational one that focuses on the measurement result and its evaluated uncertainty. However, it is not inconsistent with other concepts of uncertainty of measurement, such as
⎯ a measure of the possible error in the estimated value of the measurand as provided by the result of a measurement;
⎯ an estimate characterizing the range of values within which the true value of a measurand lies (VIM:1984, definition 3.09).
Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively. Nevertheless, whichever concept of uncertainty is adopted, an uncertainty component is always evaluated using the same data and related information. (See also E.5 .)
Lots of info in this. I can’t bold from my tablet so I will use caps (judiciously).
• parameter, … that characterizes the DISPERSION OF THE VALUES that could reasonably be attributed to the measurand
• The parameter may be, for example, a STANDARD DEVIATION (or a given multiple of it), or the half-width of an INTERVAL
• can be characterized by EXPERIMENTAL STANDARD DEVIATION
• measurement is the best estimate of the value of the measurand, and that ALL COMPONENTS of uncertainty, … CONTRIBUTE TO THE DISPERSION
• an estimate characterizing the RANGE OF VALUES WITHIN WHICH THE TRUE VALUES WITHIN WHICH THE TRUE VALUE OF A MEASURAND LIES
• The definition of uncertainty of measurement given in 2.2.3 is an OPERATIONAL one
Most of my experience has been with repeated measurements of the same thing using devices with stated precision. This fits in nicely with the concept that repeated measurements of the same thing with the same device in most texts and the GUM. You can expect a random (Gaussian) distribution surrounding a “true value” which is characterized by mean μ of that distribution. The standard deviation of that measurement distribution would be the range of values that could be expected.
Most of my attention the last two years has been how to propagate the “measurements uncertainty” of temperature measurements when measuring different things with different devices.
bdgwx referenced a document from NIST that bitch slapped me wide awake. It was TN 1900 authored by Dr. Possolo. The Example 2 in this document showed a procedure to calculate an expanded experimental uncertainty.
First, we should define what is measurements uncertainty? It is measuring the same thing multiple times in order to develop distribution surrouning a true value, i.e., μ ± ε. I am sure you and others have tired of my refrain about using measurements of different things to reduce measurement uncertainty. I have not changed my mind on that.
But?
What is experimental uncertainty? It is used to determine the range of values that can be expected from measurements that are not repeatable. TN 1900 gave exactly what I was looking for. When I went back and read the GUM in the references Dr. Possolo gave in this document it really rang a bell.
Experimental uncertainty is used for things like chemical reactions where you may run it 10 times and measure the products. Each one of those trials is a different thing and while accurate measurements for each is important, each trial will have a different result. What is more important? The accuracy of each measurement of the products or the range of expected values from multiple experiments?
I do believe this is a much better way to show the DISPERSION of values that make up the global temperature. Most folks, including lay folks, believe we know temperatures to the one thousandths of a degree. They know day vs night, day to day, month to month vary much more than that.
When calculating anomalies, you ARE subtracting two random variables. That means their variances add, even if you use RSS. It provides a more complete and more accurate picture of the values to be expected.
The procedures in the past ignoring the variance in temperatures has lead to this. Procedures that divide SEM’s by √9500 lead to uncertainties like 0.015 from temperatures recorded only in integers. That then leads to the assumption that one can also quote anomalies to the one thousands digit.
Ask any engineer ior surveyor if they believe you can add decimal digits through averaging.
Just to clarify, Andy asked for my thoughts on the matter. In essence, my response was that calculation of uncertainty in a mean depends on whether the various sources of measurement error are coherent.
If one can accurately weigh an elephant and a mouse on the same balance, the uncertainty in the mean weight is calculable and valid, and can be small, even though the natural variability — the enormous difference in masses — will be very large, because the uncertainty depends upon the instrument, not on the measured object.
I understand that in the real world, some objects can contribute to measurement uncertainty, but for purposes here I’m discounting that.
Natural variability is equivalent to the “magnitude uncertainty” ‘s‘ discussed in “Uncertainty in the Global Average Surface Air Temperature Index: A Representative Lower Limit”
Magnitude uncertainty is not due to measurement error, but rather to natural variability.
The lower limit of uncertainty I derived for the annual GMST in that paper arises from systematic (not random) measurement error. It does not diminish as 1/sqrtN.
I believe that Nicola Scarfetta is correct because the uncertainty in global annual temperatures varies from year to year and among the various compilations. This violates the statistical assumptions of random error.
“I believe that Nicola Scarfetta is correct because the uncertainty in global annual temperatures varies from year to year and among the various compilations. This violates the statistical assumptions of random error.”
My question wasn’t so much about whether you agreed with the method, and more if you agreed with the result.
As I understand it from the spreadsheet, Scarfetta’s uncertainty estimate is based on the stated annual uncertainty for HadCrut, at around ±0.03°C. If you believe the actual annual uncertainty is at least 15 times bigger, it’s difficult to see why the uncertainty for the decade wouldn’t also be 15 times bigger.
Nicola’s work is within the context set by the field.
What field would that be?
Consensus climate so-called science.
Are you saying you think Scarfetta is wrong to claim an uncertainty of 0.01, but you think it’s OK to mislead as long as it’s fro a good cause?
“Now let’s look at the dispute on how to compute the statistical error of the mean warming from 1980-1990 to 2011-2021 between Scafetta and Schmidt.”
Yer still a clown.
No. I’m saying you’ve lost the plot.
Then tell me what you think the plot is, rather than falling back on the one line insults. I get enough of those from Karlo.
Yes. The way TN1900 is set up, it measures the “magnitude uncertainty” of Tmax and of Tmin. Is it correct? It is not too dissimilar to what Dr. Taylor does in his book for small numbers of trials of a given measurand.
But no document addresses averaging trials of different things. It is like averaging the intensity of all the visible stars in the sky. Or projecting chemical reaction products from any number of sites around the world with no control over temperatures, weighing devices, or varying concentrations. Would one average them and evaluate the uncertainty by dividing by the number of sites? We have already learned that the gravity of the earth is not a constant at different points. Why would you divide measured values from different sites to get the uncertainty in measurements.
Using care, Jim, one can get physically meaningless averages with low statistical uncertainties. 🙂 Consensus climatology and welcome to it.
“Patrick Frank helped me with this post”
This is a weird article. The essential difference is that, in trying to find the differences are significant, Scafetta gave zero uncertainty to the ERA5 estimate (and later a very small one). Schmidt argued to assign an uncertainty similar to those of HADCRUT5 etc, which would reduce the significance. Pat Frank’s nonsense usually says that those uncertainties are far too small. Yet here Andy is rooting for Scafetta’s estimates, which are smaller still (but better for significance).
And, as bdgwx notes, they all use in some way the conventional division by sqrt(N) to get the uncertainty of the mean, which the Frank tribe vehemently rejects.
“Pat Frank’s nonsense”
“the Frank tribe”
Hahahahahahahahahah—so pronounceth Nitpick Nick Stokes, the world’s foremost expert on absolutely everything.
So do you think Scafetta’s estimate for uncertainty of surface average is right, as promoted by Andy et al?
“Scafetta computes a very small ERA5 error range of 0.01°C (Scafetta N. , 2022b, Appendix)”
Exactly where did I imply or state such?
It was a question. I thought you might offer a Frank tribe view on the matter. No-one else seems to.
If you had any real-world metrology experience you too should be laughing at both of these absurdly small numbers. But you instead choose to smear Pat Frank’s work as “nonsense”. I can only conclude this is either from abject ignorance, or you are trying to prop up a hidden agenda by lying.
Trendology at its acme.
WUWT isn’t laughing. It is cheering for the smaller one.
You have sampled the entire population of WUWT readers before coming to this conclusion?
I think not.
No, I read the article and comments.
You mistake disdain for Gavin Schmitt as “cheering for the smaller one.”
But you are also disdaining Andy May and Scafetta if you think Schmidt’s uncertainty is too small.
The acolyte to the rescue!
Nick is on record supposing that thermometers have perfect accuracy and infinite precision. So goes metrology.
Made me look. Apparently you have the wrong link…
You’re just going to have to read through the thread.
Pat, you made this statement way back then, about 6 years ago.
“””””There’s no getting around it, Nick. Resolution is a knowledge limit, and nothing gets it back.”””””
I have tried to explain this numerous times. None of the labs when I was in school nor any I have found online allow averaging readings to achieve a more precise value than the resolution of the measured values! The process is called is called Significant Digits Rules. You’ll find several folks do not believe it applies to averaging temperatures. Primarily because you can’t obtain anomaly values with three decimal digits. IOW, statistics overrule physical measurement resolution and DOES allow creating new measurement information out of thin air!
I have come to believe that both the GUM and NIST TN1900 address experimental uncertainty appropriately. This seems to make more sense to me when using monthly averages. This procedure uses measurement values collected under repeatable conditions, same shelter, same device, same location.
The documents use appropriate Significant Digits Rules even though they are not explicitly stated.
The procedure also provides a better view of the dispersion surrounding the mean of the data. I have not investigated procedures to use when averaging months, but expect the variance, which climate science totally ignores, to increase.
You might have a better viewpoint as an experimental scientist.
Jim — most treatments of systematic error, including in GUM, are pretty brief, because such error is of unknown magnitude and distribution, and so does not admit of closed mathematical expression.
In the absence of a rigorous description, more ad hoc methods must be used to express the resulting uncertainty. So, we use the expressions derived for random error, but judiciously. The 1/sqrtN rule is strictly inapplicable because systematic error violates the statistical assumption of normality. Variance can grow with repeated measurements.
You already know all that.
I agree that getting this pragmatic approach across to consensus climatologists and their groupies is tough to impossible.
I didn’t read your response until I had sent the latest.
One thing that bothers me is when calculating anomalies the variances are ignored. By any stretch, both monthly and baseline averages are random variables. When means of random variables are added or subtracted the variances add. This is unassailable. Yet it is never done.
Another is that Tmax and Tmin are highly correlated. This contaminates everything and makes the LLN & CTL assumptions invalid. Seasons introduce spurious trends.
I have become convinced the a GAT has so many problems it is not fit for purpose. We should be looking at Tmax & Tmin separately and only at local and regional areas. When you do this, it is hard to find areas warming sufficiently to offset those who have no to little warming.
Climate science has had to ignore a multitude of statistical rigor in order to TEASE out a value of warming that is so small it is totally inside any uncertainty interval one could imagine from the technology being used.
Dead on right Jim. Ignoring the variances when calculating anomalies has bothered me for years, too.
I’m about to submit the manuscript mentioned some time back. It deals with that bit of professional negligence. You’ll never guess what happens. 🙂
Tmin and Tmax are also contaminated with systematic error. CLT and LLN fail on those gounds, too.
Chris Essex showed years ago that the GAT was physically meaningless and — as you say — unfit for purpose. No one in the field cared.
I’m not seeing where Nick said that in the link you provided.
Nick’s assumption is in his model. I pointed it out here. He assumes perfect rounding and exact accuracy.
That’s only possible under conditions of an instrument with perfect accuracy and infinite precision.
Here is a short analysis of the temps Nick provided on that old post. I used the method from NIST TN1900. I have also added a histogram which shows that the distribution is far from normal. The expanded experimental uncertainty is no doubt needed here to appropriately define the range of temps that could be expected.
Oh my!
/saved/
Seems to be a pretty common assumption in climate science!
I think they are both incorrect. Reduction in measurement error is done through repeated measurements of the same thing in order to have a distribution surrounding the true value. Even measurands needing multiple measurements combine them in a weighted fashion.
The only way to combine measurement uncertainty of entirely different things is to directly add them.
An example is three numbers like 50., 60., 70. all ±1. The mean is:
180/3 = 60. .
Now add 1 to each value. The new mean is 61. . Subtract 1 from each and the new mean is 59. . The expected value must lay somewhere in the interval 60. ±1. You can use Root Sum Square to combine them and you end up with an uncertainty of 1.7.
So simple…
The sigma for this example with different things is 10, and there is no justification for dividing by root(N).
RSS(10, 1.7) = 10.2
“The expected value must lay somewhere in the interval 60. ±1. You can use Root Sum Square to combine them and you end up with an uncertainty of 1.7.”
If the expected value must lay in a ±1 interval, how can the uncertainty be ±1.7?
I didn’t say that RSS was the correct method or that checking the interval was correct. I said you could do it.
I believe you have been told that uncertainty grows when you average. Look at Dr. Taylor’s book. I merely pointed out, that by checking the interval calculated by assuming all measurements were at +1 or -1 from their ‘true value’.
Also, note I didn’t divide the uncertainty by 3 as is your want to do. Neither method I showed did that.
You said you could do it in order to “end up with an uncertainty of 1.7”. Why say that if you didn’t think 1.7 was the correct uncertainty?
“I believe you have been told that uncertainty grows when you average.”
Only by idiots who didn’t have a clue what they were talking about, and refused to contemplate they were wrong.
“Also, note I didn’t divide the uncertainty by 3 as is your want to do.”
Oh but you did. Three values each with an uncertainty of ±1. Add the three values and (ignoring any cancellation) the uncertainty of the sum becomes ±3. But for the average, as you say the uncertainty is ±1. Why not ±3? Because you’ve divided it by 3.
Dividing by the √N, is wrong for two reasons.
1) If the stations are samples, the SD of the sample means IS THE ERROR OF THE MEAN. You refuse to acknowledge SEM defines how close the mean of the sample means distribution is an error term thst tells you close the estimate is to the actual mean. Why you want to divide an SEM by √9500 is beyond me. It is trying to calculate the SEM of an SEM!
2) “N” is not the number of samples, it is the sample size. That means an annual average (a sample) has 12 monthly values! “N” is not the number of samples! Basically, the “N” for a 30 year baseline is 12, the number of months in each sample. If I hear dividing by the √9500 again I’ll scream loud enough you can hear me.
3) If you claim the stations are the population, there is no reason to even deal with sampling.
“Dividing by the √N, is wrong for two reasons.”
It’s what Scafetta does, applauded by Andy (and Pat Frank?). See Andy’s Fig 4,
“If you claim the stations are the population, there is no reason to even deal with sampling.”
But no-one does. The objective is an average global temperature anomaly. That can only be got by samping.
That sampling only tells you how close you are to the population mean. What if that population mean is inaccurate. How does statistical analysis handle that?
You and bdgwx and the rest are trying to tell us that there is no systematic uncertainty in any of the temperature measurements across the globe.
The issue is that you can’t average away systematic bias in a measurement. Taylor says you can’t. Bevington says you can’t. Possolo says you can’t.
Who are you to tell us that you can?
So we are discussing sampling.
Let’s look at the standard equation that shows the relationship between the population mean and the means of all the sample means.
SD of all possible sample means = Population SD / √(sample size)
From this web site (and many more)
14.5. The Variability of the Sample Mean — Computational and Inferential Thinking
Please note that the sample size does not mean the number of samples, but instead is the number of entries in each sample.
Lastly, the SD of all possible samples means is defined as the SEM.
If you are dealing with samples, you do not divide the SEM by √n. This just artificially lowers the SEM and is not correct mathematically.
The mathematically correct calculation is to MULTIPLY the SEM by the √n to determine the population SD.
I would like to point out that the SEM is not the correct statistic to use for experimental uncertainty. The SEM defines an interval that tells one how closely the sample means estimates the population mean µ. It is a statistic that can be used to judge how well the sampling estimates the population mean. It is greatly affected by the size of the samples. You can see that from the equation. The larger you make the each sample, it will reduce the SEM accordingly.
The SEM is not measurement error as defined by the GUM. The GUM defines the experimental uncertainty (used when you can’t make multiple measurements of the same thing) as the dispersion of measurements surrounding the mean, e.g., the population SD. As you can see, the SEM does not define the dispersion of measurements around the mean, it is defined by the sample size.
I see you never got a reply of any kind. from anyone!
This is a very good explanation of how the SEM should be used and construed. Good job.
On the 1001st time Jim repeated the same nonsense he always does, nobody could be bothered to correct him. Thus proving he was correct all along.
When one doesn’t know what one is talking about, an analytically correct answer can seem like nonsense.
Are you saying you think everything Jim says is correct?
The main reason I find his analysis nonsense is he never understands that the standard error of the mean does not require you to take a large number of different samples. He keeps assuming that the process of determining the SEM for say a sample size of 30, it to take thousands of different samples of size 30, and work out the standard deviation in the mean of these samples.
The big mistake he then makes is to conflate this with taking a global average. He sees each individual station as being a sample of say 30 daily values, and then assumes that if you take the standard deviation of the mean of each station will give you the standard error of the global mean.
I’ve tried to explain several times why this doesn’t work, but the most obvious reason is that it would mean that the global average temperature has the same distribution as the temperature dispersion around the globe. If 1% of your stations have a mean temperature greater than 30°C, then there’s a 1% chance that your average could be greater than 30.
In fairness it took me some time to realize this is what he’s doing as he talks in odd phrases, such as
“Please note that the sample size does not mean the number of samples, but instead is the number of entries in each sample.”
“If you are dealing with samples, you do not divide the SEM by √n.”
“The mathematically correct calculation is to MULTIPLY the SEM by the √n to determine the population SD. ”
All analytically correct, but nonsense as no-one does any of the things he’s complaining about.
“”””I’ve tried to explain several times why this doesn’t work, but the most obvious reason is that it would mean that the global average temperature has the same distribution as the temperature dispersion around the globe. “””””
And exactly what does IID mean to you?
If the distributions aren’t the same, what conclusion do you reach?
The use of the CLT as justification for computing any statistic concerning the averages of temperature automatically implies sampling, a sampling means distribution that is normal, and statistics that are estimates of the population parameters.
Does the GAT comply with the GAT?
“And exactly what does IID mean to you?”
Independent and identically distributed. I’m sure this has been explained to you before. And it’s a good reason why what you keep trying to do is wrong.
“If the distributions aren’t the same, what conclusion do you reach?”
No two stations are going to be identically distributed. You cannot use them as samples of the global average.
“The use of the CLT as justification for computing any statistic concerning the averages of temperature automatically implies sampling”
I don’t think I’ve mentioned the CLT at this point. I’m simply trying to say that a sample of stations across the globe could be considered a sample, and will have a sampling distribution. That is quite different to looking at each station as being a sample of the global average.
“a sampling means distribution that is normal”
It does not necessarily mean normal.
“will have a sampling distribution”
And is that sampling distribution normal? If it isn’t then how can the average be a good statistical descriptor?
I doubt it’s normal. That has nothing to do with how useful the mean is.
Malarky! We’ve been down this path many times. And you never get it. If the distribution is skewed or multi-modal then other statistical descriptors *must* be used to adequately describe the distribution. One – the average by itself is meaningless without a variance. Two – a skewed or multi-modal distribution does not guarantee cancellation of values on each side of the average.
If your “sample” of the global temps is not Gaussian then the global average temp is useless.
Which would be useful if I was trying to describe the distribution. We are trying to describe the uncertainty in the mean. Get it? the mean, not the distribution. If you want to look at other aspects of the distribution then there’s nothing stopping you, but it isn’t the purpose of calculating the standard error of the mean.
“If your “sample” of the global temps is not Gaussian then the global average temp is useless.”
Everything to you is “useless”. But don’t project your own limits onto others.
“””””It does not necessarily mean normal.”””””
That is a fundamental assumption with the CLT. If your ~9500 sample is not normal, then it does not meet the conclusions used with the CLT. Fundamentally, the mean and standard deviation will not be appropriate descriptors to use in making any inferences of the population distribution.
It is one reason for using the CLT. You will end up with a normal distribution where the population mean (μ±SEM) interval can be used as an unbiased estimate. In addition the equation (σ=SEM•√n)can tell you the population standard deviation. You simply can’t do that with a skewed distribution!
Have you done a frequency plot of the ~9500 values to see if it normal? If not you haven’t done your calculations with scientific rigor.
From:
http://www.smartcapitalmind.com/what-is-skewed-distribution.htm
A skewed distribution is inherently uneven in nature, so it will not follow standard normal patterns such as standard deviation. Normal distributions involve one standard deviation that applies to both sides of the curve, but skewed distributions will have different standard deviation values for each side of the curve.
From:
https://web.ma.utexas.edu/users/mks/statmistakes/skeweddistributions.html
Measures of Spread
For a normal distribution, the standard deviation is a very appropriate measure of variability (or spread) of the distribution. (Indeed, if you know a distribution is normal, then knowing its mean and standard deviation tells you exactly which normal distribution you have.) But for skewed distributions, the standard deviation gives no information on the asymmetry. It is better to use the first and third quartiles4, since these will give some sense of the asymmetry of the distribution.
“That is a fundamental assumption with the CLT. If your ~9500 sample is not normal, then it does not meet the conclusions used with the CLT.”
How many times are you going to repeat this nonsense rather than read any of your own sources. The population does not have to be normal for the CLT to work, the sample does not have to be normal for the CLT to work.
“Fundamentally, the mean and standard deviation will not be appropriate descriptors to use in making any inferences of the population distribution.”
We are not trying to make any inferences of the population distribution, apart from the mean. That’s why it’s called the Standard Deviation of the Mean. If you want you can also look at other standard errors, such as of the standard deviation, or of any parameter, but that’s not the SEM.
“In addition the equation (σ=SEM•√n)can tell you the population standard deviation.”
It’s like talking to a particularly thick brick wall. Why do you keep ignoring everything I said. By all means disagree with what I’m saying, but explain why you think I’m wrong. Don’t just endlessly repeat the nonsense.
“Have you done a frequency plot of the ~9500 values to see if it normal?”
Which 9500 values? Are you talking about all values, the gridded values, values for one day, or for a year? And are you talking about temperatures or anomalies.
Oh and it doesn’t fracking matter. The distribution does not and probably isn’t normal.
“If not you haven’t done your calculations with scientific rigor.”
I haven’t done any calculations. I have no intention of doing my own global anomaly estimates, nor an uncertainty estimate. I’m simply trying to explain where you are misunderstanding what a sampling distribution is.
“A skewed distribution is inherently uneven in nature, so it will not follow standard normal patterns such as standard deviation.”
Gosh, what a novel concept. Who would have thought that a skewed distribution was skewed.
“We are not trying to make any inferences of the population distribution, apart from the mean.”
If the mean is inaccurate then trying to get close to it is meaningless. And the SEM won’t help you find the accuracy of the mean.
In climate science everything is random, Gaussian, and cancels. It just wouldn’t work otherwise and gosh, you just can’t have that!
If sampling criteria is not followed properly you can not rely on the data.
As far as sample size, each month, from which the monthly average is calculated, has ~30 day/month.
The CLT REQUIRES samples. If you don’t do sampling correctly, you won’t end up with a normal distribution from which you can calculate statistics about the statistical parameters of the population.
“””””He keeps assuming that the process of determining the SEM for say a sample size of 30, it to take thousands of different samples of size 30, and work out the standard deviation in the mean of these samples”””””
Good grief, this is already being done. Do I need to show the several mentions of ~9500 station, each of which have monthly averages?
The variance of each station should be calculated to determine if IID is being met. The CLT has assumptions that must be met.
The distribution of the monthly sample means of all those samples had better provide a normal distribution or the CLT has no meaning!
“””””All analytically correct, but nonsense as no-one does any of the things he’s complaining about.”””””
ROFLMAO! You realize that is really a condemnation of the lack statistical rigor being applied in climate research.
Your acceptance of “no-one does” is not exactly a good excuse!
“If sampling criteria is not followed properly you can not rely on the data.”
Not true. It’s very rare you can exactly follow all sampling criteria. It’s very difficult in the real world to get a truly random sample – and weather stations are very far from random. That doesn’t mean you throw the data out with the bathwater. You do the best you can with the data. In some cases not being a random distribution will improve the data.
“As far as sample size, each month, from which the monthly average is calculated, has ~30 day/month.”
The monthly average of one station.
“The CLT REQUIRES samples.”
Not in the sense that you take it.
The CLT describes what happens to the random variable describing a single sample as the sample size increases. You can think of the random variable as indicating what the distribution of multiple samples would be, but you do not need multiple samples to use the CLT.
(I accept that the language can get confusing at this point as sample can have different uses. You can talk of a single sample made up of N values, but you can also describe each of those N values as a sample.)
“Do I need to show the several mentions of ~9500 station, each of which have monthly averages?”
Still missing the point. Each monthly average is a single value in the sampling of global temperatures for that month. It is a single sample of size ~9500. It has a standard deviation, and if you could consider it a random distribution, you could divide the Standard Deviation by √9500, to get the SEM.
What I think you believe is that each of the 30 days of data for each station is a sample, and that therefore the standard deviation of all the stations is the SEM for the global average. This cannot work, because the individual station samples are not identically distributed. They are not random samples from across the globe, they tell you about the individual stations and nothing else.
“ROFLMAO! You realize that is really a condemnation of the lack statistical rigor being applied in climate research.”
It’s nothing to do with climate research, I would hope that any half competent statistician would tell that. Assuming they could find the time to plough through your confusing word salad.
“Your acceptance of “no-one does” is not exactly a good excuse!”
Then show me someone who agrees with you, present company excepted. Someone who thinks it necessary to remind themselves that you divide the SD by the root of the sample size, and not the number of samples. Or who thinks there’s any point in specifying that you do not divide the SEM by root N. Or who sees any point in multiplying the SEM by root N to determine the SD.
Words to live by! Issued by THE data expert.
Sorry dude, I am only quoting standard sampling theory to you. There are proofs in every statistics book and all over the internet. None of them ever stated this adage.
From Wikipedia:
You are claiming one sample whose size is ~9500. Ok, what is the distribution of that sample? Is it normal? Remember, you need a lot of samples combined with a large sample size to create a normal distribution of the sample means with the CLT. Since you won’t have a sample means distribution with just one sample, then your sample must be normal to obtain statistics that you can use to estimate the population parameters.
Also, the standard deviation of your sample WILL BE the SEM! Remember, the equation is;
SEM * √n = σ
This is a different solution that what the GUM TN 1900, and Taylor recommend. Both of those claim the measured data IS THE POPULATION of information available. You use it to find the population mean, standard deviation, and from the standard deviation, you find the SEM.
Here is a statement from TN 1900 about this. The mention of “parameter” is a reference to population parameters. The distributions obtained from sampling provide statistics used to infer population parameters.
It is too bad you have declared that you have only one sample. That makes the standard deviation of that distribution the SEM.
Also, give us a hint by showing the distribution of the ~9500 data points. If will be fun to see if NH and SH land/ocean temperatures combine into a normal distribution.
Dude, I show references for everything I say. You almost never show any references from books, university classes, and documents describing the requirements of using certain statistical tools like the CLT. Rigorous use of statistics should require the requirements for each operation.
YES! The square root of one equals … one.
I learned this in junior high school.
“Sorry dude, I am only quoting standard sampling theory to you.”
Ever tried understanding what that theory is telling you rather than just quoting it.
Do you think satellite data is more or less accurate because it doesn’t randomly sample temperatures?
Sampling theory works on the assumption that you are getting a random sample. The randomness adds uncertainty to the mean. If you know your non random sample is a better representation of the population than a random sample you have better than a random sample.
“You are claiming one sample whose size is ~9500.”
You’re the one who introduced that figure, I’m just assuming it’s correct.
“Ok, what is the distribution of that sample? Is it normal?”
You’ve just produced a quote that points out to you it doesn’t have to be normal. “even if the original variables themselves are not normally distributed.”
“Remember, you need a lot of samples combined with a large sample size to create a normal distribution of the sample means with the CLT.”
As I keep telling you, and you just ignore, you do not need a lot of samples. You need one sample and the CLT to tell you the sampling distribution will tend to a normal distribution as sample size increases.
“Since you won’t have a sample means distribution with just one sample, then your sample must be normal to obtain statistics that you can use to estimate the population parameters.”
You just don;t understand this at all. Maybe I’m not explaining it well enough or you are just to invested in not understanding it.
“Also, the standard deviation of your sample WILL BE the SEM!”
Poppycock.
“Remember, the equation is;SEM * √n = σ”
Which is useless when you know σ but not the SEM. Then it’s more convenient to use the form found in every single statistics book you can find
SEM = σ / √n
“This is a different solution that what the GUM TN 1900, and Taylor recommend.”
I wonder why?
“Both of those claim the measured data IS THE POPULATION of information available.”
I’m pretty sure they don’t. If they do they are using POPULATION in an odd way.
“You use it to find the population mean, standard deviation, and from the standard deviation, you find the SEM.”
You missed the word “estimate” a few times.
“Here is a statement from TN 1900 about this.”
Do you at least understand what that says?
“It is too bad you have declared that you have only one sample.”
Why? How many samples do you think they have in that example?
“That makes the standard deviation of that distribution the SEM.”
No. It makes it the standard deviation. The SEM is obtained by dividing it by √ sample size (22 in this case).
“Also, give us a hint by showing the distribution of the ~9500 data points.”
It’s your figure. I assume this is the latest GHCN data set, but I haven’t checked if there are that many. I keep telling you they are not a random distribution. You do not get the global average by simply averaging them and you wouldn’t use just the SEM to work out the uncertainty. The uncertainty is going to depend on how you process the data.
“Dude, I show references for everything I say.”
Which you never give an impression of understanding and usually say the opposite of what you claim.
“You almost never show any references from books, university classes, and documents describing the requirements of using certain statistical tools like the CLT.”
I keep showing your own references don’t agree with you. You’ve just posted a quote from Wikipedia saying that the distribution doesn’t need to be normal, and then spend the rest of the comment saying the distribution needs to be normal.
“Do you think satellite data is more or less accurate because it doesn’t randomly sample temperatures?”
The satellite record is a METRIC. It is not a measurand. It’s uncertainty is as understated as everything else in climate science!
None of it is fit for purpose in identifying differences in the hundredths digit!
“As I keep telling you, and you just ignore, you do not need a lot of samples. You need one sample and the CLT to tell you the sampling distribution will tend to a normal distribution as sample size increases.”
You don’t get a “sampling distribution” from ONE SAMPLE!
“SEM = σ / √n”
If you don’t know σ then how do you calculate the SEM? If you know σ then what does the SEM tell you that the population standard deviation doesn’t?
Completely ignoring the question I was asking, as usual.
Yes the satellite data is a measure not the measurand.
I don’t care what uncertainty you want to attach to it. The question was if it is more or less certain because it covers the globe systematically rather than randomly?
You can estimate a sampling distribution from a single sample. That’s pretty much the point of all this statistics. If you still don’t understand this I’m sure there are plenty of references online or at your local library. But don’t let me stop you displaying your ignorance.
How do you know σ without a SEM? Why, you estimate it from the standard deviation of the sample. Virtually everything you and Jim keep quoting at me explains this.
And if you don’t know what the SEM tells you compared with σ, then you haven’t been paying attention, and I doubt you will ever understand it.
“I don’t care what uncertainty you want to attach to it. The question was if it is more or less certain because it covers the globe systematically rather than randomly?”
If it only covers PART of the globe then how can it certain that part represents the entire globe?
“You can estimate a sampling distribution from a single sample.”
But you can’t depend on the CLT to give a good estimate of the population average. Yes, you can find the sampling distribution but you can’t find the POPULATION distribution from one sample – unless your sample *is* the population.
You want to have your cake and eat it to – by trying to conflate finding the distribution of the single sample with finding the distribution of the population!
“Why, you estimate it from the standard deviation of the sample. Virtually everything you and Jim keep quoting at me explains this.”
Good luck on that! The SEM is the standard distribution of the sample means. One single sample can’t have a standard distribuiton of the sample means!
What TN1900 does is take the monthly Tmax values as a set of SAMPLES. That means MORE THAN ONE SAMPLE!
“If it only covers PART of the globe then how can it certain that part represents the entire globe? ”
Just keep dodging the question. I don’t know why I expect better. No matter how simple I try to make the point, you will always just keep dragging the example deeper into the forest.
“But you can’t depend on the CLT to give a good estimate of the population average.”
Please try to understand what the CLT actually is.
“You want to have your cake and eat it to – by trying to conflate finding the distribution of the single sample with finding the distribution of the population!”
Please stop using idioms you don’t understand.
If you don’t know the distribution of the population, the distribution of the sample is the best estimate of the population distribution.
“One single sample can’t have a standard distribuiton of the sample means!”
Gibberish. Please try to understand these things before writing the first thing that comes into your head. You clearly don;lt understand these things you claim to be an expert in. You won’t accept anything I say, but I’m sure there are plenty of simple introductions to statistics that will explain how to calculate the SEM from a sample.
Irony alert!
Iron your own lerts.
“””””You can estimate a sampling distribution from a single sample.”””””
Tell us HOW you ESTIMATE a sample means distribution from a single sample!
I want to see a mathematical proof of that. I can show you a mathematical proof of the CLT and how it converges to a normal distribution, but it requires multiple, IID samples in order to do so. ONE SINGLE SAMPLE WONT SATISFY the conditions of the CLT.
“””””How do you know σ without a SEM? Why, you estimate it from the standard deviation of the sample.”””””
If the sample isn’t normal show how you estimate it.
You are using the word ESTIMATE a whole bunch here. You need mathematical proofs backing up your statistical estimations. Without the proofs you have no way to show that your estimates can be accepted as having scientific rigor.
In climate science everything is random, Gaussian, and cancels by definition.
If that wasn’t true then combining summer NH temps with SH winter temps wouldn’t work because of them having different variances. So we just define the problem away!
“Tell us HOW you ESTIMATE a sample means distribution from a single sample!”
Take the standard deviation of the sample, divide by the square root of the sample size.
Really. I don’t know how many times I’ve told you this, and how many references you give that explain it. Yet you just keep asking the same question.
“I want to see a mathematical proof of that.”
https://en.wikipedia.org/wiki/Central_limit_theorem
Thank you for the reference but it doesn’t say what you think it says!
You want to try again? Read what this reference actually says instead of just googling and cherry-picking!.
Your reference says:. (I have capitalized the pertinent word!)
• “”””a sequence of i.i.d. random VARIABLES drawn from a distribution of””””
• “””””the sample AVERAGES”””””
Both of these are plural. You do not have plural sampleS, you have ONE sample.
Therefore, you CAN’T USE the CLT.
Warning. Don’t always trust Wikipedia. The “n” discussed here on your reference is difficult to understand. Look at the image from your reference.
See that note on the drawing, “samples of size n”. What does that tell you? There are multiple samples, each of which is made up of “n” values.
See all those X_bar’s under the Gaussian distribution? Those are the means of each of the multiple samples.
Here is a good document on sampling.
https://www.scribbr.com/methodology/sampling-methods/
“Both of these are plural. You do not have plural sampleS, you have ONE sample.”
You accuse me of cherry picking? Rather than worrying about when the plural is used, you need to understand what is being described.
“a sequence of i.i.d. random VARIABLES drawn from a distribution of”
Random variables is how you understand it mathematically. Each individual value you take from the population is a random variable. A sample of size N is N random variables taken from the same distribution (that of the population). Each value can be considered a random variable because it could have been any value from the population.
“the sample AVERAGES””
I’m not sure what you are quoting there. The only time that’s used in the Wiki is
This is the context of talking about the law of large numbers. You are looking at a single sample that is increasing in size by adding successive values. Each new value increases n by one, and the produces a new average. The sequence of all these averages converges to μ.
This is described in the previous paragraph describing a sequence of random samples, X1, X2 … Xn ,…. But as I keep trying to say “samples” in this case just means individual values taken from the population.
“See that note on the drawing, “samples of size n”. What does that tell you?”
Again, it’s illustrating what the CLT means. It’s showing what the distribution would look like IF you took an infinite number of random samples of size n.
This is no difference than any other random variable. If I tell you what the distribution of the sum of three dice was, I could illustrate that by rolling three dice hundreds of time and plotting the results. But that doesn’t mean that you have to do that for the distribution to be correct. A single roll of three dice is still an instance of a random variable from that distribution.
The advantage of understanding probability theory is that you can deduce what the distribution will be without having to repeat the experiment over and over.
“””””Again, it’s illustrating what the CLT means. It’s showing what the distribution would look like IF you took an infinite number of random samples of size n.
This is no difference than any other random variable. “”””””
You are full of crap. The CLT converges to a normal distribution as more sample of size “n” from a distribution. That is the only way to infer population parameters. Unless your single sample of ~9500 stations is normal which implies a normal population, you can not infer anything to the population.
“You are full of crap.”
Thanks for that explanation. I will now have to reconsider everything I thought I know about the subject.
“The CLT converges to a normal distribution as more sample of size “n” from a distribution.”
You’re the one who keeps pointing out that the SEM is calculated by dividing by the root of sample size not the number of samples. Why do you think that would be if the point of the CLT was the more samples you take the closer the distribution is to normal?
I mean, yes you are correct in as far as if you only base your sampling distribution on the distribution of actual samples, then you will need a lot of samples to approximate the normal distribution. But that is not what the CLT is about. It’s about convergence as sample size increases, not the number of samples. The number of samples is assumed to be infinite.
“Unless your single sample of ~9500 stations is normal which implies a normal population, you can not infer anything to the population.”
Yet somehow people manage to do it all the time.
I should say that the distribution of the population does play a part in this, in that it will determine how quickly the sampling distribution will converge to normal, and how good an estimate of σ the sample SD is.
But none of this means that you cannot use the CLT to estimate the uncertainty of a large sample from a non-normal population.
You keep saying A single large sample. You are just convincing folks familiar with sampling that you don’t have a clue!
“You are just convincing folks familiar with sampling that you don’t have a clue!”
Then provide some evidence that your method is used in practice. Show me how commen it is not to take just one sample from a population but to take multiple samples and derive the SEM from that.
From your own source, here’s how they describe taking A sample
https://www.scribbr.com/methodology/simple-random-sampling/
In particular note
Step 2: Decide on the sample size
And,
Step 4: Collect data from your sample
Nowhere does it say that once you have collected your sample, you should repeat the study multiple times, sin order to work out an estimate of the SEM.
Then there’s the page on analyzing the data
https://www.scribbr.com/category/statistics/
Note – “a sample”, “your own sample”.
“””””You need one sample and the CLT to tell you the sampling distribution will tend to a normal distribution as sample size increases.””””””
How do you get a sample means distribution WITH ONE SAMPLE?
A sample means distribution is made up of the means of numerous samples. The CLT guarantees that the sample means distribution will tend to normal with multiple samples of large enough size.
You are claiming one large sample of size ~9500. You are stuck with that and the distribution it has. The CLT will tell you nothing about the statistics of that single sample. In essence you have the population of temperatures in that one large sample. Good luck with that.
Have you checked to see if it is normal?
“Ok, what is the distribution of that sample? Is it normal?”
Where did you learn to read. You are cherry-picking phrases now in an attempt to justify your conclusions.
That is SAMPLES, plural, not singlular!
From:
6.2 The Sampling Distribution of the Sample Mean (σ Known) – Significant Statistics (vt.edu)
From the above link:
You don’t know the original population. Therefore, you have no way to know how many observations are needed. The best way to tell is do a frequency plot of your single sample. If it is normal you can probably assume the population distribution is normal. If it isn’t normal, then you can be assured your population is not normal, and a large number of samples will be needed. Yet, where are you going to draw a large number of samples from? You just put all your data into one large sample!
The CLT, doesn’t work with one sample. It requires multiple samples of size n if the population distribution is not normal.
As to the ~9500 number, it was quoted here on another thread, I think by bdgwx. That is the number of temperature stations being used.
Ultimately, you are being a troll. You can’t claim rigor in the statistical procedures being done if you don’t even know where the number ~9500 originates from.
This conversation has reached the end unless you can provide some actual illustrations of the data that you are saying is correct!
As this thread is all over the place: the quote from Wikipedia Jim is referring to is this:
Jim says “That is SAMPLES, plural, not singlular!”
As I’ve said before the use of the word sample is ambiguous. It can refer to a collection of N data points – “a sample of size N”. Or it can refer to an individual data point – “take N samples from the population to produce a sample of size N”.
The quote is using the second definition. This is clearer later on in the article.
See. Each value you draw from the population is a random sample, but the mean of N of these is a sample of size N.
Then your other quote
There it’s using sample to mean an individual sample of size N, but you keep ignoring the word “if”. This is not saying you have to draw a large number of random samples in order to determine the SEM. It’s describing what the SEM means. It means that IF you were to draw a large number of samples this is what the distribution would look like. From that same article
“If you are being asked to find the probability of the mean of a sample, then use the CLT for the mean.”
Note that page is just describing the probability that the mean of a given sample lies within a range. You need to go further to understand how to infer a population mean from a sample mean.
“ He keeps assuming that the process of determining the SEM for say a sample size of 30, it to take thousands of different samples of size 30, and work out the standard deviation in the mean of these samples.”
CLT: “In probability theory, the central limit theorem (CLT) establishes that, in many situations, for identically distributed independent samples, the standardized sample mean tends towards the standard normal distribution even if the original variables themselves are not normally distributed.” (bolding mine, tpg)
Rather than cherry picking things you need to go study Taylor, Section 5.7.
from Scribbr: “Standard error vs standard deviationStandard error and standard deviation are both measures of variability:
You keep repeating what the standrd error of the mean describes. All I’m trying to explain to you is you do not normally literally take thousands of different samples to know what it is. That’s why the equation SD / √N is so useful.
From Taylor 5.7.
From your Scribbr article
You can’t see the forest for the trees apparently.
If you don’t have the population then how do you know σ? If you don’t know σ then how do you get the SEM?
If you know σ then of what use is the SEM?
The SEM *is* the standard deviation of the sample means. Pulling multiple samples, finding their means, and that calculating their standard deviation is what tells you how close you are to the population mean.
The term SEM should never have been coined in statistics. It is misleading as all git out and it is misleading you and bdgwx.
One sample does *NOT* make a distribution from which a standard deviation can be calculated. One sample does not define how close you are to the population mean. The standard deviation of one sample is *NOT* the SEM. The CLT states that if you have multiple samples, their mean will cluster around the population average. It does *NOT*say that if you have one sample that its mean is the population mean!
How many times are you going to advertise your complete misunderstanding of statistics? I’ve answered all this above. But you only have to read any of the articles on the SEM you routinely post. What do you think SEM = σ / √N means?
Read this:
“The standard error of the mean, or simply standard error, indicates how different the population mean is likely to be from a sample mean. It tells you how much the sample mean would vary if you were to repeat a study using new samples from within a single population.
The standard error of the mean (SE or SEM) is the most commonly reported type of standard error.” (bolding mine, tpg)
“The standard error (SE)[1] of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution[2] or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM).[1]
The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.” (bolding mine)
You’re now just repeating the thing I quoted to you and ignoring the operative word – “if”. It doles not say you have to repeat the sampling, it says it will tell you want would happen if you repeated it.
Similarly in your second quote
And praytell, exactly how is the SEM calculated? Does it use the sample means distribution?
There is only one way a single sample can give an accurate estimations of the mean and σ for a population. The population must be normal AND your sample must have an identical normal distribution. In other words, a perfect population and perfect sample.
How many more times? The bog standard way to calculate a SEM is to
Every single temperature you include is a SAMPLE OF 1. That’s because each measurement is a single measurement of a different thing.
“ All I’m trying to explain to you is you do not normally literally take thousands of different samples”
Hyperbole that proves nothing. No one is saying you need thousands of different samples. You *do* need enough samples to be sure you are correctly representing the population.
“if you were to repeat a study using new samples from within a single population.”
You don’t even bother to read what you post, do you? The operative word is SAMPLES!
“Every single temperature you include is a SAMPLE OF 1.”
And combine 100 of those single samples and you have a sample of 100.
“That’s because each measurement is a single measurement of a different thing.”
That’s usually the case when sampling.
“You *do* need enough samples to be sure you are correctly representing the population.”
And how many would that be? Do the math.
“You don’t even bother to read what you post, do you? The operative word is SAMPLES!”
You think the word “samples” is more important than the words “if you were” that immediately precede it?
What he is saying is *NOT* nonsense. It only seems so to someone who doesn’t understand the difference between seeing how close you have approached the population mean and understanding the accuracy of that population mean. They are two entirely different things.
The standard deviation of the sample means is *NOT* the accuracy of the mean. The accuracy of the mean is based on the uncertainties involved in measuring the values used to calculate the mean.
The accuracy of the mean can *never* be less than the accuracy of the actual measurements.
Even Possolo in TN1900 had to assume that measurement uncertainty was insignificant in order to use his approach of evaluating the variation in the stated values as the uncertainty.
That is what you are doing and what, it seems anyway, everyone in climate science does as well. How closely you calculate the mean is irrelevant if the mean is inaccurate!
“vehemently rejects” because I am a physical methods experimental chemist, Nick, and am very familiar with systematic measurement error.
Averaging the outputs of disparate models to get an “ensemble mean” is absurd, and global average temperature does not represent “the climate”.
That’s one of Schmidt’s points. Scafetta only uses the ensemble means and ignores the uncertainty in the models.
Bellman, Not true, Scafetta used three representative runs of each model. He did only plot the means however.
But that’s not the same as looking at the uncertainty in the models.
As far as I can see he uses the ensemble mean for each of three different pathways. What he doesn’t do is use individual model runs to determine the uncertainty in the model.
“What he doesn’t do is use individual model runs to determine the uncertainty in the model.”
The common myth uniting all consensus climate modelers and their peanut gallery.
Individual models runs do not determine the uncertainty in the model. They merely determine model precision.
Uncertainty in the model is determined by parameter uncertainty and the large deficits in physical theory.
When I say the uncertainty in the model, I mean how much variation there is within individual model runs, not how true the actual model is.
The issue as I see it is if you can say a particular model is significantly wrong if it’s mean prediction for a given decade is significantly different from observations. Schmidt argues, and I would agree, that if a significant fractions of the individual runs of that model are within observations, then you have not showen sufficient reason to reject that model. If 10% of its runs are the same or colder than observed, we might just be living in that 10% of possible worlds.
Given the deficient physics in the model, they are unable to make predictions in the scientific sense.
Use of “prediction” to describe model output is an abuse of terms.
Jim Hansen admitted as much here (pdf): ““Curiously, the scenario that we described as most realistic is so far turning out to be almost dead on the money. Such close agreement is fortuitous. For example, the model used in 1988 had a sensitivity of 4.2°C for doubled CO₂, but our best estimate for true climate sensitivity is closer to 3°C for doubled CO₂.“ (my underline)
Fortuitous is the appropriate word to use when any climate model run gets close to observations.
“11-year time scale”
If you believe the planet to be more than 10k years old, then 11 years isn’t yet one good sample.
How many angels can dance on the head of a pin?
How many hundredths of a degree signal can be teased out of multiples of degrees of noise?
Thanks, Andy. A couple of further points.
The annual means shown in e.g. Figure 1 start out with their own inherent uncertainties. To start with, there’s the uncertainty of the average of all the gridcells. On top of that, there’re the errors resulting from removing the seasonal uncertainties. Then there are the errors from averaging the individual months into years.
As a result, you can’t just figure the uncertainty under the assumption that those numbers have no errors.
w.
“As a result, you can’t just figure the uncertainty under the assumption that those numbers have no errors.”
Yes. It was Scafetta who did that. Gavin objected, saying that a reasonable error estimate was 0.1°C, which is how the direct surface measures work out.
The weird thing is that we have had all the scoffing here at Gavin’s number, saying that the uncertainty can’t be lower than the uncertainty of individual thermometer measurements, say 0.5°C. But now Scafetta claims the uncertainty is 0.01°C, and everyone cheers, Pat Frank, Andy and all. Scmidt wrong again?
Nick, you aren’t paying attention. Scafetta is just using the measurement errors from HadCRUT5. You are conflating this with the total uncertainty. They are two different things. We are not concerned with natural uncertainty here, in fact we are showing that natural uncertainty exists! How big is it? I don’t think anyone really knows, and that is the whole problem with the ECS calculation.
Andy
“Scafetta is just using the measurement errors from HadCRUT5. You are conflating this with the total uncertainty. “
You keep saying that. It’s meaningless; there is only one uncertainty that matters. But it isn’t even true. Scafetta does not use measurement errors anywhere. He doesn’t quote the error of a thermometer. He uses the observed scatter of readings that make up the monthly and annual averages per location.
And when I look it up, he doesn’t even seem to say that the uncertainty is 0.01C. From his appendix 2:
He gets 0.094, very similar to Gavin’s 0.1 (and to other averages)
And BTW you might like to take up with Pat Frank and others how the measurement uncertainty comes down to 0.01C. They keep telling us it can’t be less than thermometer accuracy.
The Pat Frank method suggests the 11yr uncertainty would be ±0.90 C (95% CI). And that is only the lower limit. It is my understanding that it only includes the component arising from measurement. So Scafetta’s analysis is literally 1/90th of Pat Frank’s.
Note that the PF method uses the formula sqrt(N * u^2 / (N-1)) where N is the number of observations and u the uncertainty of an individual observation. For an 11 yr period N would be very high leading to a result of just u. And since he lists u as 0.46 C that would be expanded as 1.96 * u = 0.9 C. I will say that he says this formula comes from Bevington pg. 58 but I don’t see it on pg. 58 or anywhere in Bevington for that matter. And it is inconsistent with the formulas in Bevington, Taylor, and JCGM 100:2008.
Pat, if you’re there please chime in because the last thing I want to do is misrepresent your position.
Is that just mapping a population sd to sample sd?
Well…I hadn’t really thought about it until you said something, but yeah that is exactly what it is doing. Nice catch!
My brain was a bit fuzzy yesterday (don’t even go there), so I thought I might have imagined that.
It seems simpler just to multiply u by sqrt(N/(N-1)), but that’s just me. Without context, dunno why you’d want to do that conversion.
You’re just misrepresenting Bevington, as usual bdgwx.
Would you mind telling us where you got that formula and why is it inconsistent with the variance of the mean formula given in (4.23), the uncertainty of the mean formula given in (4.29), and the uncertainty of the mean formula given in (4.14) as obtained from the general formula (4.10)?
Noted above: 4.22. The uncertainty is not due to random error. Use of 4.23 is incorrect.
It’s eq. 4.22 in Bevington p. 58 (3rd Ed) with unit weighting. I’ve explained this to you before, bdgwx.
I’ve said this before, but 4.22 is the equation for the average variance of the data. You then have to use 4.23 to get the variance of the mean.
I think you’re misinterpreting that. It’s the ((weighted average) variance) rather than (average variance). If yotu take the weights out, it’s just the sample variance.
It’s the variance of the full data set.
For the average variance (variance of the mean per Bevington), divide by the number of data points.
It’s weird how Bevington italicized average variance. It seems like it would have made more sense to italicize weighted average make that point more clear.
I assumed the weighted was obvious, and irrelevant. The main point is that it isn’t the variance of the mean.
The “weighted” isn’t irrelevant, it’s part of the term “weighted average”. If all the weights are equal, the “weighted average” becomes irrelevant.
It’s a pedant point, but ambiguity can bite hard.
Tangentially related…
I question the use of the weighted variance formula anyway. I don’t see any reason to believe that station observations should not be treated equally. In such a case we should just use the trivial variance formula and divide it by N.
I will say that 4.22 might be useful when computing the variance of a grid mesh where cells have unequal areas. But the Frank 2010 procedure does not even attempt to propagate uncertainty through the grid mesh, which is a problem in and of itself, so it seems like using the weighted formula is overkill.
“But the Frank 2010 procedure does not even attempt to propagate uncertainty through the grid mesh,”
Incredible. Thank-you fpr demonstrating your utter lack of understanding, bdgwx.
You might as well have never read the paper, and saved yourself the time, for all the understanding you exhibit of it.
Then help me understand. Where in this series of calculation did you consider the grid mesh?
(1a) σ = 0.2 “standard error” from Folland 2001
(1b) sqrt(N * 0.2^2 / (N-1)) = 0.200
(1c) sqrt(0.200^2 + 0.200^2) = 0.283
(2a) σ = 0.254 gaussian fit based on Hubbard 2002
(2b) sqrt(N * 0.254^2 / (N-1)) = 0.254
(2c) sqrt(0.254^2 + 0.254^2) = 0.359
(3) sqrt(0 283^2 + 0 359^2) = 0.46
Translation from bgwxyz-ese:
“Please give me more ammunition for another fun round of Stump The Professor (I have no interest in learning about reality)”
The grid mesh is irrelevant.
Of course it is relevant. That’s how a global average temperature is calculated. Specifically it is the area weighted average of the grid cells. If you’re going to do a type B evaluation then you have to propagate the uncertainty temporally from daily observations to monthly and then yearly observations and spatially from multiple stations into a grid cell and then multiple grid cells into a global value.
The weighted average of a grid cell is just as much garbage as the global average.
See the attached picture of the temperatures in northeast KS for the morning of 4/19.
What do you get when you average all those temps? Something that tells you what is going on in the entire region? Those temps vary from 72 to 67 and the 67 is in the UHI region of Kansas City!
“If you’re going to do a type B evaluation then you have to propagate the uncertainty temporally from daily observations to monthly and then yearly observations and spatially from multiple stations into a grid cell and then multiple grid cells into a global value.”
You should propagate the uncertainties of each individual station daily observations clear through to the global value.
I’m going to let you pick that fight with Scafetta and Andy May alone.
Not a problem. I can show them the current temps in NE Kansas any time they want and they can explain how the average of those temps are meaningful.
Exactly. These guys are ever so excited to find a way to find a method that reduces uncertainty to a point where you can get an anomaly like 0.001 ±0.0001. The fact that this equation with unit weights still shows adding uncertainties is a disappointment to them I’m sure.
They still haven’t figured out that a function like:
f(x,y,z) = (x+y+z)/n = μ, and for uncertainty:
u = √(u_x + u_y + u_z)
“These guys are ever so excited to find a way to find a method that reduces uncertainty to a point where you can get an anomaly like 0.001 ±0.0001.”
Who on earth is claiming an uncertainty of ±0.0001? In this article, even looking at a decade, there are only two uncertainties quoted. ±0.1, which is considered far too large, and ±0.01, which is considered correct.
“The fact that this equation with unit weights still shows adding uncertainties is a disappointment to them I’m sure.”
You do realize that you are still dividing by the sum of the weights?
“They still haven’t figured out that a function like:
f(x,y,z) = (x+y+z)/n = μ”
Seriously? You think none of us realize that the average of three values is the sum divided by three?
”
and for uncertainty
u = √(u_x + u_y + u_z)”
And you still haven’t figured out you are wrong. That’s the uncertainty of the sum. For the average you divide by n.
Show me where any function with a constant on the bottom also has the uncertainty divided by that constant!
You are declaring the FUNCTION to be the (Σnᵢ / n). Find an example in the GUM, Taylor, Bevington where the uncertainty calculation includes dividing by that “n” divisor in a function.
You are making the assertion that since the function is called an AVERAGE, that the uncertainty must be an average also! That is an assumption with no support.
How about a triangle? {A = (bh) / 2}? Do you divide the uncertainty by 2 as you would with the average? NO!
To be correct, since uncertainties always add, the uncertainty should be u = √((Σuₙ_ᵢ)² + u(n)²). Now tell me what the uncertainty of “n” is!
The derivation of the uncertainty calculation function is totally different than the derivation of a function. Function definitions do not directly transfer into the uncertainty calculations, only the uncertainty of each component.
What’s the point? I’ve been trying to show you all these things for the past 2 years – it just goes straight out the other ear.
If you really thin that uncertainty increases with sample size you need to explain to Pat Frank why he’s wrong. You need to address why you think the premise of this article is wrong. You need to explain why it will be impossible to ever know if the models were wrong.
The irony of this article is that it really demonstrates what I’ve been thinking for a long time – uncertainty is not your friend. If you would prefer not to worry about the future, not to spend any money reducing a problem, then claiming there are huge uncertainties in what we know is not going to help your cause. The less we know the worse the potential problem.
“uncertainty is not your friend”
hahahahahahahha
bellcurvewhineman is on a roll today…
Average uncertainty is *NOT* uncertainty of the average.
Do I need to draw you a picture?
“Average uncertainty is *NOT* uncertainty of the average.”
Correct. Why do you keep repeating this nonsense as iof it was something I believe?
Because of posts like this from you!
————————————————-
Lets assume that the main uncertainty was due to a systematic error, with the rest coming from rounding to the nearest integer. Say there is a systematic error of +5°C to all readings both for max and min. If the maximum temperature was exactly 20°C and this is read as 25°, and the minimum temperature was 10°C which is read as 15°C, then the true median or mean of the two is (20 + 10) / 2 = 15. And your measured value is (25 + 15) / 2 = 20. The error is 5°C which is the average of the two systematic errors.
————————————————
You missed the qualification about this being under the assumption that all uncertainty is systematic.
If as Pat Frank pretends all errors are sysyematic, then and only then is the uncertainty of the average the same as the average uncertainty.
“If as Pat Frank pretends all errors are sysyematic,…”
A complete misrepresentation.
“You missed the qualification about this being under the assumption that all uncertainty is systematic.”
It doesn’t MATTER! Uncertainty is uncertainty! It all adds.
And no, the uncertainty of the average is *NOT* the same thing as the average uncertainty. If I take two boards whose systematic error is ±1″ and put them together then is my uncertainty of the average ±1″ or ±2″? Assume no random uncertainty.
Ans: One board can be 1″ too short and the other 1″ to short, either is equally possible – a total uncertainty of -2″. They could both be 1″ too long and the uncertainty then becomes +2″.
The average uncertainty is ±1″. So what? It doesn’t describe the physical world in any way! In the real world I need to know that my boards could be too short or too long so I can adjust my design.
“If as Pat Frank pretends all errors are sysyematic, then and only then is the uncertainty of the average the same as the average uncertainty.”
You have no idea what you are talking about. Remember the equation that ẟtotal = ẟrandom + ẟsystematic?
What if you *KNOW* what the systematic bias is?
“It doesn’t MATTER! Uncertainty is uncertainty! It all adds.”
But one adds by simple addition and one adds in quadrature.
And then when you divide by sample size, one gives you the average uncertainty and the other gives you something smaller.
“And no, the uncertainty of the average is *NOT* the same thing as the average uncertainty.”
Good. So we’re agreed on that.
“If I take two boards whose systematic error is ±1″ and put them together then is my uncertainty of the average ±1″ or ±2″? ”
±1, obviously.
“One board can be 1″ too short and the other 1″ to short, either is equally possible”
Then it wasn’t a systematic error.
“They could both be 1″ too long and the uncertainty then becomes +2″.”
You really need to have a word with Pat Frank about this. He wrote a whole paper about how when dealing with “adjudged” or systematic uncertainties the uncertainty of the average will be the same as the average uncertainty.
That’s why for instance he says the uncertainty of a 30 year period will be the same as that for a one year period. Not as you would think 30 times as large.
“±1, obviously.”
beeswax sez you can’t use plus/minus, cuz it ain’t algebra. Y’all need to get on the same wavelength.
Are you just insane. Why on earth do you think ± can’t be used to express an interval all of a sudden?
I mean it’s obvious you are just using these rants to deflect from Tim’s inability to add up. But pretending you don’t understand what ± means in this context is a extreme even for you.
It’s because TG told me I need to start using ± in place of the + operator in algebra equations because + assumes the rhs operand is always positive. When I said that ± isn’t an operator and that + does not assume anything about the sign of the rhs operand karlomonte created the absurd argument that you cannot use ± to express uncertainty intervals and then wanted me to defend his absurd argument.
Yes, I released that trawling through all of yesterday’s posts this morning.
It’s the problem of these systematic errors that can be different each year, but still can’t be considered random. The odd thing is that Pat Frank seems to consider the uncertainty of the anomaly as being made up of two random uncertainties, given he adds them in quadrature.
Even more fun was Tim claiming that the average of two systematic uncertainties could be twice as big as the individual uncertainties on the grounds that one could be positive and one negative. Not quite understanding what happens when you add a positive and negative number together.
“The odd thing is that Pat Frank seems to consider the uncertainty of the anomaly as being made up of two random uncertainties, given he adds them in quadrature.”
V.R. Vasquez & W.B. Whiting Accounting for Both Random Errors and Systematic Errors in Uncertainty Propagation Analysis of Computer Models Involving Experimental Measurements with Monte Carlo Methods. Risk Analysis. 2006; 25(6): 1669-1681.
Page 1672ff: “When several sources of systematic errors are
identified, β is suggested to be calculated as a mean of bias limits or additive correction factors as follows:
[β ≈ √∑(φ)²,] where φ is the bias range within the error source.”
The same rules of addition in quadrature apply to the uncertainties arising from systematic error.
The odd thing about you and bdgwx, Bellman, is that neither of you ever know what you’re talking about.
The even odder thing is that even after repeated demonstrations of that fact, neither of you ever think the better of declaiming in ignorance.
“When several sources of systematic errors are
identified, β is suggested to be calculated as a mean of bias limits or additive correction factors as follows:”
So when you combine the uncertainties over the base period with the uncertainties for a single year to make an anomaly, are they two different sources of systematic errors?
When you combine 30 years to make the base period are they all different sources of systematic errors or are they all the same source?
“The same rules of addition in quadrature apply to the uncertainties arising from systematic error.”
I agree, if they can be considered different systematic errors, but that means they are effectively random errors.
“The odd thing about you and bdgwx, Bellman, is that neither of you ever know what you’re talking about.”
The odd thing is no-one here seems to be capable of explaining why I’m wrong without throwing out these petty insults.
“but that means they are effectively random errors.”
No. It does not.
Uncertainties are associated with measured magnitudes. When two such magnitudes are added or subtracted, the uncertainties combine in quadrature.
Given two temperatures measured using the same sensor, the uncertainty in their sum is their individual uncertainties combined in quadrature.
Why you’re wrong has been explained ad nauseam, Bellman. You continue as though nothing had ever been offered you.
It’s not an insult to observe that you don’t know what you’re talking about. It’s a fact. Relating a fact is not an insult. Or are you ignorant of that fact, as well.
A one hit, full nail drive!!!!
“Given two temperatures measured using the same sensor, the uncertainty in their sum is their individual uncertainties combined in quadrature.”
Then you’re are correct, I really don’t understand your paper. If you are adding uncertainties from the same sensor in quadrature, how do you end up saying the average does not decrease with sample size?
Try finding the expanded experimental uncertainty of Tmax and Tmin. Let Tmax = 74 and Tmin = 62.
Use TN 1900 as the procedure. Remember, an average is a statistical parameter and every average has an associated variance/standard deviation.
Variance = {Σ₁²(xᵢ – x_bar)²} / (2 -1)
Standard Deviation = √{Σ₁²(xᵢ – x_bar)²} / (2 -1)
The expanded experimental uncertainty will make you smile. In case you don’t have it, here it is.
You went through all this a couple of months ago, and I said at the time why I don’t think this works. max and min are not two random values from a distribution they are the extreme values from the daily distribution of temperatures. Any uncertainty estimate treating them as a sample of 2 will be meaningless.
But let’s just pretend and see if we get a value that makes sense.
TMean = 68
Sample SD, will be (74 – 62) / √2 = 8.49.
The SEM is then SD / √2 = (74 – 62) / 2 = 6.
Now we apply the coverage factor, and I guess you want to use a student-t distribution with DF = 1, which for a 95% confidence interval is 12.71. (This does make the assumption that the distribution of daily values is normal, which it isn’t.)
So our 95% interval is 68 ± 76 = (-8, 144).
For extra fun the 99% interval is 68 ± 382.
This might make sense if you assume that you just have two random values from an unknown random distribution. But all it’s really telling you is you can’t deduce much with a sample size of two.
But as I say max and min are not two random values, they are the high and low points of the daily temperature cycle. How you determine the actual uncertainty is going to depend on what you are trying to measure.
If you want to know the traditional TMean value (max + min) / 2, then the only uncertainty is the measurement uncertainty.
If you want to know how close it’s likely to be to the true daily average (based on the integral of all temperatures throughout the day) that’s a lot trickier. Maybe comparing high resolution data with max and min values, as I was doing using USCRN data. This will vary from location to location and from season to season, and in many cases there will be a systematic as well as a random error.
If you want to know what the average daily mean values are for a month, you do the TN1900 method just using the daily average values.
You make an unwarranted assumption. There is no restriction in the GUM or TN 1900 on using two values to find an experimental uncertainty.
The fact that the T table is has such a high value simply tells you that the uncertainty is very high.
You didn’t even take into account that these are highly correlated values and need a covariance added into the uncertainty. This is an important part of finding uncertainty which is never done.
That is one reason I choose to keep Tmax and Tmin separate and climate science should do the same. Using covariance with a full month of Tavg will only increase the experimental uncertainty. Another factor is that months are correlated also, at least in a hemisphere. In addition, one can see what is happening to each curve.
Funny that you complain about using extreme values to calculate an experimental uncertainty, but it is ok to use them for calculating uncertainty when you average. Just a little hypocrisy! If they aren’t fit for purpose, they aren’t fit for purpose. To remove all doubt, treat them separately. Climate scientists at some point are going to be required to do this.
Another comment based on your failure to read or understand what I wrote.
I am not assuming you cannot use the SEM on a sample of two values, it’s just that the result will usually be a very high level of uncertaint. This follow from the fact that it’s a very small sample, but more importantly if you have to estimate the population deviation from two random values. Trying to do that is very problematic as the values could come from anywhere on the spectrum. You might get two values that are identical and conclude there was zero uncertainty in your average.
But your problem here is that you are not doing anything that resembles the conditions for the calculations. You should be able to see that. You bang on enough about requiring IID and having Gaussian distributions. None of this holds when you take the maximum and minimum values from a bounded non-Gaussian distribution as your sample
of two.
Your uncertainty estimates will be impossibly large because they don’t know that the two values are the largest and smallest values in the distribution. They don’t know that there are not temperatures during the day that are much bigger than the maximum or much smaller than the minimum.
You on the other hand do know that. An if you thought about what the values mean, rather than blindly plugging them into any old equation, you should realise that the average cannot be bigger than maximum or smaller than minimum, baring measurement errors.
“””””I am not assuming you cannot use the SEM on a sample of two values, it’s just that the result will usually be a very high level of uncertaint. This follow from the fact that it’s a very small sample, but more importantly if you have to estimate the population deviation from two random values. “””””
Everything you are saying applies to the mean also. Daytime and nighttime distributions ARE DIFFERENT. You advocate that finding the mean of two different distributions tells you something of great value only because the arithmetic is easy. That the other statistic parameters are large should say something to you about the value of the mean.
In which universe does this happen?
Why any universe where averaging can increase resolution beyond a measuring device’s inherent limitations and uncertainty is infinitely reduced if you just measure enough different things!
Hallelujah!!
The Promised Land, trail to which was blazed by Elvis Presley.
“In which universe does this happen?”
None. That’s why I mentioned it. You really have to keep your trolling under control. Baring measurement errors means that measurement errors are the things you have to worry about.
Then WTH did you write it?
Uncertainty is not error, you still can’t get past Lesson One.
I see somebody’s desperate for attention this evening.
“Then WTH did you write it?”
Because I meant it.
The actual average of a day cannot be greater than the maximum temperature of that day or smaller than the minimum temperature of the day. I probably should have left it as that, but knowing how prone you are to little tantrums if I don’t mention every conceivable eventuality, no matter how improbable I thought I better add “barring measurement errors.”.
But of course that’s still not good enough, because in karlotroll land, “barring” means I’m saying measurement errors are impossible, rather than saying you would need a measurement error of some description to get an average greater than the maximum.
Let me help (I know, its hopeless):
uncertain | ˌənˈsərtn |
adjective
not able to be relied on; not known or definite: an uncertain future.
Is this the part where I say “don’t whine”. I’m still trying to perfect the karlomonte debating technique.
I’m not sure if your definition of uncertainty is very helpful. We are talking about measurement uncertainty here, not an everyday usage of the word.
Try the GUM’s defintion
“how do you end up saying the average does not decrease with sample size?”
The average of what?
The sample.
“The average of what?”
Whatever you want. You use thesame argument when talking about averaging samples over the globe, or averaging different time periods. In all cases you say the uncertainty of the average is effectively the same as the average uncertainty, and emphasise that it doesn’t reduce with √N.
Yeah, so? The statistical assumptions of random error are violated. No 1/√N.
The “so what” is you saying above
If you sum in quadrature the uncertainty in their average will reduce by √N.
Two temps summed:
T1 + T2 = T
The uncertainty of a measurement sum is:
δT = √(δT1 + δT2)
Why would you divide the uncertainty by (1 / n)
You never answered this. It is very important. Do you average the 2 weights AND divide their RSS uncertainties by 2 when you give the customer a quote? How often will the average weight exceed your quoted range? Half the time maybe?
Let’s do a standard quality assurance problem. My assembly line creates 1000 widgets a day. Each widget is designed to weigh 5 units and is made up of two materials of equal weight with an uncertainty in each of 0.5 units. What should I quote the uncertainty of the average weight to be for any two widgets?
“Why would you divide the uncertainty by (1 / n)
You never answered this.”
I’ve explained it to you dozens of times over the past 2 years. Why do you think you’ll understand it any better at this late stage of the conversation.
If you think you shouldn’t divide by 2, complain to Pat Frank who does just that throughout his paper. If he didn’t the uncertainty over 1000s of stations and tens of years would be much much bigger than ±0.46°C.
“What should I quote the uncertainty of the average weight to be for any two widgets?”
The usual problem – I’ll have to make assumptions and then you’ll claim I know nothing because my assumptions are wrong.
Nice dance around a simple problem. Two objects, simple numbers. Too bad you can’t make your interpretations make it work.
What is the mean weight?
What uncertainty in the average weight would you quote to a customer?
Fine, on all the usual assumptions, especially that the uncertainties are independent, that there as no drift in the instruments, that you calibrate the machinery constantly, that nobody sabotages the equipment, that there is no shrinkage in the manufacturing process, and all the other things you will claim are impossible to assume.
And also assuming you mean the standard uncertainty.
Then the uncertainty of the widget is √2 * 0.5 ~= 0.71 units.
You might report this using one of the GUM formats as
5.00(71) units.
So, did I pass?
“Two temps summed:
T1 + T2 = T
The uncertainty of a measurement sum is:
δT = √(δT1 + δT2) ** this should be √(δT1^2 + δT2^2)
Why would you divide the uncertainty by (1 / n)
You never answered this.”
In case anyone hasn’t spotted the trick here – I asked “If you sum in quadrature the uncertainty in their average will reduce by √N.”
Jim is just talking about the sum – adding two temperatures and saying the uncertainties are added in quadrature (δT = √(δT1 + δT2)). But I was asking about the uncertainty of the average. (The sum of two temperatures is meaningless in any case).
(T1 + T2) / 2 = T_avg
As we’ve discussed numerous times, but the Gormans still don’t accept, when you divide a sum to by a value to get an average, you have to divide the uncertainty of the sum by that value.
δT_avg = δT / 2 = √(δT1^2 + δT2^2 ) / 2
and if T1 = T2, this reduces to
δT_avg = √(2 * δT1^2) / 2 = δT1 / √2
If I build two units for a client I don’t give them an “average” unit. I give them two separate units whose variance in weight in based on the uncertainties surrounding the two units. If I build them a third unit then that 3rd unit should fit within the specifications I give the client for the possible weight the 3rd unit might have.
That possible weight is based on the experimental uncertainty I derive from the first two units, the TOTAL uncertainty, not the uncertainty of the average.
Think of it in terms of volume associated with that weight. He is using the contents of the units to fill molds for a product he is making. He doesn’t care about how closely you calculate the average value of the contents of the units. He wants to know the total uncertainty in the expected volume so he can plan his pours without having too little to finish or too much he has to dispose of.
You continually show a disregard for the real world. You somehow think that how closely you can calculate the average actually means something in the real world.
The average value *is* important. But it’s the uncertainty surrounding that average that is meaningful, not how closely you can calculate the average!
“If I build two units for a client I don’t give them an “average” unit.”
Then it’s irrelevant to a discussion about averages.
You missed the whole point, as usual. No one cares about the median or mid-point. They care about the individual units associated with those. Including temperature. It’s Tmax and Tmin that are the important values, not the mid-point. The mid-point can’t distinguish between different climates.
“No one cares about the median or mid-point.”
We’re talking about the mean, not the median, and people care about it. This whole article is about the uncertainty of a global mean. Pat Frank’s paper is about
UNCERTAINTY IN THE GLOBAL AVERAGE SURFACE AIR
TEMPERATURE INDEX
That is not how you did the problem I posed!
You said the uncertainty is 0.71.
So let’s do it your way.
δW_avg = δW/ 2 = √(δW1^2 + δW2^2 ) / 2
δW_avg = √ (0.5² + 0.5²) / 2 = √0.5 /2 = 0.707 / 2 = 0.4
Please explain the discrepancy! IOW, why divide by 2 for temp?
One’s a sum the other’s an average.
That isn’t an explanation!
You quoted “5” as the mean. That is, (5 + 5) / 2 = 5. It is a sum and should be divided by two to get the mean. Just like the sum of temperatures divided by 2.
In order to explain the difference you need something more than an obtuse reason.
I think you stumbled on why uncertainties aren’t divided by √2 or even 2.
Sorry I misremembered your convoluted question. Yolu asked
I assumed, giving your emphasis on each widget as being made up of two different materials, each with it’s own uncertainty, you wanted to know the total uncertainty of each widget. I’d missed the point that for some reason you wanted to know the uncertainty of the average of two widgets.
Of course, the phrase “uncertainty of the average weight” is deliberately ambiguous.
If you mean the uncertainty in the weight of an individual average widget then it’s the same as before 0.71.
If you mean the uncertainty of the average weight of the two then the the average of 2 widgets will still be 5, and the uncertainty of the average will be the uncertainty of one divided by √2 (again assuming independent uncertainties) That is
0.71 / √2 = 0.50
This follows for the fact that the uncertainty of the total weight of the two would be
0.71 * √2 = 1.00
Note, that if you were talking about temperatures as we were originally, the total of two temperature readings would be meaningless, as it’s an intensive property. So talking about the uncertainty of the sum would also be irrelevant.
You *really* don’t get it at all, do you? You don’t add the temperatures! You add their uncertainties!
“You don’t add the temperatures!”
It’s literally what Jim said
“Two temps summed:
T1 + T2 = T”
It’s literally what Pat Frank said that started this thread.
“Given two temperatures measured using the same sensor, the uncertainty in their sum is their individual uncertainties combined in quadrature.”
You are getting further and further from reality.
Let me explain. To get 0.71 you obviously did:
√(δW1 + δW2 = √(0.5² + 0.5²) = 0.707 => 0.71
That is the RSS method of determining a combined uncertainty of independent measurements with normal distributions.
Why you did (√2 • 0.5) I have no idea! You got the right answer for the wrong reason.
It appears you are first calculating the combined uncertainty, i.e., 0.71, and then reducing it by √2. “0.71 / √2 = 0.50”. That appears to be how to determine an estimated SEM using the following formula:
SEM = σ / √n
What would this SEM tell you? It says with a sample size of two, your mean would lay in an interval of ±0.5 units surrounding the mean.
What you are doing here is combining two measurements with “assumed” normal distributions. Their average could be between (0.71 – 1). RSS would give 0.71, while simple addition would give 1.0.
Think about this from a probability standpoint! Would you really expect the average weight to NEVER exceed the value of just one?
Now we see why I’m reluctant to answer some badly worded toy example.
“Let me explain. To get 0.71 you obviously did:
√(δW1 + δW2 = √(0.5² + 0.5²) = 0.707 => 0.71”
You said the widget was made of two materials with an uncertainty of 0.5. I assumed you meant each material had an uncertainty of 0.5, and wanted to know the uncertainty of a single widget. Why else mention the two materials?
If you mean each widget has an uncertainty of 0.5, then that equation is the uncertainty of the weight of two widgets combined. Though you should say √(δW1² + δW2²).
“Why you did (√2 • 0.5) I have no idea! You got the right answer for the wrong reason.”
How long have you been doing this? It’s standard algebra.
√(0.5² + 0.5²) = √(2 * 0.5²) = √2 * √0.5² = √2 * 0.5
More generally, this gives you the equation for a sum of N values from the same distribution.
√(σ1² + … + σN² ) = √(N * σ²) = √N * √σ² = √2 * σ
And for an average
√(σ1² + … + σN² ) / N = √(N * σ²) / N = (√N / N) * √σ² = σ / √N
“It appears you are first calculating the combined uncertainty, i.e., 0.71, and then reducing it by √2. “0.71 / √2 = 0.50”
See above. If the uncertainty was for a single widget than the uncertainty of an average of two widgets would be the uncertainty of the sum of two, divided by 2.
(√2 * 0.5) / 2 = 0.5 / √2 = 0.35
“What would this SEM tell you?”
It would tell you the uncertainty of the mean of two widgets. More usefully perhaps it would tell you the standard deviation of a sample of size two, if for some reason you wanted to assess the average weight of a widget based on a sample of just two (and assuming you already knew that the standard deviation of the widgets was 0.5).
“It says with a sample size of two, your mean would lay in an interval of ±0.5 units surrounding the mean.”
I’m assuming your 0.5 value was the standard uncertainty. The SEM is only going to tell you what the standard uncertainty of the mean of two widgets. The expanded uncertainty will be bigger – and depend on the distribution of your uncertainty.
“What you are doing here is combining two measurements with “assumed” normal distributions.”
And there you go. First you insist I have to answer a badly specified problem, then complain that I made assumptions. But in this case, I did not make that particular assumption, just that there was a standard uncertainty of 0.5.
“Their average could be between (0.71 – 1). RSS would give 0.71, while simple addition would give 1.0.”
As I said, I assumed the uncertainties were random and independent because you refused to give any other information.
Second the average could not be between (0.71 – 1), whatever you mean by that. In a worse case, assuming every widget was +0.5 overweight, the error ion the average would still only be +0.5. Which is to say the uncertainty of the average would be between 0.35 and 0.5.
“Think about this from a probability standpoint!”
That’s what I keep doing, only to be told that probability has nothing to do with uncertainty.
“Would you really expect the average weight to NEVER exceed the value of just one?”
Of course the average weight could be bigger than the weight of one widget, if the one widget was below average weight.
The point is that the average weight can’t be bigger than the heaviest widget, and in all probability, assuming randomness in the weights will be a a bit less than heaviest.
“It would tell you the uncertainty of the mean of two widgets.”
In the real world, no one wants to know the uncertainty of the average. They want to know the uncertainty associated with the unit you give them.
You *still* aren’t living in the real world.
” First you insist I have to answer a badly specified problem, then complain that I made assumptions”
This is just a whine. The problem was posed correctly. *YOU* made assumptions to make the problem fit what you wanted to say.
“In the real world, no one wants to know the uncertainty of the average. They want to know the uncertainty associated with the unit you give them. ”
It was Jim who wanted to know what the uncertainty of the average was. I said it was a dumb question. All you are saying is that the weight of widgets are a bad analogy for global temperature. You might not want to know the average weight of a widget, but we do want to know the average global anomaly.
Dodge the question. Not unexpected.
Only if you can’t understand the difference between adding and dividing.
That’s just it Pat. If u(Σ[xi]/N) > u(x)/sqrt(N) then u(xj – Σ[xi]/N) < sqrt[u(xj)^2 + u(Σ[xi]/N)^2].
In other words, if the uncertainty of the average isn’t scaled with 1/sqrt(N) then the uncertainty of a subtraction must be less than root sum squared.
This is an indisputable mathematical fact. It is an inevitably from the law of propagation of uncertainty from Bevington 3.13 or GUM 16.
I encourage you to prove this yourself. Solve Bevington 3.13 or GUM 16 algebraically for both y = Σ[xi]/N and y = a – b. Then try it numerically via the NIST uncertainty machine with r(xi, xj) > 0.
The “average” uncertainty of the average of ~9500 stations!
Don’t you know if you run a reaction 1000 times you can add 3 decimal digits of resolution to your measurements of the products and reduce the uncertainty by 10^-3!
We can physically measure the distance to the nearest star down to the tenths of a meter if we just take enough measurements to reduce the precision and uncertainty.
These people STILL don’t understand that uncertainty is not error. Yet they keep yammering on a subject about which they know next to nothing.
“uncertainty is not error.”
Error and uncertainty are related. Uncertainty is caused by error and uncertainty describes the extent of the errors. You can try to redefine uncertainty to avoid using the term error as the GUM does, but the results are the same.
From Pat Frank’s paper:
Um, is English your third or fourth language?
“Related” does not mean “equal”.
Perhaps you need to spend time in an on-line dictionary with beeswax.
And error is still not uncertainty. Until you figure this out, you will remain drifting without sail or oar.
““Related” does not mean “equal”.”
It’s almost as if I deliberately used the word “related” to avoid the word “equal”.
All you have to do is open the book and read.
But you are stuck in these trendology lies, unable to admit the truth.
Let’s open some of these books then.
Let’s start with Taylor’s An Introduction To Error Analysis.
Then there’s the book Pat Frank uses to justify his equation.
Bevington: Data Reduction and Error Analysis for the Physical Sciences
Our interest is in uncertainties introduced by random fluctuations in our measurements, and systematic errors that limit the precision and accuracy of our results in more or less well-defined ways. Generally, we refer to the uncertainties as the errors in our results, and the procedure for estimating them as error analysis.
Or the GUM
That’s calibration uncertainty, determined by instrumental test against a reference that provides a known value.
In a field measurement, one doesn’t know the physically true value. Therefore, the measurement error cannot be known.
One then must apply the instrumental calibration uncertainty to the field measurement as an indicator of reliability.
Of course. But the inevitabilities of this are:
1) Anomalies for that station and that station only are computed from a subtraction of two values both of which share the same calibration error so a lot (not all) of the systematic component of uncertainty gets eliminated even before stations are grouped/aggregated for averaging.
2) The systematic uncertainty still remaining for the station specific anomalies would be different for different stations. So in the new grouped/aggregated context that component of uncertainty switches from systematic to random like what the GUM says is possible.
3) Some of the systematic uncertainty arising from the gridding, infilling, and averaging methodology gets eliminated via the re-anomalization at these higher level steps.
Don’t hear what hasn’t been said. It has not been said that all systematic uncertainty gets eliminated. In particular it is the fact that both station systematic error and methodological systematic error changes with time that some systematic uncertainty remains and propagates through the selection, gridding, infilling, and averaging steps. An example of a station systematic error that propagates through would be if the calibration drifted but the homogenization could not detect it. An example of a methodological system error that propagates through would be how NOAA does not infill the polar regions but BEST does. That creates a bias between the two datasets.
You are subtracting two random variables to create a third random variable. They each have their own combined uncertainty variance. At best you might claim that they should be added by RSS. Otherwise their uncertainties simply add.
Taylor says:
What constitutes a better estimate of δq? The answer depends on precisely what we mean by uncertainties (that is, what we mean by the statement that q is “probably somewhere between qbest – δq and qbest + δq. It also depends on the statistical laws governing our errors in measurement. Chapter 5 discusses the normal, or Gauss, distribution, which describes measurements subject to random uncertainties. It shows that if the measurements of x and y are made independently and are both governed by the normal distribution, then the uncertainty in q = x + y is given by
δq = √(δx)² + (δx)² (3.13)
Taylor goes on to say that it is difficult to decide how to treat systematic uncertainty. You can simply add it to the uncertainty that is determined statistically or you can add with RSS.
You have made no logical argument for this, only an assertion. The GUM in E.3 simply says that uncertainty should be treated the same regardless if it is Type A or Type B.
Again, as assertion with no logical underpinning. You need to have a much better argument with some data and math to back it up.
I have attached an image of temperature analysis using TN 1900 methods for Topeka Forbes and the month of May since 1952.
Your search for a way to reduce measurement uncertainty to the point where current anomaly determination can be said to be statistically significant is not going to work as more and more local/regional analysis’s are completed. We have already seen no increase in a Japanese location. I assure you there are more being done all the time. The expanded experimental uncertainty the data is showing will make global anomalies a thing of the past.
“1) Anomalies for that station and that station only are computed from a subtraction of two values both of which share the same calibration error so a lot (not all) of the systematic component of uncertainty gets eliminated even before stations are grouped/aggregated for averaging.”
We keep going over this. The uncertainty has two components.
u_total = u_random + u_systematic
“3) Some of the systematic uncertainty arising from the gridding, infilling, and averaging methodology gets eliminated via the re-anomalization at these higher level steps.”
The operative word here is “Some”. How much is some? If the systematic component changes over time how does that affect the analysis of the time series? Systematic bias is *NOT* always calibration error. It can be micro-climate variation. How does that get subtracted out?
Where does climate science include the remaining systematic bias in their uncertainty? All I ever see is analysis of the stated values. Where do *YOU* include it in your analyses?
“Don’t you know if you run a reaction 1000 times you can add 3 decimal digits of resolution to your measurements of the products and reduce the uncertainty by 10^-3!”
How? Even if you ignore all other factors a sample of 1000 would only reduce the uncertainty by 1 / sqrt(1000) = 0.03, which is still a lot bigger than 0.001.
Meanwhile you still claim that taking 1000 measurements will increase the uncertainty by a factor of 30, or possibly 1000.
“””””I agree, if they can be considered different systematic errors, but that means they are effectively random errors.”””””
Why is your question important? You posted about E.3 in the GUM. Did you read it closely? ALL UNCERTAINTIES CAN BE TREATED EQUALLY.
It is important to note that in the Vasquez uncertainty budget β is the systematic component where β is “a fixed bias error” and is “assumed to be a constant background for all the observable events”.
The fact that β can be composed of several error sources combined via equation (2) β ≈ √[∑(φ)²] does not change the fact that β is still fixed and constant for all observations.
What this means is that for any two metrics they would both share the same systematic component of uncertainty β which means there will be correlation r > 0 between the two. And when y = a – b you’ll notice Vasquez (6) (which is also GUM (16)) says the correlation term is negative meaning it reduces uncertainty when a subtraction is involved. It is important to note that an anomaly involves the subtraction of the baseline from the observation. This is the law of propagation of uncertainty demonstration of the removal of the β component when dealing with anomalies.
I encourage you to play around with NIST uncertainty machine and see how this works with measurement models of the form y = a – b and choosing r(a, b) > 0.
That is true only if the correlation of the inputs is zero (ie r(xi, xj) = 0). If they share the same bias β then r(xi, xj) > 0. The magnitude of r(xi, xj) depends on how the uncertainty budget is formulated (ie Vasquez (1) vs (4)) and how much β is contributing relative to ε. For those who don’t have access to the publication Vasquez defines two uncertainty budgets. (1) is u_k = β + ε_k and (4) is u_k = sqrt[β^2 + ε_k^2].
The point is the equation of use, bdgwx. Temperature sensor field calibrations invariably demonstrate deterministically variable systematic error. The assumption you require is violated. QED my usage.
I already pointed out to you that Vasquez & Whiting recommend eq 2 when “several sources of systematic errors are identified.” But you conveniently overlooked that.
“y = a – b” The correlation of a and b is independent of the operator.
“an anomaly involves the subtraction of the baseline” An anomaly involves subtraction of an empirical normal, not a baseline. An empirical normal retains the mean uncertainty of the entering measurements.
Subtraction of a monthly or an annual mean from the normal requires addition of the respective ± uncertainties in quadrature.
“If they share the same bias β then r(xi, xj) > 0.” Correlation only is discoverable in data series. A constant bias offset will not affect inter-series correlation.
“Vasquez defines two uncertainty budgets. (1) is u_k = β + ε_k and (4) is u_k = sqrt[β^2 + ε_k^2].” V&W (1) describes the systematic and random error of a single observation. V&W (4) describes calculation of the uncertainty for sets of observations containing both sorts of error.
You’re wrong on every single point, bdgwx.
He will never grasp this.
Make sure you “play around” with the NIST Uncertainty Machine!
I don’t know what you mean by “deterministically variable”. But if it is variable in the sense that each station exhibits its own but different systematic error then when viewed in the context of multiple stations aggregated together there may be a component of that error that is fixed and constant β and certainly a component that is random ε_k.
I didn’t overlook. I posted equation (2)…literally.
I know.
You call it “empirical normal”. I call it “baseline”. We’re talking about the same thing.
Patently False. The empirical normal or baseline must have the uncertainty of its constituents propagated into it in accordance with GUM equation 10 or 16 or Bevington equation 3.13 or 3.14 or any equation derived from them and which produce a combined uncertainty. It most definitely does NOT “retain the mean uncertainty”. And just to be clear “mean uncertainty” is Σ[u(xi)]/N and is different than “uncertainty of the mean” which is u(Σ[xi]/N).
If and only if their respective uncertainties are uncorrelated such that r(xi, xj) = 0. If the uncertainties are correlated such that r(xi, xj) > 0 like would be the case if they shared a component of uncertainty then you must subtract the correlated component of uncertainty. That is not optional as described by Bevington 3.13 or GUM 16.
Sure, it can be discoverable in a data series via their covariance r(yi, zi) = s(yi, zi)/(s(yi)s(zi)). But that’s not the only way. Another way is to use r(xi, xj) = u(xi)δj/u(xj)δi which is useful in GUM 16 or Vasquez 6. For example, if β contributes 50% to the uncertainty then r(xi, xj) = 0.5. If β contributes 100% to the uncertainty then r(xi, xj) = 1.0. So for the model y = f(a, b) = a – b here are the results using GUM equation 16.
r = 0 then u(y) = sqrt[u(a)^2 + u(b)^2]
r = 0.5 then u(y) = sqrt[u(a)^2 + u(b)^2 – 2*u(a)*u(b)*0.5]
r = 1.0 then u(y) = abs[u(a) – u(b)]
The r = 1.0 derivation is interesting because sqrt[u(a)^2 + u(b)^2 – 2*u(a)*u(b)*1.0] simplifies to u(a) – u(b) since a^2 + b^2 + 2ab = (a+b)^2. That’s right. When u(a) = u(b) = β + ε_k and ε_k = 0 such that all uncertainty is systematic then it completely and entirely cancels when doing a subtraction.
Funny because I’m actually doing the derivations. I verified them algebraically with a computer algebra system and numerically with the NIST uncertainty machine. It is a fact. When your uncertainty budget contains a β term it necessarily cancels when doing a subtraction. This can be proven both with trivial algebra or with the law of propagation of uncertainty like what I did above.
You don’t have the first clue about “bias” error.
And you ran away from this point:
Yet another indication that you and curveman still don’t comprehend this very basic idea, that uncertainty is not error.
I suggest using an on-line dictionary (in between stuffing numbers into the NIST Uncertainty Machine, of course).
An empirical normal is not a baseline. It’s a mean.
Means are conditioned with the uncertainties of the measurements that go into it.
When constructing a temperature anomaly, the uncertainty in the normal combines in quadrature with the uncertainty in the requisite temperature.
When “several sources of systematic errors are identified,” β is not a constant.
You’re still wrong on every single point, bdgwx.
You USE the plus sign to always indicate that “e” is positive.
If “e” is something negative and sometimes positive then you do NOT always get cancellation or “e”.
(+) – (+) cancels. (-) – (-) cancels.
(+) – (-) does *NOT* cancel. (-) – (+) does *NOT* cancel.
Therefore your assertion that systematic uncertainty always cancels because of the “algebra” is obviously incorrect.
It’s just that simple.
I’ll reply if I get a chance but I am swamped with work so it might take a while or it might not happen at all. Ce le vie!
No Tim. That’s not how algebra works.
No, “e” can be either positive or negative.
That means you can have (A + e) or (A – e).
THAT’S algebra. That’s why “e” is always shown as +/-.
You want to always want to consider “e” as positive so you can assert the systematic bias in an anomaly calculation cancels.
You simply can’t admit that such an assertion is NOT algebra, can you?
BTW, what is the sqrt(Variance)?
Complex?
No it isn’t. In algebra if you want to add y to x and set that equal to z you simply write x+y=z and leave it at that. It doesn’t matter if x or y is positive or negative. The algebra works out all the same. What you definitely don’t do is write x±y=z. I’ll repeat again and again if I have to. 1) The symbol ± is not a valid algebra operation. 2) The operation + does not assume anything about the sign of the operand on the right hand side.
Did you learn this from the NIST Uncertainty Machine?
hehehehehehehhehe
Projection time!
“How about a triangle? {A = (bh) / 2}? Do you divide the uncertainty by 2 as you would with the average? NO!”
YES.
“You are making the assertion that since the function is called an AVERAGE, that the uncertainty must be an average also!”
No, I’m saying that if you scale any measurement by an exact value you also have to scale the uncertainty by the same value. The fact that you are scaling the sum by 1/N to get the average also means you scale the uncertainty of the sum by 1/N to get the uncertainty of the average. It applies just as much if you divide the diameter by 2 to get the radius. You also have to divide the uncertainty of the diameter by 2.
“That is an assumption with no support.”
How much support do you need.
I’ve pointed you to Taylor page 54. I’ve shown how this follows from the general rules for propagating uncertainty when multiplying and dividing measurements. I’ve shown how all this can be derived from the general uncertainty equation (e.g. GUM equation 10.), I’ve shown how this can be derived from the rules for combining random variables. And I’ve suggested how you can test this for yourself using montecarlo methods. I’ve tried to explain why what you are doing leads to mathematically impossible answers.
But it’s hopeless when you just rewrite and misunderstand any equation if it does not leads to the answer you want. For example
“To be correct, since uncertainties always add, the uncertainty should be u = √((Σuₙ_ᵢ)² + u(n)²). Now tell me what the uncertainty of “n” is!”
You keep on writing this nonsense no matter how many times we explain why it is wrong. You are quoting the propagation rule for adding and subtraction. The rules for multiplication and division require relative uncertainties to be added.
If you are dividing a value by n the correct equation is
u_avg / avg = √((u_sum / sum)² + (u(n) / n)²)
And as u(n) = 0
u_avg / avg = u_sum / sum
And as avg = sum / n
u_avg = avg * u_sum / sum
= (sum / n) * u_sum / sum
= u_sum / n
And in general
“No, I’m saying that if you scale any measurement by an exact value you also have to scale the uncertainty by the same value.”
Who is scaling measurements? Did you read the example in Bevington? The measurements aren’t scaled, the uncertainty is!
You *still* haven’t figured out 3.9 yet!
It’s a simple relationship between the number of items you have and their total uncertainty! If you have B items their uncertainty adds – by adding their relative uncertainty!
How can you confuse such simple concepts?
“Who is scaling measurements? ”
Anyone who’s converting a diameter to a radius, or a diameter to a circumference. Anyone who’s calculating the are of a triangle by multiplying base and height and then divides by 2. And of course, anyone who wants to convert a sum of multiple values into an average.
“If you have B items their uncertainty adds – by adding their relative uncertainty!”
And if B = π? Or 1/2, or 1/200? Multiplication is not just repeated adding. I would have assumed you learnt that by now.
Do you *ever* stop to think about what you are posting?
“Anyone who’s converting a diameter to a radius”
That’s not “scaling” a measurement! The measurand is the diameter! You are using the measurand to calculate another value. The uncertainty associated with the measurand, i.e. the diameter, propagates to the calculated value. The uncertainty of pi is zero, so when you propagate its uncertainty to the circumference it contributes nothing to the uncertainty of the calculated circumference!
“Anyone who’s calculating the are of a triangle by multiplying base and height”
The uncertainty in the result is propagated from the uncertainties of the height and the base. The constant 2 has no uncertainty and therefore doesn’t propagate into the uncertainty of the result.
Take a rectangular table where one end is of length x and the other of 2x. The area is 2x^2. Does the factor 2 contribute to the uncertainty in the area? If you think so then go study Taylor, Chapters 2 and 3 again – it’s obvious you haven’t bothered to actually study them so far, all you’ve done is cherry pick things you think you can use to show people are wrong without actually understanding what you are cherry picking.
“That’s not “scaling” a measurement! The measurand is the diameter! You are using the measurand to calculate another value. ”
What on earth do you think scaling means? Of course you are not literally making a measurand smaller, you are applying a transformation on one value to get another value.
“The uncertainty associated with the measurand, i.e. the diameter, propagates to the calculated value”
Yes, halfing it.
“The uncertainty of pi is zero, so when you propagate its uncertainty to the circumference it contributes nothing to the uncertainty of the calculated circumference!”
You multiply the uncertainty of the diameter by π to get the uncertainty of the circumstance. Why do you think that’s not a contribution.
At this point I keep having to wonder if you really don’t understand your own prefers text books, or if you are just trolling.
unfreakingbelievable!
10x = x + x +x + x +x +x +x +x + x + x
Where is the scaling in that?
“Yes, halfing it.”
nope!
Can’t you do simple math?
“10x = x + x +x + x +x +x +x +x + x + x”
Those of us with more advanced maths degrees learn that there is a better way of multiplying a number by 10.
“Where is the scaling in that?”
You are asking where the scaling is in multiplying a number by 10? Maybe this page will help, though you might find it a little advanced.
https://www.bbc.co.uk/bitesize/topics/z36tyrd/articles/zb7hm39
“Can’t you do simple math?”
No. Only complicated stuff, but I’m pretty sure the radius is still half the diameter.
Snooty bellcurvewhinerman returns!
Stop whining. I’ve had enough to day of people pretending that they don’t understand how scaling works.
Hey! Make sure you don’t use plus/minus!
Yur the expert, so if it’s now wrong to use ± ion MontyKlown world, what am I supposed to use instead. [-n,+n]?
Coherence much?
You don’t want me to use ± to express an interval, but you think I’m being incoherent?
Is this the ambiguous terminology trap again?
It seems absolute uncertainties would scale, but relative uncertainty would be scale invariant.
Bevington 3.18.
Yes!
Following the procedure in Bevington 3.2 we let A = f(b, h) = (bh)/2. The partial derivatives are ∂A/∂b = (1/2)h and ∂A/∂h = (1/2)b. Then applying 3.14 we have σ_A^2 = σ_b^2*(1/2)h + σ_h^2*(1/2)b. That is σ_A^2 = (hσ_b^2 + bσ_h^2) / 2. This can also be seen via Bevington’s multiplication rule via 3.21, 3.22, and 3.23.
The fact that u has different units than u_x, u_y, and u_z should be a clue that something went horribly wrong here. Anyway…why not just follow the procedure Bevington outlines in section 3.2 for this exact problem?
I’ll give you a shorter answer than the one to bellman.
Your hands are tied when you define the average function. The components of that function (or any function) do not transfer directly into the uncertainty equations, only their uncertainty. Therefore, since “n” is a count, it has no uncertainty, and does not affect the uncertainty at all.
You need to find an example where a constant or count contained in a function has been used to divide or multiply an uncertainty calculation and the reason for doing so.
Bevington does that already with his weighted sums and differences rule via 3.18, 3.19, and 3.20 all derived from 3.13.
If x f(u, v) = au + bv then σ_x^2 = a^2*σ_u^2 + b^2*σ_v^2 + 2abσ_xv^2. The astute reader will recognize this as derivation of 3.13. Anyway, when a = 1/2 and b = 1/2 then x = f(u, v) is is the average of u and v and when there is no correlation between u and u then 2abσ_xv^2 = 0 resulting in σ_x^2 = (σ_u^2 + σ_v^2) / 2.
“divide by the number of data points.”
Only for random error.
Yes
I’ll revise that. Variance covers recorded excursions from the mean. It’s a measure of dispersion.
Pat,
They are *NEVER* going to figure this out!
They deny vehemently that they assume no systematic bias in any of the data but is just shines through in everything they do!
No matter how many times they are told that they are assuming that all measurement uncertainty is random, Gaussian, and cancels they just keep on doing it.
If they would *explicitly* state this assumption every time they make a post maybe it would finally sink in!
Eq. 4.23 is valid only when the error is random. Figure it out.
As is 4.22.
It’s not even about measurement uncertainty, it’s about the variance in the data, and you can’t detect systematic errors by looking at the variance in the data.
That is correct you can not detect systematic errors by any statistical analysis.
You can have a measurement like
5×10⁻⁶ ±10⁻⁷ with a systematic error of +0.03. Think you can ever find that error through statistics using either the data AND/OR 1000 measurements of the same thing?
“That is correct you can not detect systematic errors by any statistical analysis.”
That’s a rather sweeping statement. I’m no statistician but I think I could come up with some methods to detect and estimate systematic error.
“5×10⁻⁶ ±10⁻⁷ with a systematic error of +0.03. Think you can ever find that error through statistics using either the data AND/OR 1000 measurements of the same thing?”
I think in that case it would be pretty obvious. If the thing you are measuring is of the order 10⁻⁶ and you get a result of 0.03, there’s an obvious problem with your instrument.
More generally I would try measuring the same thing with different instruments or measure against known standards.
Your example does not involve detection using statistics. You’ve negated your own argument.
I assume that you would be using statistics as part of the process.
E.g. if you measure the same object with two different instruments one, then you won’t know if any difference is the result of random or systematic error. But if you measure the object with each instrument several times and compare the average difference, and find the difference is statistically significant – then you have evidence of a systematic error in at least one of the instruments.
You’re approaching a calibration experiment. That’s not detection by statistics.
They could BOTH have systematic bias due to being out of calibration. All you know is that they don’t read the same.
How do you decide which one has what systematic bias?
“More generally I would try measuring the same thing with different instruments or measure against known standards.”
As usual you were cherry-picking when quoting Bevington. Did you bother to read the example he gave after your quote?
It was how to use 4.22 and 4.23 to calculate uncertainty when measuring the same thing with different devices.
It does *NOT* allow identifying systematic bias.
And how may temperature stations get calibrated (your “known standards) before they take every measurement?
“As usual you were cherry-picking when quoting Bevington.”
I wasn’t quoting anyone. I was thinking of the top of my head.
“It does *NOT* allow identifying systematic bias.”
Which is why I made no mention of those equations.
Systematic errors are revealed by calibration experiments. They are central to my analysis.
Your comments, and bdgwx’, are indistinguishable from those of someone who’s never read the paper.
Then why did you bother to quote Bevington? His Eq 4.22, etc only apply when there is no systematic uncertainty.
It was Pat Frank who referenced 4.22 as justification on how to deal with systematic uncertainty. I’m saying it’s a mistake to use that equation as justification.
I find it strange, because the result may be reasonable if you assume there is unknown systematic error across all instruments, but using that equation as justification for your equation is completely wrong.
Pat Frank is telling me he used that equation because his paper is on systematic uncertainty.
Read the example in the book. It is designed when you have data groups with different uncertainties.
Dr. Frank has told you unit uncertainties. Does that mean anything to you?
“It is designed when you have data groups with different uncertainties.”
Indeed, hence the weightings. What has that got to do with them not being systematic errors?
“Dr. Frank has told you unit uncertainties. Does that mean anything to you?”
It does not, and “unit” is not one of the names I’ve heard him give to these uncertainties. What do you think “unit uncertainty” means, or are you thinking of “unit weights”?
You have solidified my expectation that you have no high level physical science training at all that your math is lacking.
You could have just asked me. I’ve made no secret of the fact I have no high level science training, and not much in the way of statistics.
And frankly, if you and your group are examples of what that training does, I’m quite relieved about it.
Now, do you have a point, or are you just throwing out random insults?
Guys, everybody seems to be talking past each other again and getting cranky as a result. That doesn’t really enhance understanding or improve the state of knowledge.
To try to reduce the mutual misunderstandings, here’s a little mind experiment.
Feel free to give your answers, and the reasons for those answers.
Also feel free to add more questions, etc.
We have a single site with maximum and minimum thermometers. Let’s say they are separate instruments.
They read to 1 degree C, and have been freshly calibrated.
We take daily maximum and minimum readings at the same time every day. I’m Australian, so make it 9 am. The maximum is attributed to the previous day.
We start on the 1st of April, and finish on the 1st of May, and discard the 31st March max and 1st May min.
The same observer takes each reading.
Calibrations are performed at the end of the trial, and both instruments pass.
1/ What are the uncertainty intervals each day for the max and min?
2/ What is the uncertainty interval each day for the median?
3/ What are the uncertainty intervals for the monthly maximum and minimum?
4/ What are the uncertainty intervals for the monthly mean maximum and minimum?
5/ What is the uncertainty interval for monthly mean median?
Also, feel free to cuss me out for being presumptuous, ignorant, arrogant, etc 🙂
To put some arbitrary figures on, readings taken on odd numbered days are max 25, min 10. Readings on even numbered days are max 22, min 7. Don’t forget the max is attributed to the previous day.
I made odd days 25/10 and even days 22/7. I used 30 days worth of data. I used NIST TN 1900 as the guide for calculating the expanded experimental uncertainty for Tmin and Tmax. Tavg is a special problem due to correlation. I used this from the GUM to calculate the standard uncertainty of correlated variables with the special condition of a correlation of +1.
5.2 Correlated input quantities
5.2.2
NOTE 1 For the very special case where all of the input estimates are correlated with correlation coefficients r(xi, xj) = +1
The combined standard uncertainty uc(y) is thus simply a linear sum of terms representing the variation of the output estimate y generated by the standard uncertainty of each input estimate xi (see 5.1.3). [This linear sum should not be confused with the general law of error propagation although it has a similar form; standard uncertainties are not errors (see E.3.2).]
I cut some of the spreadsheet off.
Also from the TN 1900.
For example, proceeding as in the GUM (4.2.3, 4.4.3, G.3.2), the average of the m = 22 daily readings is t̄ = 25.6 ◦C, and the standard deviation is s = 4.1 ◦C. Therefore, the √ standard uncertainty associated with the average is u(r) = s∕ m = 0.872 ◦C. The coverage factor for 95% coverage probability is k = 2.08, which is the 97.5th percentile of Student’s t distribution with 21 degrees of freedom. In this conformity, the shortest 95% coverage interval is t̄ ± ks∕√ n = (23.8 ◦C, 27.4 ◦C).
“In this conformity, the shortest 95% coverage interval is t̄ ± ks∕√ n = (23.8 ◦C, 27.4 ◦C).”
Something’s not right here. Half the maximum temperatures were 22 and the other half 25, yet you calculate that the average max could be higher than 27.4.
Your own figure for an expanded uncertainty of 1.07 would imply a range of [22.4, 24.6]°C.
This still seems way to high given that the range is close to the range of all the maximum temperatures.
The main problem I expect is you are just adding the full range of uncertainty from rounding to the expanded uncertainty from sampling. At the very least these should be added in quadrature assuming they are independent of each other. That would make your total uncertainty ±0.76.
But then you are treating the ±0.5 rounding error as systematic. Possible given the artificial nature of the problem. But if you assume each rounding error was random, your rounding uncertainty for the month becomes approximately 0.5 / √30 ~= 0.1, and combining that with the sampling uncertainty in quadrature gives √(0.57² + 0.1²) = 0.58.
So that would make the 95% confidence range [22.9, 24.1]°C.
The numbers are right there. Put them in a spreadsheet and see what you get.
You have never read or studied TN 1900 have you. I did not use the reading error, I assumed measurement error and systematic error was negligible as in TN 2900, and proceeded to compute the expanded experimental error just as shown in TN 2900. The calculations are shown in the notes.
I carried two decimal places to prevent rounding errors and rounded the final interval temperature to the units digit. You’ll notice that TN 1900 rounded to one decimal.
Have you ever heard of Significant Digits? Make sure that your final answer uses them.
If you wish to combine the experimental uncertainty with a systematic reading uncertainty please be my guest. It only makes the uncertainty wider.
This is just an example of how anomalies to the one thousandths digit are so far inside a proper uncertainty interval that they can not be considered anything but statistically insignificant.
“The numbers are right there. Put them in a spreadsheet and see what you get.”
What’s the point. I’ve already said where I think you are wrong and the reasons. Blindly adding numbers into a spreadsheet is not useful.
“I did not use the reading error, I assumed measurement error and systematic error was negligible as in TN 2900”
That was the point of my joke. I made exactly the same assumptions and was hit with numerous hostile comments insisting that it was impossible to make such an assumption, and that it showed I knew nothing of the real world.
“Have you ever heard of Significant Digits?”
Never heard of them. Maybe you should mention them more.
Seriously, I don’t care in this hypothetical artificial question. I simplified the figures quite a lot in order to err on the side of getting the largest interval and gave the answer to 0.1°C. The problem has nothing to do with a lack of rounding. It’s the assumptions you made in combining the different uncertainties.
You can believe you know the average to to a precision an order of magnitude better than any of the measurements, but you will be mistaken. I’m glad you aren’t building a house for me or overhauling an engine. I would certainly not want to be on a rocket going to the moon with parts you have machined.
You are contradicting a NIST PhD who has written books on uncertainty. You are contradicting a PhD experimental chemist who makes a living doing measurements. You got a big pair, but they aren’t fully loaded.
Just keep focusing on the trivial points rather than question if your calculations were correct.
Fine, then say the confidence interval for the monthly maximum was
[23, 24]°C, rather than your claim of (23.8 ◦C, 27.4 ◦C).
The extra decimal point isn’t the issue. It’s the fact your numbers are wrong.
Have you lost your mind?
23.8 ◦C, 27.4 ◦C is the range for the interval determined from the temperatures in TN 1900.
Here are my intervals for the temperatures I stated in the post.
Tmax => 22 – 25
Tmin => 7 – 9
Tavg => 15 – 17
“Have you lost your mind?”
No I just corrected your mistakes. (which might be an act of insanity around these parts).
But checking back I see my mistake. I assumed when you said “the shortest 95% coverage interval is t̄ ± ks∕√ n = (23.8 ◦C, 27.4 ◦C)”, you were describing your own results not those taken from the example in TN1900. Sorry.
Your quoted uncertainty as I take it is ±1.1, where as mine is ±0.6.
Expect to be attacked vigorously for assuming there are no systematic errors. Any time now I expect.
If you don’t calibrate before each measurement then you can’t know the exact uncertainty, it has to be estimated.
The uncertainty of the median temp will be the sum (either directly or in root-sum-square) of the uncertainties of the two measurements.
The uncertainties associated with any mean is the sum (direct or rss) of the data elements making up the average. Dividing the total uncertainty by N only gives you the average uncertainty, not the uncertainty of the average.
“The uncertainty of the median temp will be the sum (either directly or in root-sum-square) of the uncertainties of the two measurements.”
How?
Lets assume that the main uncertainty was due to a systematic error, with the rest coming from rounding to the nearest integer. Say there is a systematic error of +5°C to all readings both for max and min. If the maximum temperature was exactly 20°C and this is read as 25°, and the minimum temperature was 10°C which is read as 15°C, then the true median or mean of the two is (20 + 10) / 2 = 15. And your measured value is (25 + 15) / 2 = 20. The error is 5°C which is the average of the two systematic errors.
How is it possible that you could get an error equal to the sum of the two systematic errors?
Because you assumed the systematic bias is only positive!
Uncertainty is stated as a +/- interval unless you can justify the use of an asymmetric interval.
So what you should be looking at is 20C +/- 5C ==> 15C – 25C. The minimum temp would be 10C +/- 5C ==> 5C – 15C.
The interval in which the average might lie is 5C to 25C. The uncertainty has grown from +/- 5C to +/- 10C!
Remember, the systematic bias is UNKNOWN. It is part of an uncertainty *interval*.
Taylor goes into this in detail in Chptrs 2 and 3. Why do you keep insisting on cherry picking from Taylor instead of actually studying it as you would in a high school/college course? Including doing the homework!
If the systematic bias can be both positive and negative then it isn’t systematic.
In the case of the min max thermometer you could treat them as two separate instruments each with their own different systematic bias, but as soon as you combine their measurements you would have to look at their uncertainties as being random. This is why the GUM says it’s not easy to just talk about random v systematic uncertainties as depending on context random cand be systematic and systematic random.
“as soon as you combine their measurements you would have to look at their uncertainties as being random.”
Wrong.
Again, unfreakingbelievable! Your lack of physical science knowledge is showing again.
If a temperature measuring device has hysteresis, i.e.it reads differently when the temp is going up then when the temp is going up, then why can’t one be a positive systematic bias and the other a negative systematic bias?
All of the components in even an electronic device have temperature coefficients and thermal inertia making hysteresis a real thing!
“but as soon as you combine their measurements you would have to look at their uncertainties as being random. “
Malarky! Where do you get this crap? Systematic biases don’t all of a sudden become random error!
You are stuck on the assumption that systematic bias can be analyzed statistically and accounted for – even after being given quotes from three experts that it is either very difficult or impossible. Yet you continue to try and search for a method to do so. You are going to have to do much better than this if you are going to undo what Taylor, Bevington, and Possolo have asserted!
“systematic and systematic random”
OMG! Where does the GUM say that? Reference please?
He’s back to fanning from the hip again.
I think that is what he does *all* the time. The mark of a troll who is only looking for attention.
Absolutely.
“He’s a troll, Tim.” — Pat Frank
This quote still cracks me up.
See my comment above. It doesn’t matter if the two systematic errors are positive and negative. When you add them that just means some cancellation. The biggest error is if both are positive or both are negative, and that still means that the uncertainty is given by the average uncertainty.
And you still have nary a klue about uncertainty…
The bellcurvewhineman hamster wheel:
“averaging reduces uncertainty” /squeak/
“averaging reduces uncertainty” /squeak/
“averaging reduces uncertainty” /squeak/
“averaging reduces uncertainty” /squeak/
“averaging reduces uncertainty” /squeak/
“averaging reduces uncertainty” /squeak/
…
Around and around it goes, its a perpetual motion machine!
Irony Alert. Stop whining. Squeak.
Have I got the full Karlo style of intelligent argument correctly?
hehehehehehehehehehe
Mockery is not your strongsuit.
That’s me told.
Unless actions are taken to calibrate prior to a measurement, then minimize systematic error, one can not know the value let alone knowing the sign.
I don’t understand how anyone who has ever made measurements for profit can possibly believe measurements of different things can be averaged and reduce uncertainty or error.
I’ll repeat what I have said before about measuring the eccentricity of an engine piston bore in several places and orientations and do the same for each bore. What mechanic would average those measurements and then say I know each with increased precision?
Geez, you could use old inside calipers, an old wood ruler, and multiple measurements to end up knowing the eccentricity to 0.00001 inches of uncertainty.
Uncertainties add, they add regardless of the type. I can’t measure a basketball with a micrometer and a baseball with a ruler, average them, and say the uncertainty of each measurement has been reduced by a factor of two just because I averaged them.
His agenda outweighs the truth, so he has to keep pushing the same nonsense over and over. Without tiny “uncertainty” numbers for his trends, he is bankrupt.
“Uncertainties add, they add regardless of the type. I can’t measure a basketball with a micrometer and a baseball with a ruler, average them, and say the uncertainty of each measurement has been reduced by a factor of two just because I averaged them.”
You can if you are a climate scientist or statistician!
“OMG! Where does the GUM say that? Reference please?”
Note to 3.3.3
E.3.6
Cherry picking—the GUM is trying to tell you that the “some publications” are old-fashioned.
But you just look for the bits that tell you averaging reduces certainty…
I’ll just go ahead and assume the E.3.6 quote is out-of-context, not even going to bother looking it up.
It is. I was reading this a few days ago. It is mentioned as reference in several places in sections way earlier. Guess what E.3 is about? “Justification for treating all uncertainty components equally. ” IOW, intervals within which the measurement is unknown, i.e.,uncertainty, whether random, systematic, Type A or Type B, a priori, statistical or varience of measurements, etc.
If you don’t like the answer you shouldn’t have asked the question.
Ah yes, of course. 30 years or so ago treating both equally as just uncertainty was a radical and heretical idea, especially to hardened statisticians. There is still today one standards organization (that I won’t name) that still adheres to “precision and bias”, and has never embraced the GUM. They have a whole committee dedicated to statistics that oversees their methods, but as time goes by it all becomes more and more esoteric and isolated.
Having been splashed in the face with the ice-cold water of reality, the trendologists have subtly modified their tune from “anomalies cancel all bias” to “throw enough different measurements together in a big pot and the biases become random and cancel.”
It is all nothing but vapor and handwaving.
“Cherry picking—the GUM is trying to tell you that the “some publications” are old-fashioned.”
I was asked politely (OMG! Where does the GUM say that? Reference please?), where Gum said under some circumstances systematic uncertainties may become random uncertainties. I provided the references, but of course you will now try to change the subject.
“not even going to bother looking it up.”
Of course you won;t. Why put your faith to the test?
See above, mr. experto.
Yep. And because it is astronomically unlikely that all stations have the exact same systematic error it is necessarily the case that when processed in the context of an average of many stations it becomes, at least partially, a random component.
See? I am right again, this is the new tune of the trendologists.
“astronomically unlikely”
Better put than my dated reference to chimps typing the encyclopedia Britannica….
Chimps typing an encyclopedia is good too. And more imaginative which counts for something.
“So what you should be looking at is 20C +/- 5C ==> 15C – 25C. The minimum temp would be 10C +/- 5C ==> 5C – 15C.
The interval in which the average might lie is 5C to 25C. The uncertainty has grown from +/- 5C to +/- 10C!”
I’m an idiot. Or wasn’t paying enough attention when answering this on my phone.
Tim’s calculations are completely wrong in any event.
Adding a positive and negative value can’t be bigger than adding two positive or two negative values. The average of the uncertainties still defines the range.
Max is in the range 15 – 25, Min 5 to 15.
Smallest sum is 15 + 5 = 20. Average is 10
Biggest sum is 25 + 15 = 40. Average is 20
Range of average is 10 – 20, not 5 to 25.
Uncertainty interval is still ±5.
And way way larger than ±0.01°C…
Well aren’t you a smart boy today. Yes, 5 is a bigger number than 0.01. It’s also bigger than 1. Can you find all the other numbers it’s bigger than?
Keep spouting these amazing facts and maybe nobody will notice Tim’s mistake in thinking that 5 + -5 is twice as big as 5 + 5.
“Guys, everybody seems to be talking past each other again and getting cranky as a result. That doesn’t really enhance understanding or improve the state of knowledge.”
True, and was ever thus.
“What are the uncertainty intervals each day for the max and min?”
Impossible to be know for sure without more information. If you can assume that all random errors are a lot smaller than the 1°C resolution, and that there are no systematic errors, and that there is no special reason for the actual temperature to favor a particular spot in the integer, then the obvious assumption would be that the rounding was the main source of uncertainty. In that case I would assume we are talking about a rectangular error distribution, with an interval of ±0.5°C.
What the actual value of the uncertainty is, will depend on how you are defining the uncertainty. You could define it in terms of the absolute range, in which case it would be ±0.5. If you want the standard uncertainty you would need the standard deviation of a rectangular distribution which I think would give you 1/√12 ~= ±0.289°C.
“What is the uncertainty interval each day for the median?”
Do you mean the median of the max and min values, or the actual daily median? And why talk about the median rather than the mean?
If you just want the mid point value of max and min, and assume that the two measurements are independent and no systematic errors, the standard uncertainty should be the uncertainty for max and min divided by √2 ~= ±0.204°C. A 95% confidence interval would require looking at the combined distribution of two rectangular distributions, which I think should be a triangular distribution, but I’ll have to check how that would affect the result.
“What are the uncertainty intervals for the monthly maximum and minimum?”
If you mean the actual warmest and coldest temperatures during the month, I’m guessing they would be the same as for any daily max and min. Maybe there’s some complicated reason why that wouldn’t be the case, but I can’t think of it at the moment.
“What are the uncertainty intervals for the monthly mean maximum and minimum?”
At the risk of bringing all this up again, it depends on what you actually want to measure. If you want an exact average, then the uncertainty depends on just the measurement uncertainty.
If you want to know the uncertainty in the average as a potential value (not sure if that’s the best wording) you can use the standard error of the mean, based on the standard deviation of the temperatures.
In either case the calculation is the same (again assuming no systematic errors), dividing the standard deviation by the root of the number of days, it’s just you are dividing a different thing in each case. One is the daily measurement uncertainty, the other is the deviation of all values.
(You could also argue that the uncertainty of a daily max and min is not the measurement uncertainty, but the deviation of all values across the month.)
It really depends on what and why you are measuring.
“and that there are no systematic errors”
hahahahahahahahaha
In what universe?
In the universe where that assumption holds. That’s the point of stating your assumptions. I’m not saying it is true, I’m trying to answer a hypothetical question which inevitably requires making simplifying assumptions. I’m sorry if these concepts tax your troll brain.
“tax your troll brain” — I really don’t care what you think of me (and I’m not going to give you any satisfaction by pontificating about my metrology qualifications).
“In the universe where that assumption holds. “ — the point that people have been to get you to acknowledge (and Stokes of course) is that this universe is the null set, It Does Not Exist. As much as you would like to, you cannot just ignore bias and sweep it under the carpet.
Experience is a hard taskmaster which has demonstrated to my satisfaction that attempting to give you clues is quite pointless.
And you call me a “troll”.
Next rant here…
Why didn’t you answer what the values are when there *are* systematic biases in the measurements?
Because it was already a very long set of answers to a set of hypothetical questions, and I hoped that for once the answers might lead to genuine discussions rather than the normal name calling.
Systematic errors are complicated, especially when you are giving no information apart from the claim that it has passed a calibration tests.
You didn’t even attempt to explain how you would deal with the systematic errors, and just claimed you should add the uncertainty of the max and min to get the uncertainty of the mean.
It was intentionally left as a black box, to see what different assumptions would be made.
Explicitly stating the assumptions is better than leaving them implicit. Thank you for providing detail to support your answers.
This is why calibration laboratories are required to perform formal uncertainty analyses for measurement values they report. See ISO 17025.
“no systematic errors,:”
“no systematic errors,”
“again assuming no systematic errors”
And what are the answers if there *ARE* systematic errors?
Are we back to “all measurement error is random, Gaussian, and cancels” again?
Yep, its a hamster wheel.
See what Old Cocky means by everyone being cranky.
You missed the context when I said “Impossible to be know for sure without more information.” If you can supply me with all the calibration details uncertainty specifications and details of the honesty of the observer, then maybe I’ll go into more detail.
Note, it’s mentioned in the description of the question that
So I think it might be reasonable to assume there are no serious systematic errors.
“Are we back to “all measurement error is random, Gaussian, and cancels” again?”
I specifically said the uncertainties would be rectangular and so NOT GAUSSIAN.
“So I think it might be reasonable to assume there are no serious systematic errors.”
And you would still be wrong.
“So I think it might be reasonable to assume there are no serious systematic errors.”
Therein lies the assumption of someone with no experience in the physical sciences at all.
Exactly what calibrations were done? Especially if they were done in the field. There aren’t many portable calibration labs going around to field temp measurement sites.
Get a grip.
Part 2:
“If you can assume that all random errors are a lot smaller than the 1°C resolution, and that there are no systematic errors, and that there is no special reason for the actual temperature to favor a particular spot in the integer, then the obvious assumption would be that the rounding was the main source of uncertainty.”
So what if you don;t make those assumptions. In particular, what would the answers be if there was some systematic error in one or both instruments?
That’s more difficult without knowing any details of this set up, and this is all outside any area of my expertise – I’m sure the experts will be quick to correct any mistakes I make. But, to me the first issue with systematic is what do you know and what can you assume.
There are two scenarios here. One you know there is a systematic error in your instrument (or from any other source). This might be obtained from calibration experiments, or other statistical analysis. In that case, you don’t actually have “uncertainty”. You know there is an error and the obvious approach to correct for it. This might be as simple as saying we know the instrument is always 1°C too warm, so just subtract 1 from all measurements. This does not mean you are not introducing more uncertainty into the result. You need to add an extra term into the uncertainty analysis, but that is covering the uncertainty in your adjustment, not the statistical uncertainty.
The bigger problem is when you have known unknowns. You know (or just guess) there is some systematic error, you just don’t know what it is. The obvious way of dealing with that is to include it as a dependent uncertainty in your calculations. This is what Frank does, I think. Assumes there is an unknown systematic error in all instruments, which might be somewhere in the interval of ±0.46°C. When you include this as a dependent uncertainty the result will be an uncertainty of ±0.46°C in the mean.
The big question I would have though is how you can know the extent of any systematic error without knowing what the actual error is?
“But, to me the first issue with systematic is what do you know and what can you assume.”
How do you know what the systematic error might be in a field measurement device?
“One you know there is a systematic error in your instrument (or from any other source). This might be obtained from calibration experiments, or other statistical analysis.”
How many times does it have to be pointed out to you that systematic bias cannot be analyzed statistically? Taylor says so. Bevington says so. Possolo says so.
Calibration only works in a controlled environment using standards whose own calibration can be traced. How many field measurement sites have calibration labs co-located?
“You know there is an error and the obvious approach to correct for it.”
The correction should be applied to the measuring device and not to the data. I know some people will modify their data but that is, in essence, creating an entirely new record. In the case of calibration drift or environmental changes you are faced with a changing systematic bias. How do you correct a set of measurements taken over a period time when you have a time-varying systematic bias?
“This does not mean you are not introducing more uncertainty into the result. “
Of course it means you are introducing more uncertainty. Anytime you fiddle with the stated value you are introducing more uncertainty. The proper way to handle this is to revise your uncertainty interval to make it larger. Remember (ẟ_total)^2 = (ẟ_random)^2 + (ẟ_sys)^2. How do you know what ẟ_total is when you don’t actually know one of the contributing factors? The answer is to calibrate the measuring device!
“ You need to add an extra term into the uncertainty analysis, but that is covering the uncertainty in your adjustment, not the statistical uncertainty.”
Huh? What is “statistical uncertainty”? If you don’t know both ẟ_random and ẟ_systematic how do you know what extra term to put in the uncertainty analysis?
“The bigger problem is when you have known unknowns. You know (or just guess) there is some systematic error, you just don’t know what it is”
Now you are getting there. How do you know what the systematic bias is in a field measuring device?
“This is what Frank does, I think. Assumes there is an unknown systematic error in all instruments, which might be somewhere in the interval of ±0.46°C. When you include this as a dependent uncertainty the result will be an uncertainty of ±0.46°C in the mean.”
Hallelujah! You are finally starting to get it. Don’t lose it!
When an instrument is specified as having an uncertainty of +/- 0.5C that includes both random and systematic uncertainties. You just don’t know the ratio that applies. If your measurements all have that +/- 0.5C uncertainty then that propagates onto the uncertainty of the mean. Since you are adding measurements of different things by different instruments the uncertainties ADD, they don’t average.
The average uncertainty is *NOT* the uncertainty of the average. Is that finally starting to make some sense?
It’s why when you have two boards b1 +/- u1 and b2 +/- u2, the uncertainty of the mean is u1 + u2, not (u1 + u2)/2. The uncertainty interval expands, it doesn’t contract. This applies to temperatures just as much as it applies to boards.
When you find a mid-range temperature (t1 + t2)/2 the uncertainty of the average becomes u1 + u2. The uncertainty range is from -u1 to +u2.
It’s why assuming that all measurement uncertainty is random, Gaussian, and cancels is just so wrong. But it *is* what climate science does.
“The big question I would have though is how you can know the extent of any systematic error without knowing what the actual error is?”
YOU DON’T! Not in a field measurement device. The best you can do is hope the total uncertainty given by the manufacturer is the appropriate interval to use. Either that or use your own educated judgement to pick a total value that makes sense. Picking a value of 0 for systematic bias is only fooling yourself. Assuming systematic bias can be identified statistically, at least in a field measurement device, is just wrong. It is why the total uncertainty *MUST* be appropriately propagated. Otherwise you are just committing a fraud on the users of your analysis.
Even calibration labs have all sorts of intervals that insert Type B values into their results and have to be counted—time, temperature, different operators, are typical. Even precision resistors have temperature coefficients and drift (i.e. time) specs that are usually given as plus or minus, you don’t even know the sign! All instrumentation, anything with a calibration sticker, has to be on a recalibration schedule—the longer the schedule, the larger the Type B interval must be.
If you go out and buy a nice RM Young temperature-humidity probe for your weather station, it comes with a set of nice, sexy error specs, but it doesn’t have a datalogger to digitize its outputs. You have to provide one for yourself. And it likely won’t be a $1200 Fluke 8808A DMM so its error specs won’t be as nice. You then get to figure out how your datalogger works and develop an uncertainty analysis for the numbers the RM Young probe will generate. And I can guarantee it won’t be ±10 milli-Kelvin!
“Assumes there is an unknown systematic error in all instruments,…”
Another obvious misrepresentation. My assessments strictly include the systematic error revealed by field calibration experiments. No assumptions at all.
One gets the distinct impression, yet again Bellman, that you criticize the work without having read it.
Are you surprised? He cherry picks from all kinds of sources without actually doing the homework, such as in Taylor and Bevington. Basically he’s a troll, trying to see how many clicks he can get.
That is true for Hubbard & Lin 2002. But you also include Folland et al. 2001 which is random error as shown in Brohan et al. 2006.
Except…the manner in which you combined the uncertainty is equivalent to assuming correlation r(xi, xj) = 1. See GUM equation 16 and especially note 1 underneath it. Alternatively Bevington 3.13 can be used and yields the same results. Whether you realized it or not you are declaring implicitly that all error from the Folland 2001 (σ = 0.2 C) and Hubbard 2002 (σ = 0.254 C) uncertainty distribution is exactly the same. Assuming for a moment that is true (it would be infinitesimally unlikely) then that means the baseline would be contaminated with the same error as the individual observation thus cancelling the error completely. Try it with GUM equation 16. Or you can use the NIST uncertainty machine since it allows entering the correlation matrix.
“My assessments strictly include the systematic error revealed by field calibration experiments.”
But you keep wrapping up all these uncertainties with the claim that this makes the uncertainty of the mean of any number of these errors will be the same as the average uncertainty. I don’t see how that can be if you are not assuming the same error for every station over any time period. That’s my problem with the analysis. Not that there is no systematic error, but that it is assumed to negate any division by sample size.
Hence you claim that the uncertainty over a 30 year period is identical to a 1 year period, is identical to a single station on a single day.
I’m still waiting for you to say if you agree or disagree with the claims made in this post. If you think surface data has an uncertainty of at least ±0.46°C, and if you think this isn’t reduced by looking at a multi year average, can the claim of an uncertainty for the 11 years of ±0.01°C be justified?
Just add more points, it goes down to ±0.000001°C.
Success achieved!
(and you used the illegal plus/minus sign, again)
So we are agreed. Scafetta and May are wrong.
“illegal plus/minus sign”
Clown or troll? You decide.
Here boy, have a bone!
/pats head/
I really should have used “midpoint” rather than “mrdian”, as you used later.
For each day, the midpoint of the max and min values.
Midpoint would have been better. I’ve been a bit gunshy of using the term “mean” since being roundly castigated for not using it in the one true meaning of “expected value” (probabilistic mean)
Yes
±0.5 C rectangular. The standard uncertainty is thus u = 1/sqrt(3) = 0.289 C.
Letting y = f(Tmin, Tmax) = (Tmin+Tmax)/2 and proceeding as Bevington 3.14 or GUM 10 we compute u(y) = sqrt[ (1/2*u(Tmin))^2 + (1/2*u(Tmax))^2 ] = 0.204 C.
±0.5 C rectangular. The standard uncertainty is thus u = 1/sqrt(3) = 0.289 C.
Letting y = f(Tobs1, … Tobs30) = Σ[Tobsi, 1, 30] / 30 and proceeding as Bevington 3.14 or GUM 10 we compute u(y) = sqrt[ 30 * (1/30*u(Tobs)) ] = 0.05 C given u(Tobs) = 0.289 C.
Letting y = f(Tmed1, … Tmed30) = Σ[Tmedi, 1, 30] / 30 and proceeding as Bevington 3.14 or GUM 10 we compute u(y) = sqrt[ 30 * (1/30*u(Tmed)) ] = 0.04 C given Tmed = 0.204 C.
All answers assume uncorrelated errors of the observations.
“±0.5 C rectangular.”
Why do you assume rectangular? If it is all random then it is probably Gaussian, measurements further from the mean grow less likely.
“Letting y = f(Tmin, Tmax) = (Tmin+Tmax)/2 and proceeding as Bevington 3.14 or GUM 10 we compute u(y) = sqrt[ (1/2*u(Tmin))^2 + (1/2*u(Tmax))^2 ] = 0.204 C.”
Nope. This is so simple I can’t believe you keep getting it wrong.
If Tmax is 60 +/- 1 and Tmin is 40 +/- 1 then the range of possible values is:
59 to 61 and 39 to 41. The average can range from (59+39)/2 = 49 to (61+41) = 51. That’s an uncertainty interval of 2. Your uncertainty interval has grown from 1 for each element to 2 for their combination!
The average uncertainty is *NOT* the uncertainty of the result, i.e. the average.
The average uncertainty is *NOT* the uncertainty of the average.
Your problem is that you think the uncertainty has to be divided by the number of terms. IT DOESN’T WORK THAT WAY!
It reads to the nearest 1 C.
Bevington 3.14, Taylor 3.47, and GUM 10 all say that is how you do it.
That’s a different scenario. I’m working off of what old cocky posted which stated that the instrument reads to the nearest 1 C. That means it is ±0.5 C rectangular.
Yeah. I know. The question was regarding the uncertainty of the uncertainty of the average u(Σ[xi]/N); not the average uncertainty Σ[u(xi)]/N. So I computed u(Σ[xi]/N) like what was requested.
I’m just following the procedure set by Bevington, Taylor, and the GUM. I also verified the results using the NIST uncertainty machine.
“It reads to the nearest 1 C.”
So what? That doesn’t mean its a rectangular distribution!
“Bevington 3.14, Taylor 3.47, and GUM 10 all say that is how you do it.”
You are misapplying *all* of these. You are no better than bellman. You haven’t worked out a single example in any of them, just cherry picked things you think might support your assertions – without really understanding what you are quoting!
As people keep trying to tell you the AVERAGE IS NOT A FUNCTION! A function relates one measurand to another. The average is *NOT* a measurand, it is a STATISTICAL DESCRIPTOR. Statistical descriptors are not functions.
” I’m working off of what old cocky posted which stated that the instrument reads to the nearest 1 C. That means it is ±0.5 C rectangular.”
Your lack of knowledge concerning physical science is showing again. If you are reading to the nearest 1C then you *still* don’t know what the underlying distribution of the uncertainty is. All you know is that your last significant digit is in the units digit. The tenths digit could be anywhere from 0.1 to 0.4 or 0.11 to 0.49. Or it could be 0.5 to 0.9 or 0.51 to 0.99.
YOU DON’T KNOW WHERE IN THOSE INTERVALS THE TRUE VALUE LIES. Not every value has to have the same probability.
You *still* don’t have the concept of measurement uncertainty straight!
“I’m just following the procedure set by Bevington, Taylor, and the GUM. I also verified the results using the NIST uncertainty machine.”
You are finding the interval in which the population average lies, not the uncertainty of the average. Until you get that straight you don’t have a clue as to what any of these people are doing!
Sure it does. See JCGM 100:2008 F.2.2.1.
Sure it is. y = f(x1, …xN) = Σ[xi, 1, N] / N. See JCGM 200:2012 2.48 and 2.49.
Sure it is. See JCGM 200:2012 2.3
Yep. And don’t forget about 0.0 to 0.1. All with equal probability. That makes it rectangular. See JCGM 100:2008 F.2.2.1.
Yep. That’s why it is a component of uncertainty. See JCGM 200:2012 2.26.
I’m using the definition from JCGM 200:2012 2.26.
I’m finding u(y) where y = y = f(x1, …xN) = Σ[xi, 1, N] / N. y is the average. u(y) is the uncertainty of that average. See JCGM 100:2008 section 5.
cherry picking again
F.2.2.1 => The resolution of a digital indication
Like all measurements are based on a device with a digital indication!
Cherry picking again. 200:2102 2.48 and 2.49 never mention the average as being a measurand.
Measurands EXIST. There is no guarantee than an average eixsts anywhere except in a statisticians mind.
I’m not even going to bother with the rest. You are cherry picking crap hoping someone will buy it. Peddle it elsewhere!
It also does not mention voltage as being a measurand. Does that mean voltage is not a measurand?
Funny because NIST TN 1900 E2 says in no uncertain terms (the pun was intended) that the measurand is the mean in that example.
I’m not sure it’s necessarily rectangular, because I specified separate max and min thermometers.
In any case, I really only wanted to tease out underlying assumptions, not open up a new battlefront.
It was meant to be “Oh, so that’s where you’re coming from”.
The right question at the wrong time, perhaps 🙁
The thermometers being separate is irrelevant. If they report to the nearest to the integer than the uncertainty interval is ±0.5 C with the standard uncertainty being u = 0.5/sqrt(3) = 0.289 C. This is true for both thermometers. This is documented in JCGM 100:2008 F.2.2.1.
But is it? I can see that the ui for each thermometer will be rectangular, but is the combination still rectangular, and is it the same as if a single minimax thermometer was used?
My suspicion is that each thermometer may have a different systematic error.
I’m not claiming to be any sort of expert, but sometimes that different perspective can help.
If you’re talking about a combination of the two like the average (Tmin+Tmax)/2 then that ends up being triangular with u = 0.204 C. The more you combine the more gaussian it becomes. After combing 30 Tmin and 30 Tmax in this manner it becomes indistinguishable from a normal distribution with u = 0.037 C. But individually the observations are rectangular regardless of whether by one or two instruments.
You are correct; the 0.29°C value for the integer resolution limit is only one component of the overall uncertainty. Other sources also must be quantified and included in the combined temperature uncertainty for that instrument.
“I also verified the results using the NIST uncertainty machine.”
Well pin a bright shiny star on your vest!
Isn’t the average 50 +/- 1? The interval appears to be the same as those for the min and max.
What if those uncertainties are *not* the same? What does the uncertainty interval become?
Less than one.
As far as I can tell:
If Tmax is 60 +/- 4 and Tmin is 40 +/- 2 then the range of possible values is:
56 to 64 and 38 to 42. The average can range from (56+38)/2 = 47 to (64+42) = 53. That’s an uncertainty interval of 6.
In this case, the interval surrounding the mean (53 – 47) is the same as the sum of (64 – 60) + (40 – 38).
You could say that the interval around the mean is the mean of the intervals around the max and min.
Assuming we’re still talking about instruments that read to the nearest integer it’s actually 40 ± 0.5 (rectangular) ± 0.289 (standard) and 60 ± 0.5 (rectangular) ± 0.289 (standard) with the mean being 50 ± (triangular) ± 0.204 (standard). This is easily proven with the NIST uncertainty machine.
There are too many things going on to keep track of them all. Numbering of comments or better indentation levels would reduce the potential confusion a bit.
Somewhere along the line, we got a new example wth a lower bound range of 39 to 41, and an upper bound range of 59 to 61.
This was just working from the extremes.
Why are thou using the illegal plus/minus sign, thou hypocrite?
“Guys, everybody seems to be talking past each other again and getting cranky as a result. That doesn’t really enhance understanding or improve the state of knowledge.”
OC, this conversation has been going around the circle for a good 5 years. It’s always the same.
bdgwx and Bellman (and others) raise the same dead-letter questions every time the subject comes up. Explanations are given. Tim and Jim Gorman have exhausted themselves in the effort.
But it never makes an impression. Leave them unchallenged, and they’ll claim victory. And that’s their hope.
First, I’m asking new questions. This is the first I heard that your sqrt(N*σ/(N-1)) came from Bevington 4.22. Second, Now I’m following up with your claim that 4.22 is for systematic error and 4.23 is for random error. And I’m asking because I can’t find anywhere that Bevington says 4.22 is used systematic uncertainty or even that it can even be used for the uncertainty of the average at all. You’re explanations so far have been lacking.
Tim/Jim made 23 algebra mistakes in a single derivation alone. Now some of these mistakes are difficult to spot like calculating partial derivatives incorrectly. But some of these mistakes include conflating sums (Σ[xi]) with averages (Σ[xi]/N) and conflating the quotient operator (/) with an addition operator (+). More mistakes are then made defending the previous mistakes. So if you’re position aligns with theirs then you might want to seriously reconsider your position.
I know. It’s interesting, technical, and frustrating.
That’s why I’m trying to get down to the core aspects to attempt to find where the assumptions differ.
Thanks for the responses.
I’ll try to clarify questions asked in those responses as I come to them.
I see you *still* don’t get it!
Bevington is finding the standard deviation associated with the variance of the stated values around the population mean! That is simply not the same thing as the uncertainty of the population mean.
You keep denying that you assume that all measurement uncertainty is random, Gaussian, and cancels leaving the stated values by themselves to be analyzed. Yet that is *EXACTLY* what this section is Bevington has assumed. Bevington explicitly states that in the very first part of his book! And yet, for some reason, you keep on ignoring that basic fact.
Not only that but this only applies when you are MEASURING THE SAME THING. Look at the example following. It is using 40 measurements of the SAME THING coupled with 10 additional measurements of the SAME THING using a better device!
This simply does not apply when it comes to averaging temperatures to create a global average temperature. It doesn’t even apply when you are calculating a baseline average temp at the same location over a number of years!
If you follow the assumptions Possolo made in TN1900:
then you can justify the method he used to calculate a monthly average Tmax. These assumptions simply can’t be justified for using measurements taken over a span of years. Too many outside confounding factors impact the monthly averages of a time span of years to make the assumption you are measuring the same thing multiple times.
Taylor, in his Principal Definitions and Equations of Chapter 4 at the end of Chap. 4 states:
“The average uncertainty of the individual measurements x1, x2, …, xN is given by the standard deviation ….”
σ_x = sqrt[ (1/N-1) Σ(x_i – ẋ)^2 ]
Note carefully that this is not the same thing as x1 ± u1, x2 ± u2, …., xN ± uN.
Taylor goes on to state: “As long as systematic uncertainties are negligible, the uncertainty in our best estimate for x (namely ẋ) is the standard deviation of the mean, or SDOM.”
“If there are appreciable systematic errors, then σ_ ẋ gives the random component of the uncertainty in our best estimate for x”
“If you have some way to estimate the systematic component”
ẟx_tot = sqrt[ ẟx_ran^2 + ẟx_sys^2 ]
Why does everyone trying to justify the “average global temperature) insist on ignoring ẟx_sys?
“I see you *still* don’t get it!”
“Bevington is finding the standard deviation associated with the variance of the stated values around the population mean! That is simply not the same thing as the uncertainty of the population mean.”
What bit of that do you think I don’t get. It’s exactly what I and bdgwx have been trying to explain to Pat Frank.
“You keep denying that you assume that all measurement uncertainty is random, Gaussian, and cancels leaving the stated values by themselves to be analyzed.”
And I will keep denying it – because as you must realize by now it’s a pack of lies.
“Yet that is *EXACTLY* what this section is Bevington has assumed.”
Take it up with Pat Frank. He’s the only one quoting the equation to justify his own analysis.
“Not only that but this only applies when you are MEASURING THE SAME THING.”
Point this put to Pat Frank.
“This simply does not apply when it comes to averaging temperatures to create a global average temperature.”
It’s almost as if you think Pat Frank was wrong to use that equation as justification.
“What bit of that do you think I don’t get. It’s exactly what I and bdgwx have been trying to explain to Pat Frank.”
The fact that you always assume there is no systematic bias in measurements – so you don’t have to address the real world, hard issues with the average global temperature.
“And I will keep denying it – because as you must realize by now it’s a pack of lies.”
Malarky! In your answer to old cocky you never once attempted to discuss how to handle measurement uncertainty with systematic bias.
—————————
“no systematic errors,:”
“no systematic errors,”
“again assuming no systematic errors”
—————————
Not one single word about handling systematic bias which *always* exists. You just always assume – “all measurement uncertainty is random, Gaussian, and cancels.
You can’t get away from it. It’s just built into your world view and you can’t let it go.
“Take it up with Pat Frank. He’s the only one quoting the equation to justify his own analysis.”
Yet *you* are the one that quoted Bevington. A typical cherry-pick.
I have no problem with anything Pat has done.
“The fact that you always assume there is no systematic bias in measurements”
More lies. I’ve discussed systematic error many many times. I do not always assume there are no systematic errors. Assuming something for the sake of a particular argument, does not mean a you think they do not exist. You were the one who bought up the claim that the sum of 100 thermometers, each with random independent uncertainties of ±0.5°C, would have a combined uncertainty of ±5°C. That’s you assuming there are no systematic errors. It’s a useful thing to do when discussing how random errors work. It does not mean you were claiming systematic errors can never exist.
“More lies. I’ve discussed systematic error many many times.”
The issue is that you NEVER USE IT! You always assume that measurement uncertainty is random, Gaussian, and cancels – in essence that no measurement uncertainty exists.
“That’s you assuming there are no systematic errors.”
You can’t even understand that the total uncertainty +/- 0.5C includes BOTH ẟ_random and ẟ_systematic!
“It’s a useful thing to do when discussing how random errors work.”
Useful for what? The real world? How can that be when the real world *always* includes systematic biases. Climate science is the REAL WORLD, not statistics blackboard world.
Then you used the wrong equation. 4.22 is an intermediate step when using relative (as opposed to absolute) uncertainties. Bevington tells you that you have to take that result and divide by N to get the variance of the mean. See equation 4.23. And if there is any question on what to do then example 4.2 should drive the point home when Bevington says “To find the error in the mean the student could calculate σ from her data by Equation 4.22 and use Equation 4.23 to estimate σ_u”. Note that σ_u is the uncertainty of the mean and must be computed via 4.23. That step is not optional.
The issue here is that you are STILL trying to say that how close you get to the population mean is a measure of the uncertainty of that population mean. IT IS NOT. The standard deviation of the sample means can be ZERO, i.e. equal to the population mean, while the population mean might be as inaccurate as all git-out because of the measurement uncertainties associated with the data elements themselves!
If the population mean is 5 but the true value is 10 it says the population mean is wildly inaccurate due to measurement uncertainty. The standard deviation of the sample means may come out to 5 +/- .000001 AND WILL TELL YOU NOTHING ABOUT HOW ACCURATE THE POPULATION MEAN IS.
It is the accuracy of the population mean that is important. If a temp measuring station gives temperatures with an uncertainty of +/- 0.5C then it doesn’t matter how temp data values you collect in order to calculate an estimate of the population mean, that population mean will *always* have an measurement uncertainty of at least +/- 0.5C. It can’t be anything less than that!
You keep wanting to use the assumption that all measurement uncertainty is random, Gaussian, and cancels leaving nothing but 100% accurate stated values.
You and Bellman deny that you do this but it comes through in every possible assertion you make.
If you actually go through Bevington as a learning exercise instead of cherry picking bits and pieces you will find that his analyses are done assuming NO SYSTEMATIC BIAS in the data, only random error. He specifically talks about this in Chapter 1.
This simply doesn’t apply to the development of a global average value using measurement devices with systematic bias.
You are blinded by your lack of knowledge of metrology and measurement uncertainty. You were taught in an environment of all statistical data being nothing but stated values in a population, with *NO* uncertainty. Therefore the population mean becomes 100% accurate and how close you get to the population mean, i.e. the SEM, is a measure of the uncertainty you have about the mean of the sample means.
It’s the blind leading the blind in climate science. Statisticians that know nothing of the real world and scientists that can’t properly judge the value of statistical descriptors. I would trust anyone in climate science to design a bridge span, be they a statistician or a scientist. It would wind up 2feet short or 2ft long at the end of the span!
Not even close. I made it clear what the issue is. The issue is that Pat Frank used equation 4.22 to assess the uncertainty of the average. Bevington says that 4.22 is but an intermediate step and that you must divide that result by N as was done in 4.23. Frank then claims that 4.22 is the equation to use for systematic error and 4.23 is the equation to use for random error. No justification has been provided of Frank’s claim. I’ve searched the Bevington text and find no such justification. In fact, all I see is the opposite.
Which is beyond ironic since Pat Frank is telling me that his paper is focused on systematic error and that’s why he chose equation 4.22 instead of 4.23.
“Bevington says that 4.22 is but an intermediate step ...”
He says that nowhere. 4.22 is “the average variance of the data.”
“...and that you must divide that result by N as was done in 4.23.”
The variance of the mean when all measurement error is random.
F. Back of the class, bdgwx.
Except, of course, when he says “The variance of the mean can then be determined by substituting the expression σ^2 from Equation (4.22) into Equation (4.14): σ_u^2 = σ^2/N (4.23)” and when he says “To find the error in the mean the student could calculate σ from her data by Equation (4.22) and use Equation (4.23) to estimate σ_u.”
It’s actually the “weighted average variance of the data”.
‘It’s actually the “weighted average variance of the data”.’
bellman fell into the same trap, so it may be lack of clarity in Bevington.
Bevington is dealing with random error. His 4.23 is valid only for random error.
My paper deals with systematic error. It’s not random.
I’ve explained this fundamental categorical unbridgeable impenetrable distinction to you ad nauseam.
And you still don’t get it.
This is strange logic. You accept that 4.23 is for a random variable yet believe 4.22, which 4.23 is dependent upon, somehow isn’t. How does dividing 4.22 by N suddenly make it random? And why are you using 4.22, which is only the sample variance, as if it were the variance of the mean anyway?
Your hubris seems to be boundless.
Your wrongness is refractory, bdgwx. There’s no hubris in remarking that patient and superabundant repetition has got nowhere with you. It’s merely the truth.
You’re wrong and will evidently never figure that out.
Just to add, dividing by N does not make 4.22 random. Pre-existent knowledge that the error is random justifies use of 4.23.
When error is not random, one stops at 4.22, because in that case dividing the variance by N cannot be statistically derived.
Treatment of systematic error cannot be statistically derived, period. One is left with approximate methods of estimating uncertainty. One engages the part of statistics useful to that estimate.
Where in 4.22 do you see measurement uncertainty mentioned?
x_i is a stated value of a measurement. It is not an uncertainty value. u’ is the average of stated measurement values, not of the measurement uncertainties. N is the number of stated measurement values.
Endemic in all this is the assumption that all measurement uncertainty is random, Gaussian, and cancels leaving on the stated values to be analyzed.
In other words you are assuming that all temperature measurement stated values are 100% accurate. And then you base the rest of your analysis on that ssumption.
Why do you think that there is measurement uncertainty that must be propagated throughout this entire process?
I suggest you read Pg 3 in Bevington’s book FOR UNDERSTANDING where he discusses systematic and random errors.
Bevington states: (re systematic error) “Errors of this type are not easy to detect and not easily studied by statistical analysis.”
You and your climate science buddies try to get around this by assuming there is no systematic bias in any temperature measurement. It’s a farce from the word go.
Nowhere. Perhaps you can explain it to Pat Frank. He’s not listening to Bellman or me on this matter.
He never will.
Eq. 4.22 is used to find the standard deviation of a set of measurement errors. Use of weights governs fractional contributions, not relative versus absolute.
Once again, Bevington is concerned with random error. Random error is the only sort of error to which 4.23 applies.
My analysis concerns systematic error. It’s not random error. Eq. 4.23 does not apply. Applying 4.23 to systematic error is wrong. You’re wrong.
You’re like a stuck phonograph record, bdgwx, playing the same BS groove over and over again.
Bevington says 4.22 and 4.23 are used (in combination) to determine the combined uncertainty when the absolute uncertainty of the measurements are not known. See example 4.2. I’m not saying 4.22 cannot be used for other purposes, but that was the clearly the intent of section. And the section is titled “Relative Uncertainties” afterall. Of course that detail isn’t all that important in this context of this discussion. I’m just saying…
Why? Where does Bevington say 4.22 is the uncertainty of the mean for error that is not random?
How can you be so dense?
The example shows how to combine measurement uncertainty, not calculate the average uncertainty.
I know.
And for the umpteenth time…we’re all discussing the uncertainty of the average u(Σ[xi]/N). Nobody, except apparently you and Jim, cares about the average uncertainty Σ[u(xi)]/N.
If you would just READ Bevington instead of trying to cherry pick bits and pieces you would understand that Chapter 4 is concerned only with stated values assumed to be 100% accurate, i.e. no systematic uncertainty.
Page 53: “We have assumed that all data points x_i were drawn from the same parent distribution and were thus obtained with an uncertainty characterized by the same standard deviation σ.”
“Each of these data points contributes to the determination of mean u’ and therefore each data point contributes some uncertainty to the determination of final results. A histogram of our data points would follow the Gaussian shape, peaking at the value u’ and exhibiting a width corresponding to the standard deviation ±σ.”
If you read on in the text you will see that he uses (∂u/x). Not one single mention of systematic uncertainty anywhere. NONE. His analysis is based on there being random error only, and that random error is defined by the variation in the stated values and not in the uncertainty intervals associated with the measurements.
On Page 54, Bevington states: “We obtain ….. for the estimated error in the mean σ_u. Thus, the standard deviation of or determination of the mean u’ and therefore, the precicision of our estimate of the quantity u, improves as the square root of the number of measurements.”
If it isn’t obvious to you from these kinds of statements that Possolo is calculating how closely you can get to the population mean from your sampling of the parent distribution then you are being deliberately obtuse and willfully ignorant.
How close you get to the population mean is USELESS if there is systematic bias in the measurements since the accuracy of the population mean will be compromised.
You are trying to substitute the SEM for the accuracy of the population mean. THEY ARE NOT THE SAME THING!
Climate science, and that includes you, lives and dies by the assumption that all measurement uncertainty is random, Gaussian, and cancels thus allowing the variation of the stated values to determine the accuracy of the mean.
It is statistically wrong. Period. Exclamation Point. The use of the term “uncertainty of the mean” has led far too many statisticians down the primrose path to perdition. It isn’t the uncertainty of the mean. It is the standard deviation of the sample means. It only tells you how close you are to the population mean. IT DOES NOT TELL YOU THE ACCURACY OF THAT POPULATION MEAN.
If the accuracy of the population mean has a large uncertainty then trying to identify ever smaller increments of differences in different population means winds up getting subsumed into the uncertainty interval and you can never know if that difference actually exists or not.
I know. Perhaps you can explain it to Pat Frank. He’s not listening to Bellman or me on this matter.
They don’t care—they truly have no interest. All they care about are seeing small U numbers, which is why bellcurve goes ape when I repost the UAH averages with more realistic Us.
4.22 calculates an empirical SD for any set of data.
To be pedantic it calculates the sample variance which can be more easily be done via 1.9 if the weights are all the same.
Anyway, what does that have to do with systematic error? And why do you think that represents the variance of the average?
Bevington’s “uncertainty of the mean” is still nothing more than the interval in which the population mean might lie. It is *NOT* the uncertainty of the population mean!
You can calculate the interval in which the population average might lie down the .000001, i.e. almost virtually certain of what the population average is, AND STILL NOT KNOW THE ACCURACY OF THE MEAN ITSELF!
The population mean can be 10 while the true value is 15, i.e. a huge uncertainty interval. You can get as close to 5 as you want with big samples but it won’t lessen the actual uncertainty associated with that mean!
I’ve done no cheering.
Willis,
No one is arguing that the ERA5 values have no uncertainty. But the measurement, coverage, and seasonal bias uncertainty studies done all agree that the yearly values are in the 0.02 to 0.04 range as described in the post around Figure 6. As already explained ad nauseum in the comments, natural variability is irrelevant in this test because the models assume that natural variability is zero.
Since clearly natural variability is a factor over these two decades, we know the model assumptions are incorrect and that Schmidt’s calculation of error from total standard deviation is also incorrect. Another source of error must be used to check the model calculations and Scafetta used HadCRUT5 errors. It is a good choice in my opinion.
It’s because natural variation does not factor into modeled ECS but does factor into the observation that you must treat it as a component of uncertainty on the observation side.
FWIW I think Schmidt’s method for assessing uncertainty, while easy to implement, does overestimate the natural variation component a bit. I did a quick test earlier today with actual quantifications of the effects of ENSO, AMO, volcanic, and solar variations and I was getting values less than ±0.1 C, but I hesitate to post it right now as I want to double and triple check things first. Unfortunately I’m going to be busy the next two days so it might be a few days before I can post it.
I think the best source on this is the paper by Morice et al on HADCRUT. The main source of error in average anomaly is not natural variation, or measurement error, but spatial sampling error.
Yeah, I have that paper in my stash.
But I don’t think it is relevant. The reason is because HadCRUT (and other datasets) are only focused on the uncertainty of their products. These products are designed to be accurate enough to see the natural variation. That’s why we see upticks during El Nino and downticks during La Nina. We don’t want their uncertainty assessments to include natural variation because natural variation does not (at least not significantly) impact the quality of their product.
The reason why natural variation is important for Schmidt though is because he is doing a comparison of the global average temperature (GAT), which is affected by natural variation, with the ECS, which is not affected by natural variation. There will be some noise in the observed GAT due to the inherent uncertainty of the dataset. But there will also be noise in the true GAT (and thus observed GAT) due to natural variation. If you don’t account for the fact that the GAT wiggles due to natural variation then you may draw the wrong conclusion when comparing the ECS to it.
I agree that a different uncertainty is needed. Normally you want to know how the temperature might have been if measured differently, especially if you sampled different places. But to test the difference with a GCM result, you need also to include how the temperature might be different if there had been different weather, eg ENSO. That is because models have weather, but it is not synchronised with that on Earth. If you can say that neither sampling nor weather could account for the difference, then you can say it is significant.
Try tea leaves.
The main source of natural variation should be the variance as calculated in TN 1900. When subtracting two random variables, the variances add. That should be how the variance of anomalies should be calculated.
If you do incorporate it into the observation side, it invalidates the ECS calculation, and you get the same result.
Let me see if I can explain it generally.
Consider a hypothetical cyclic process such that y = v*sin(x). Over large domains of x it is the case that avg(y) = 0. The function y does not increase due to the cyclic v*sin(x) term, but it does exhibit a lot of variation.
Now consider that a forced change term is added to the function y such that y = b*x + v*sin(x). The function y now exhibits an increase due to the forced change term. In addition it is still exhibiting a lot of variation.
At some point a clever scientist comes along as says I can predict the process and determines that it behaves like y = b*x. He knows that the predictions aren’t exactly right because of variation he has not modeled. Specifically the variation is large with v = 30 but he doesn’t fully understand that. Nevertheless he estimates the forced change term as having b = 3.
Another scientists comes along and tests the model and the estimation of b. He has collected data from x = 1 to x = 7. At x = 1 y = 28.2. At x =7 y = 40.7. That gives a slope of b = (40.7 – 28.2) / (7 – 1) = 2.1. Because the observations suggest b = 2.1 this scientist says the other overestimated b. The problem…this scientist didn’t consider the uncertainty that the 30*sin(x) variation term was causing.
With v = 30 the variation causes an uncertainty in the function y of ±41.6 (95% CI) meaning that the observation evidence of b has a rather large uncertainty of ±1.9. The observation should have been qualified as b = 2.1 ± 1.9 meaning that it is consistent with the original estimate of b = 3. BTW…I actually did the type A evaluation of uncertainty here so these are correct values.
Think of b in this case as the ECS for the climate. Just the like the scientist who estimated b = 3 without consideration of variation so too is ECS considered without variation. In both case the variation nets to zero over long periods so does not impact the long term evolution of the system. And in both cases observational tests of the predicted variable must be considered in the context of variation that is actually occurring.
Remember the statistics term “IID”.
Independent and identically distributed random variables.
Exercises like division by sqrtN for error estimates require IID.
The numbers being analysed by Schmidt and Scafetta are NOT IID.
The whole premise of the paper is wrong, surely?
Geoff S
Which paper? Gavin didn’t do that. The only time he mentioned sqrt(N-1) was in taking the mean over different models.
Yep, both results look way too small.
There seems to be some overthinking going on, mucking about with decadal averages.
Surely this is a matter of testing whether sets of time series differ at a selected significance level. If they differ, then you start looking at whether the lines are fuzzy.
There’s also the question as to whether to compare model run by model run, or consolidate the runs. Ideally, both.
A quick visual check says that pre-1995 is too good a fit in all the model runs, so discard that. 2000 might be a better starting point than 1995, which was a peak in the observations.
Post 2000:
1.8 < ECS <= 3.0 is flatter than ERA5, but probably not significant.
3.0 < ECS <= 4.5 is near enough the same slope.
4.5 < ECS <= 6.0 is steeper than ERA5, and it’s probably significant.
That’s where we get into fuzzing the ERA5 line.
Sorry, feeling a little unwell today, so a bit iconoclastic and argumentative.
Sigh – not only out of sorts, but more typos than usual 🙁
After letting it percolate for a while.
1/ Comparing 11-year periods is still an odd thing to do.
2/ 2011-2021 has the 2016-17 el nino, which pushes observations up. If you’re going to cherry-pick, use 2000-2010
3/ Above an ECS of 3.5, around 2/3 of the model run temperatures are above the ERA5 upper 95% bound. A 90% CI makes a little difference.
5/ Using bookend 11-year averages is an odd way to compare noisy time series.
“2011-2021 has the 2016-17 el nino, which pushes observations up.”
But also a lot of La Niñas. Looking at the average ENSO value over that period its slightly negative.
Don’t tell CMoB 🙂
Do you reckon the 2011-2021 slope is lower than 2000-2021. It looked slightly higher by eye.
Yeah, it has las nina at the start and end, so ENSO could well be negative. The short-term ENSO temperature effect seems biased towards el nino rises giving a saw-tooth effect. Again, just visual.
I just did a type A evaluation of ERA wrt to the mean of HadCRUT, GISTEMP, and BEST given an 11yr period. The result was ±0.017 C (95% CI). This would include correlation effects so it is expected to be a bit higher than the type B method Scafetta did since he assumed the years were uncorrelated. Note that the equation in figure 4 is just a special case of JCGM 100:2008 equation (13) with the correlation matrix set to r(x_i, x_j) = 0. So yeah, Scafetta could be underestimating the measurement uncertainty a bit, but it’s not a lot.
17 milli-Kelvin — great work!
According to Andy, Scafetta is claiming 10 milli-kelvin.
And just as absurd.
Why does the trendology tribe never report values such standard deviations or even numbers of points of all the averaging of averaged temperature average gyrations?
When this kind of issue comes up I am always reminded of the quote attributed to Ernest Rutherford, “If your experiment needs statistics, you ought to have done a better experiment.”
sheesh- it’s like the Medieval scholars arguing over how many angels can dance on the head of a pin.
Regarding “If we assume that Scafetta’s estimate correct, figures 1 and 2 show that all climate model simulations (the green dots in figure 2) for the 21 climate models with ECS >3°C and the great majority of their simulation members (the black dots) are obviously too warm at a statistically significant level. Whereas, assuming Schmidt’s estimate correct, figure 2 suggests that three climate models with ECS>3°C partially fall within the ERA5 margin of error while the other 18 climate models run too hot.”: Doesn’t this mean about 85% agreement between results using Scarfetta’s error estimate and Schmidt’s error estimate? So, why a headline and general tone of the article saying this is a big difference?
Another thing: I want an explanation for statement here of 21 models with ECS >3°C while Figure 1 indicates the number of models with ECS >3°C is 25.
Something else: This article indicates that ECS is actually in the range of 1.8 to 3 °C (apparently per 2xCO2). Don’t the main article writers at WUWT usually claim it’s much less? ECS much above 1.1°C per 2xCO2 requires net positive feedback.
This article only addresses the model-calculated ECS values listed in Scafetta’s paper, see his Table 1. There are 38 of them. Quite a few result in an ECS over 3, but I didn’t count them.
If climate sensitivity to CO2 is computed with observational data, as opposed to model results, the values are almost all less than 3, some are less than 1.
Climate Sensitivity to CO2, what do we know? Part 1. – Andy May Petrophysicist
Remember ECS is a totally model-based number that cannot be measured in the real world. The real-world values are best approximated by the model variable called TCR, or the transient climate response. TCR is much lower than ECS.
Andy May April 13, 2023 6:46 pm
What autocorrelation does to uncertainty depends on the uncertainty you are talking about. If it is the uncertainty of the next value in a series, yes, that is reduced.
But the uncertainty of say the mean of a series is increased by autocorrelation, sometimes very greatly.
For example, the post-1950 HadCRUT5 temperature with seasonal variations removed has a high Hurst exponent (large autocorrelation).
If we don’t correct for autocorrelation, the standard error of the mean (SEM) of that series is 0.01.
But once we adjust for autocorrelation, the SEM is 0.11, more than ten times as much.
w.
Since the uncertainty in temperature measurement, and the uncertainty of many iterative steps of a climate model that in any way uses any physical weather measurement, is so much greater than the claimed differences in results, how can this be anything except a theological dispute? Sure, there are different views about what are proper statistical calculations, but how can the numbers they are calculating upon have any real world meaning?
I apologize if this has already been beaten to death in the comments but while the scientifically oriented articles are generally so much more interesting than the politically oriented ones, given the concerns I outlined above, I can’t see plowing through all the arguments about the details. I will follow up to see if anyone has addressed my concern.
When calculating anomalies, absolute temps are used.
Let’s look at 15 ± 0.01. That is an interval of 14.99 – 15.01. ROFL!
Let’s look at an anomaly of 2.5 ± 0.01.
That’s an interval of 2.49 – 2.51. ROFLMAO!
Even NOAA shows CRN stations at ±0.3. And somehow through the magic of averaging we end up knowing the final temperature with an uncertainty of 0.01?
Does anyone stand back and look at these and say hold on common sense says playing with statistics is giving unreasonable answers.
Come on, some of you lab rats. If you had a voltmeter or balance beam scale that was only accurate to 0.3 volts or grams. Would your boss or customer or lab professor let you claim uncertainty of 0.01 in an average of 1000 different voltage sources or rocks?
Unable to formulate a rational response, the trendologists instead reach for the downvote button.
Measurement uncertainties add, the ALWAYS add. The only way to reduce uncertainty is to use better equipment. Unless you are a climate scientist where assuming there is no measurement uncertainty is accepted or where averaging uncertainty away is accepted.
If you can’t measure the temperature to an accuracy of better than +/- 0.3C then how can you possibly know that an average has less uncertainty than that?
All you can know is how closely you have come to the population average – but you will *never* know if that population average is accurate or not!
That global average can bounce around inside that +/- 0.3C uncertainty for infinite time and you’ll never be able to be certain about what is actually going on!
They will never acknowledge this because it would collapse their entire act.
“Schmidt’s calculation of the standard deviation of the mean (SDOM) is based on the erroneous premise that he is making multiple measurements of the same thing”
Andy,
This gets really silly. You say that Schmidt in Fig 3 has it wrong, but Scafetta in Fig 4 has it right. But they are the same, with Schmidt making one minor and correct refinement. Here is Schmidt’s:
Ringed in red is the calculation of sd of items, and in blue the factor to convert to standard error of the mean. Schmidt correctly divides by sqrt(N-1) to correct for a missing degree of freedom in the mean.
And here is Scafetta’s
The only difference is that he doesn’t correct for d.o.f. – a minor error.
It is also odd that this article makes a fuss of Scafetta better handling the uncertainty because, as is claimed anyway, he did not assume independence of the years and instead assumed there was autocorrelation. Yet bizarrely he still used the formula σ/√N which assumes independence or uncorrelated inputs. The formula he should have used if he wanted to account for the correlation is GUM equation (16) with the appropriate correlation matrix r(x_i, x_j) where at least some of the r(x_i, x_j) values are non-zero.
“Ringed in red is the calculation of sd of items, and in blue the factor to convert to standard error of the mean.”
Slight correction, I think it’s the first 1/√N, that is the conversion to the standard error. The √(N – 1) is part of the SD calculation.
“The only difference is that he doesn’t correct for d.o.f. – a minor error.”
I don’t think that’s the difference. Schmidt is taking the standard deviation of the annual anomalies, where as Scafetta is using the measurement uncertainties of each annual anomaly. At least that’s what May’s spreadsheet appears to show.
I guess it could be interpreted different ways. The population σ divides by sqrt(N) whereas sample σ divides by sqrt(N-1). Then when averaging 1/sqrt(N) would be with N dof and 1/sqrt(N-1) would be with N-1 dof. So I guess you could say it is either u(avg) via σ_population and dof(N-1) or via σ_sample and dof(N).
Nick, you are supposed to be a mathematician.
Both formula’s shown assume that the averages being dealt with are a “population”. IOW, the entire population of all temperatures.
That is the only reason you would be dividing σ by “N” or “N-1” to calculate the SEM. The formula for the relationship is {SEM = σ/√n}
When you have the entire population the SEM is useless. The mean µ does not have to be estimated. It is what it is, and the SEM won’t make the calculation of µ any more accurate. The calculation of σ using the traditional formula will provide the dispersion of the values surrounding the mean. The following document is pertinent.
Everything I read, you have said, and other references refer to station averages as “samples”. The standard deviation of a mean of sample means IS THE SEM. You do not divide it again by √n in order to calculate ANYTHING. The standard error of the mean indicates how different the population mean is likely to be from a sample mean. It is not a value that assesses the measurement uncertainty or the dispersion of values surrounding the actual data distribution.
You must decide, are station groups samples? Are station averages considered the entire population. I’ll tell you that temperatures being treated as and averaged by groups is exactly how one treats samples.
Someone must determine if the value of “n” being used is the size of the samples, i.e. 12 for an annual value or if it is the number of samples ~9500!
So much of what has been done in the past is a mishmash of statistical operations that are not done in a standard order.
Lastly, in order for any of this to even begin to be done properly when referencing the LLN and CLT, independence must be dealt with. Highly correlated data like Tmax and Tmin are not transformed into non-correlated data by a simple arithmetic average. That means everything from Tavg is contaminated with correlated data. That is being ignored like high school kids just learning how to average data.
“Both formula’s shown assume that the averages being dealt with are a “population”. IOW, the entire population of all temperatures.”
What do you think the entire population is in each case.
“That is the only reason you would be dividing σ by “N” or “N-1” to calculate the SEM.”
You are dividing by √N, not N. And it’s the sample standard deviation not σ, or in the second case σ_ξ which according tot he spread sheet is the average measurement uncertainty for all the years.
“When you have the entire population the SEM is useless.”
In the first case the population is taken as all possible values for each year. That is each year is assumed to be a random variable around the actual 11 year mean (this is the same as in TN1900 example 2).
In the second case the population is the actual annual value with the full range of possible measurement errors. This is equivalent to propagating the measurement uncertainties for an average.
“The standard deviation of a mean of sample means IS THE SEM.”
You are still making the same mistake I addressed in the comments to your previous nonsense. (And it’s also irrelevant to the discussion of the uncertainties discussed in this post).
What I think you believe, is that as each annual value in a single station could be considered a sample, and has a mean, then if you take the standard deviation of all stations for the year, that standard deviation is the standard error of the global average. This is not, and is obviously not, the case.
A sampling distribution is made up of all possible values of the mean, taking from equivalent samples. It does not consist of the a load of entirely different samples, each taken from a different distribution.
The fact you believe the SD of station means is obvious from your next statement.,
“You do not divide it again by √n in order to calculate ANYTHING.”
Indeed, you do not divide the SEM by √n. But you do divide the SD by √n to get the SEM.
“It is not a value that assesses the measurement uncertainty”
It is if the population is the assumed distribution of measurement errors as being done in figure 4.
“Someone must determine if the value of “n” being used is the size of the samples, i.e. 12 for an annual value or if it is the number of samples ~9500!”
It depends on what sample you are talking about. If you have 12 monthly values than the sample is those 12 monthly values. If you have 9500 average values, then the sample is those values, and the size is 9500.
“Highly correlated data like Tmax and Tmin are not transformed into non-correlated data by a simple arithmetic average.”
They could be, but why would you want to. I keep asking you what correlation you are talking about and why you think it matters, but you never explain and just keep invoking the correlation as if it means the statistics don’t work.
“That means everything from Tavg is contaminated with correlated data.”
It matters not a jot that Tavg is the average of two highly correlated data points. In fact if they weren’t Tavg wouldn’t mean much. The correlation juts means that when it’s a hot day both max and min are high and so the average is high.
This comment is perhaps off-topic. The farthest I got was film school 50 years ago, so I don’t have the background to really appreciate the science, but I enjoy reading everything Pat Frank writes (like his comments here), just to observe the language, grammar and punctuation. He writes very clearly and concisely, I’m not sure anyone else I’ve read on this site comes close.
Pat does write very well. It’s a great skill to have.