## A more detailed reply to Pat Frank – Part 1

Posted by Jeff Id on August 8, 2011

We’ve been discussing Pat Franks recent temperature uncertainty publications for quite some time on this other thread. I’m not pleased to say that nearly zero ground has been made in understanding the problems in this work and even some very sharp people have missed the mark. I’ll give it an hour this morning, and work away at it for a bit until I’m finished. Math isn’t always boring but we have already covered this stuff so you guys might not like it. The paper being discussed is online here.

There are multiple problems with the paper but I’ll start from the beginning and work my way through. There are several statistical analyses presented which are broken into different cases. Case 1 is fine, Case 2 has some problems. Case 2 however, is not used in the conclusions but is merely presented as an example. However, the error in the example is repeated throughout the paper.

Note the definition of tau at the beginning, N true temperature magnitudes. Not N temperature measurements with error but ‘true’ temperatures. Equations 4 and 5 are non-controversial for noise which is normally distributed and random. No disagreements here.

The problem with Case 2 starts on page 972 word three “‘further’ source of uncertainty” now emerges from the condition that tau’s are different. Equation 6 takes the spread of individual temperature stations and incorporates them into an error calculation s which is the standard deviation of inherently different temperatures. In other words, Pat is accidentally making the claim that there is an uncertainty s created by the delta tau from the mean of the TRUE temperature magnitudes. This is weather noise, and this is false.

This equation is apparently not used in calculating the uncertainty though so my recommendation is to ignore section 2 entirely.

Later in case 3 we have this:

Note the statement of lack of knowledge concerning stationarity and true magnitudes of noise variance. The problem here though is i n the final paragraph where again the ‘true’ temeprature magnitudes have a standard deviation which is represented again as s. This standard deviation is the difference betweeen true temepratures at different stations so it is again ‘weather noise’. Again, ‘s’ is defined as a ‘magnitude’ uncertainty and is incorrectly calculated. Having known magnitudes for tau that are different does not create uncertainty and sigma bar total in this equation includes the different true temperature magnitudes as an uncertainty, this is both confusing and incorrect.

Then we get this quote farther along:

This is where Pat makes the claims that stationary noise variances are unjustified. This is key to the conclusions and equations presented throughout the paper. I also believe that has problems but they are more subtle than the more obvious problems of the previous equations. We’ll get into that tomorrow.

I get that Pat hasn’t included weather noise in his final calculations for Table 1,2 and the figures, but he has incorrectly defined ‘s’ throughout the paper – and that does include the distributions of true temperature anomaly magnitude, and thus ‘weather noise’. What’s more is that the lack of improvement in uncertainty of the mean is a powerful assumption which can be disproven as time permitting, I will show next time.

## steve fitzpatrick said

Jeff,

My hat is off to you for taking the time to go through this in (painful) detail. I hope it makes a difference.

## Crashex said

Jeff,

These discussions all describe Uncertainty as an individual parameter. I recall measurement uncertainty as a combination of the bias error and the precision error of the measurement.

For a stable environment, a single sensor and a single measurement produced an uncertainty that combined both. Making multiple measurements of that stable condition with the same sensor, reduced

the precision error component, but did not change the bias error. Using multiple sensors to take the measurement, reduced the the inherent bias error of the result. And taking multiple measurements using multiple sensors reduced both the bias and precision error components.

This discussion seems to evaluate using different sensors to make measurements of different conditions and averaging those parameters. Can you comment on how these component parts of uncertainty are considered in this discussion?

## Jeff id said

Crashex,

I will in the next post. All of the uncertainties combine as you and Pat say. The next problem I have is the characterization of the uncertainty and the overly general description of systematic error.

## Pat Frank said

Jeff, you wrote: “

Equation 6 takes(emphasis added)”the spread of individual temperature stationsand incorporates them into an error calculation s which is the standard deviation of inherently different temperatures.No, it doesn’t. Equation 6 has

nothingto do with real station temperatures or with individual temperature stations. It hasnothingto do with real-world temperatures.I’ve repeated this point over and over again, and neither you nor Lucia, nor Steve, seem able to grasp this.

Equation 6 and Case 2 are

notreal world. They refer to a rigorously limited statistical case of signal averaging, period. Within the context of Case 2, they have nothing whatever to do with physically real temperatures, with real climate stations or with real station temperatures. It’sa statistical model. Case 2onlydescribes how the statistical equations change when the conditions are strictly limited to considering measurements of signals with inherently different magnitudes but stationary noise.It’s really incredible to me that you can’t seem to understand the application and meaning of a strictly statistical model, when it’s being used to show

onlyhow the statistical equations evolve under axiomatically changing bounds.## Pat Frank said

I’ve posted a term concordance in the prior thread to assist anyone wanting to reach a full understanding of the objections Jeff, Lucia, and Steve Fitzpatrick have concerning Case 2, as applied to my paper 1.

I’m glad to see in any case, Jeff, that you’ve retracted your prior claim that I misrepresented weather noise as an error variance: “

I get that Pat hasn’t included weather noise in his final calculations for Table 1,2 and the figures“.I’m guessing you retracted the claim because, after a full month of asserting that error, you finally got around to actually reading the full paper. That came, apparently, after I requested yesterday that you actually point out your claimed error in the three Figures and two Tables.

You’ve made several interpretative errors in your latest critique. I’ll get to them another time.

## Pat Frank said

Crashex, you’re right. But bias errors from multiple sensors cancel out only when the biases go in opposite directions about the mean. If one is perfectly lucky, the bias errors are randomly distributed around the true mean. However, the only way to know the distribution of the bias errors is to have measured them empirically.

Hubbard and Lin have developed a real-time method to determine the sensor bias errors at the same time as the air temperatures are being recorded. That allows them to remove these errors from the measured air temperature. If their method is applied to the sensors measuring air temperatures in the future, we’ll get a very much more accurate record.

However, sensor bias error measurements were never carried out in any systematic way for surface climate stations during the 20th century. I see no way to recover an accurate air temperature record for the 20th century, or the early 21st. Perhaps the best that could be done is to get a good estimate of the systematic error by setting up Hubbard and Lin type experiments testing the accuracy of, e.g., LIG thermometers in a CRS screen, at representative spots around the US (or better, around the globe).

## Jeff Id said

Pat,

And it is incorrect no matter how you interpret it physically in your mind, so what should I write. The sections I have pointed out state explicitly that different temperature magnitudes result in a ‘magnitude uncertainty’ of the mean defined as the standard deviation of the true magnitudes, this is not correct.

## Jeff Id said

No Pat, I haven’t. That is the entire point I have made here. You need to stop worrying about whether I give a real world example of temperature sensors and focus on the specific error. I’ve pointed it out with the cup examples, I’ve pointed it out with the equations, I’ve given physical meaning with temperature stations, and I’ve discussed it in general terms. It is an error that does exist in the paper and yes I do see that you have substituted different numbers in the paper which do not include this error to my knowledge but that doesn’t stop the error from permeating the discussion portion.

## steve fitzpatrick said

Pat Frank,

When you say in Case 2:”the N true temperature magnitudes, Tau_i, vary inherently”, what you are describing is weather noise (for Earth). The variation is due to the natural behavior of the system. If there was no weather, the Tau_i would not vary. That you do not choose to call it “weather noise” makes no difference.

You go on to say: “a further source of uncertainty now emerges from the condition tau_i tau_j. The mean temperature, T_bar, will have an additional uncertainty, +/- s, reflecting the fact that the tau_i magnitudes are inherently different.”

It is clear to me (and lots of others) that you are assigning an uncertainty contribution in the mean calculation based on “inherently different” varying temperatures (tau_i). That variation (those “inherently different” temperatures)

is weather noise. So yes, weather noise makes the measured average temperature (even in the absence of a long term temperature trend) jump about a bit. We already new that. That does not mean that there is any additional uncertainty in an individual “true mean” value beyond uncertainty due to the measurement process itself (that is, due to imperfect/incomplete sampling/measurement of the system).There is in fact uncertainty about whether or not a particular mean value lies close to the longer term trend in the mean, but that has nothing to do with uncertainty in that particular measured mean value. You want to dig into a blizzard of equations to prove you are correct; I just want to point out that the

starting assumptionsyou base everything on are incorrect. The starting assumptions convolute inherent variation with uncertainty in the measured mean. So in spite of the use of rather elegant equations, your entire effort is based on an incorrect understanding of the physical process; you automatically draw incorrect conclusions.Several people have tried to explain this to you already, but I figured I would give it one last try.

## steve fitzpatrick said

Above the “tau_i tau_j” should read “tau_i is not equal to tau_j” . HTML has it’s limits.

## Jon said

So if you have no measurement uncertainty, the estimate of the ‘true mean’ is not uncertain? I’m afraid I don’t follow. If you sample a varying signal exactly–no measurement uncertainty–your estimate of the mean is going to have an uncertainty because you’re making finite, discrete samples of a continuously varying phenomena.

What am I missing, joining this conversation late?

## Joel Heinrich said

Jon:

So if you have no measurement uncertainty, the estimate of the ‘true mean’ is not uncertain?Yes, exactly.

If you sample a varying signal exactly–no measurement uncertainty–your estimate of the mean is going to have an uncertainty because you’re making finite, discrete samples of a continuously varying phenomena.No. Your mean will vary over time and the average of the mean over time will have an uncertainty, but

at the moment of the measurementthe mean will be exact.The mean and the average of the mean over time are two different things.

## RomanM said

All of the discussion of weather and signals is a red herring in the context of the cases presented by Pat in his paper. These bring their own baggage and obscure the relatively simple issues involved in his models.

In case 2, we have a sequence of random variables t_i = tau_i + n_i. In his model, the (possibly different) tau_i are constants and the n_i are independent random variables with means equal to 0 and a common standard deviation sigma.

He now defines the random variable T_bar (which is equal to the mean of the t_i), but does not explicitly state what it is an estimate of. In case 1, the tau_i are all equal to the “constant temperature” tau_c. However, in case 2, we no longer have such a value. A simple calculation shows that T_bar has a mean of tau_bar (the average of the tau_i) and a variance equal to (sigma^2)/N.

T_bar is therefore an unbiased estimator for the parameter tau_bar. Furthermore, the variance of T_bar is exactly the same as that in the earlier case 1 with no “magnitude uncertainty” involvement. What Pat also does not seem to realize is that when sigma is not known and when no particular structure has been specified for the relationships among the tau_i, it is not possible to estimate sigma from the given t_i data.

So what exactly does Pat’s SD (which is the merely the standard deviation of the t_i) estimate? The answer is fairly simple. Consider the situation where rather than than calculating the average T_bar, we decide to estimate tau_bar by randomly selecting one of the N measurements t_i and using that as the estimate of tau_bar. If you then calculate the error bound for the result, the “magnitude uncertainty” does indeed play a role and you basically end up with the error bound that Pat incorrectly ascribes to T_bar. It should be fairly obvious that using the mean T_bar rather than a single randomly chosen measurement will produce a smaller uncertainty.

There was one other thing in the paper that caught my eye. For case 3a, there is a formula for sigma_mu-squared which apparently comes from reference 17, p. 57. I was unable to get a copy of the reference to see how this formula was derived, but the result seems strange. If any single one of the sigma_i in the formula is equal to zero (or close to zero), sigma_mu is equal to zero (or close to zero) with the values of the other sigma_i becoming completely irrelevant. Perhaps Pat could indicate to me where I might find an accessible copy of that derivation.

## Jeff Id said

Roman,

As I have written so many times now in #2 Pat has estimated the probability that an individual tau deviates x from the mean not the uncertainty of Tbar. The next bit is a little more cute (besides your 3a find) hopefully you can show up after I write that one.

It is nice not to be wrong every time.

## steve fitzpatrick said

Jon #11,

We are saying the same thing. I said there was no uncertainty in the mean, “beyond uncertainty due to the measurement process itself (that is, due to imperfect/incomplete sampling/measurement of the system).”

So yes, if your sampling is less than 100% complete (and it must be), or your thermometers are drifting in calibration (and they will), there will be (of course) some uncertainty in the mean. The issue here is that Pat uses that conventional uncertainty, and then adds another (very odd) ‘uncertainty’, simply because what is being measured is a heterogeneous system. That is the part that is very wrong. There is no heretofore unrecognized uncertainty in the mean of a heterogeneous system.

## lucia said

The estimate of the true mean is uncertain, but it is not equal to the standard deviation of the temperatures. That is: Pat’s equation incorrectly describes the error in Tbar. This is not the same as saying the error in Tbar is zero.

What’s important to recognize is Pat’s discussion in case 2 is wrong independent of any application to climate science. The standard deviation in the population describes

something, it just doesn’t describe “the magnitude uncertainty in the mean”.## slimething said

What if the instrumentation is faulty? Where I work, my job is to assure the instruments are calibrated and in good working order according to the manufacturer’s specifications. It sure seems like there is not much interest in the metrological area, which IMO should be the very first consideration before measurements are taken. I suspect there are many more similar examples that haven’t been addressed, based on 25 years of experience measuring stuff, statistics aside.

http://is.gd/3WJnu7

## lucia said

slimething–

Pat’s example in case 2 is a hypothetical in which the instrumentation is not faulty. If it’s faulty, you have to deal with that– but Pat’s cases 2 & 3b do not address this.

## Pat Frank said

#13, Roman, you wrote, “

All of the discussion of weather and signals is a red herring in the context of the cases presented by Pat in his paper.”You are

soright on there. In the previous post, Jeff and Lucia made an insistent big deal that the variances I presented in paper 1 were merely weather noise. When I finally asked them to demonstrate that error in any of the 2 Tables and 3 Figures of paper 1, they were unable to do so. Since then, neither one of them have owned up to their insistent and mistaken claim of a weather noise error in the variances I reported.The basic error Jeff, Lucia, and Steve Fitzpatrick are making is shown perfectly in Steve’s #9. He writes, “

When you say in Case 2:”the N true temperature magnitudes, Tau_i, vary inherently”, what you are describing is weather noise (for Earth).”But where in Case 2, do I ever state that the t_i or the tau_i represent physically real temperature observables? Nowhere, that’s where.

Steve goes on, “

If there was no weather, the Tau_i would not vary.” But the tau_i are not about weather. The tau_i are about statistics, and in a mathematical analysis of signal averaging statistics the tau_i can vary as I define it to vary. Steve is incorrectly imposing a physical meaning onto my words, which arestrictly and onlyabout signal averaging statistics. Steve’s imposed meaning is false.The variation in tau_i doesn’t carry with it any real physical meaning. It’s merely an expanded mathematical condition so as to explore the statistical consequences. I’m not discussing weather. I’m not discussing physically real temperatures. I’m discussing the evolution of signal averaging statistics as the mathematical conditions change.

I actually pointed this out in detail in post #169 in the previous thread. I suggested that throughout Section 2, “

Wherever the word “temperature” appears, substitute ‘intensity.’” I showed that the three cases could just as well have been written using y_i to represent magnitudes (intensities) instead of t_i. Instead of T_bar, use Y_bar. Instead of tau_i use upsilon_i.The statistical meaning stays the same, the significance to the estimated read error of Folland, etal, 2001 remains identical (discussed in Section 3), and there’s no temptation to impose a false meaning of “weather noise” on the Cases because there’s no reference to t’s or to temperature.

But Jeff, Lucia, and Steve F completely ignored that post and merely continued their insistence on what is not in evidence.

The title at the top of Section 2 is “

2. SIGNAL AVERAGING.” It’snot, ‘how to estimate weather noise,’not‘treatments of weather noise,’not‘how to model real daily temperatures,’ etc. None of that.Here’s the first paragraph under that heading: “

The error in an observable due to random noise can be made negligible by averaging repetitive measurements [14, 15]; a technique that is exploited to excellent effect in spectroscopy [16].signal averagingThree cases below show when noise reduction byis appropriate, and when it is not. The statistical model in B06 is then appraised in light of these cases.”Once again, I specify that the section is about signal averaging. Nowhere do I specify that I’ll be discussing weather noise, or that anything I include is applicable to weather noise.

Immediately following that opening the paper proceeded to, “

2.1 Case 1” I wrote that, “In(emphasis added).” Case 1 logically follows the opening remarks and is about signal averaging. Case 1 is clearly about statistics, and not about physically real temperatures.signal-averagingrepetitive measurements of a constant temperature, the measurement ina random noise model…Immediately after Case 1 is, “

2.2. Case 2, “Now suppose the conditions of Case 1 are changed…,” i.e., the statistical case is expanded to include a new condition. Case 2 follows immediately after Case 1 and necessarily includes the identical logic.It is about signal averaging statistics. Still not word one about weather noise. It’s all about the basic signal averaging statistics under strictly specified statistical (mathematical) conditions. It’s

nothingabout physically real temperatures or about weather noise.Steve is imposing his internalized meaning of real surface-of-the-earth temperatures onto my statistical Case 2. This is a Case 2 that is strictly about the statistics of signal averaging and strictly specifies conditions of a set of mathematical t_i and tau_i that are never represented as physically real. The

onlyproperties they have are that the tau_i vary inherently and the n_i are stationary. Those conditions exhaust Case 2. Those conditions do not include climate physics or physical observables. Case 2 isnotabout weather noise and has no connection to weather noise.Steve Fitzpatrick’s insistent inferences, and Jeff’s, and Lucia’s, are wrong and tendentious in the extreme.

If I wanted to analyze weather noise, or if I had intended to specify that the analysis was about weather noise, I’d have discussed actual weather noise in the paper and gone on from there. But I didn’t.

Further, there is not the slightest hint in the written content of the Cases that an implication about weather noise was mistakenly included. There isn’t a hint of physics anywhere. Only signal averaging statistics is included. Only mathematical evolution is invoked throughout. Steve F., Jeff, and Lucia are reading in what is not in evidence.

Steve Fitzpatrick, Jeff, and Lucia have not only chased after a red herring, it’s a red herring of their own making. Dragging it behind them, they’ve continued running in circles around it.

All of them have insisted on imposing their own false and novel meanings onto my words. All of them have insisted on projecting their mistakes onto me. None of them have exhibited the grace to allow me to mean exactly and only what I wrote.

## Pat Frank said

#13, Roman, you further wrote, “

In case 2, we have a sequence of random variables t_i = tau_i + n_i.”Where does Case 2 state that the t_i or the tau_i are random variables?

Their distributional properties are nowhere defined. All you know is that the tau_i have inherently different magnitudes, and that the n_i represent stationary noise.

The t_i and tau_i may be

arbitraryintensity observables, but the meaning of “arbitrary” is different from “random.”You wrote, “

He now defines the random variable T_bar (which is equal to the mean of the t_i), but does not explicitly state what it is an estimate of.”Could that be because I’m discussing signal averaging in general, and not any specific observable?

I define what T_bar is: the mean of the t_i. In a strictly statistical discussion, that’s all that’s necessary. So, you’re right: there’s no description of what it’s an estimator of.

I’d like to thank you for your observation, because it has entirely destroyed the criticism that Steve F., Jeff, and Lucia have made about Case 2. They have insistently imposed that T_bar is an estimator of weather noise. As you correctly surmise, it is not. Their claim is false and tendentiously imposed.

I have already pointed that out to Steve, Jeff, and Lucia an uncountable number of times. They have intransigently dismissed all of my protests. Maybe they’ll believe you.

You wrote, “

A simple calculation shows that T_bar has a mean of tau_bar (the average of the tau_i) and a variance equal to (sigma^2)/N.”You should have written that T_bar (and tau_bar) has a

noisevariance equal to (sigma^2)/N. Without that specification to noise, your statement is incomplete as written. Acompletedescription of the variance of T_bar and tau_bar includes the inherent variation of the tau_i about the mean.You wrote, “

Furthermore, the [Case 2] variance of T_bar is exactly the same as that in the earlier case 1 with no “magnitude uncertainty” involvement.”Again, you neglected to specify that you’re discussing the

noisevariance, Roman. I explicitly stated that the noise variances are identical in Case 1 and Case 2.You’re supposing a mistake in noise error variance where none exists, without being careful enough to qualify that it’s noise you’re discussing.

Magnitude uncertainty is never described as an error variance in the mean, or as part of the noise variance. I specifically stated that it is a measure of the variation about the mean due to the inherently different magnitudes of the tau_i.

Inherent variation in magnitude is introduced with the tau_i in Case 2.

When noise sigma_mu and magnitude uncertainty are discussed, the distinction between them and between their meanings must always be made clear. This I do everywhere in the paper. You’re

notdoing that here.You wrote, “

What Pat also does not seem to realize is that when sigma is not known and when no particular structure has been specified for the relationships among the tau_i, it is not possible to estimate sigma from the given t_i data.”First, your “

is unjustified. You didn’t qualify “alsodoes not seem to realizevariance” withnoise. This lack of specificity vitiates your analysis and obviates your first purported error. The “also” is thus unjustified in purporting a second error because there is no first error.Second, none of the cases in my paper include the conditions where, “

sigma is not known.(bold added)” Cases 1 – 3b all include a defined relationship among the tau_i.andwhen no particular structure has been specified for the relationships among the tau_i,So, you have supposed a mental error because I didn’t discuss a condition

that has nothing to do with any of the Cases.You wrote, “

So what exactly does Pat’s SD (which is the merely the standard deviation of the t_i) estimate?”Here, you have also chosen to focus on the equation for “SD” as though it represents the full meaning of Case 2, but which appears only in the last paragraph of Case 2. In starting out, Case 2 sets itself to separately analyze the consequences of the random noise, n_i, and the inherently different tau_i. Eq. (6) shows how each of these separately behave.

So, jumping down below eq. 6 to an overall SD as representing the message of Case 2 is a little bit tendentious, because in doing so you by-pass and ignore the meaning of Case 2.

Then, here’s what you wrote about the meaning of that SD: “

Consider the situation where rather than than calculating the average T_bar, we decide to estimate tau_bar by randomly selecting one of the N measurements t_i and using that as the estimate of tau_bar. If you then calculate the error bound for the result, the “magnitude uncertainty” does indeed play a role and you basically end up with the error bound that Pat incorrectly ascribes to T_bar.”But I don’t describe the Case 2 SD as an “

error bound“, Roman. Why do you? Why isn’t yours a mistaken conception?Here’s what the paper says, starting immediately before SD is presented: “

The usual way to represent the(bold added)”uncertaintyin averages of inherently varying magnitudes is with the standard deviation (SD) of the total scatter about the mean, e.g., SD = sqrt[(sum over N)(t_i - T_bar)^2/(N-1)].That is, the SD is calculated in the full defined knowledge that the magnitudes of tau_i vary inherently and, thus, so do the t_i. The SD is the common equation for the empirical SD of any set of averaged measurements. The first empirical SD is always the total scatter about the mean.

The SD is not strictly an error bound. Please, Roman, you’re not going to follow Lucia’s false lead and claim that every mention of uncertainty in paper 1 is really a mention of measurement error, are you?

You wrote, “

T_bar is therefore an unbiased estimator for the parameter tau_bar.”Right. Not an estimator of mean weather noise. Steve, Lucia, and Jeff: say bye-bye now to your insistent “weather noise” mistake, will you?

Let’s now look at your SD variant, Roman. Let your mean estimator, as chosen from the set of t_i, be represented as t_j (t-jay). That is, under your scheme let t_j be chosen to represent T_bar.

In eliminating T_bar from the analysis, the loss of one degree of freedom is avoided. So “N-1″ no longer applies and is properly replaced by “N.” The equation under Case 2 changes to, SD = sqrt[(sum over N)(t_i - t_j)/N], where t_j is also included as one of the t_i.

Since the t_i are not known to be randomly distributed, the decremental 1/sqrtN cannot be applied to SD, and SD represents a simple average of the new (asymmetric) spread of the t_i around t_j.

You finished with, “

It should be fairly obvious that using the mean T_bar rather than a single randomly chosen measurement will produce a smaller uncertainty.”And so what? The total spread about T_bar includes both the noise error sigma/sqrtN, and the magnitude uncertainty due to the inherently variable tau_i, (+/-)s. And (+/-)s is not an error bound.

In the sentence immediately after the equation for SD, I wrote that, “

If the sensor (+/-)sigma has been measured independently, then (+/-)s can be extracted as (+/-)s = (+/-)SD(-/+)(sigma_n/sqrtN)…”So, under Case 2 I specify that the SD must be different from magnitude uncertainty (because SD includes sigma), and that magnitude uncertainty can be extracted from SD if the sigma is independently known.

I also specify magnitude uncertainty as, “

a measure of how well a mean represents the state of the system.” andnotas an “error bound,” as you have it.SD is represented as the total empirical uncertainty in a set of measured observables. And so it is. If further knowledge is in hand, i.e., of sigma, the SD can be parsed into sigma and (+/-)s, where (+/-)s is the spread about the mean due to the inherently different magnitudes of the tau_i.

Magnitude uncertainty is

nota measure of non-knowledge of the mean value. Right there in Case 2, it’s defined as, “a measure of how well a mean represents the state of the system.” That’s not an error bound. It’snota measure of how poorly the mean is known. It’s a measure of the inherent heterogeneity of an inherently heterogeneous system.One might know a mean to arbitrary precision. If the system displays inherent scatter, there will always a non-zero magnitude variance around the mean. That magnitude variance does not affect your knowledge of the mean value. It represents knowledge of the variation of the sub-states about the mean.

Where’s my mistake?

You asked about, “

a formula for sigma_mu-squared which apparently comes from reference 17, p. 57” because, “If any single one of the sigma_i in the formula is equal to zero (or close to zero), sigma_mu is equal to zero (or close to zero) with the values of the other sigma_i becoming completely irrelevant.” and you wanted to know “where I might find an accessible copy of that derivation.”That equation is given on page 57, but the explicit derivation begins on page 53 under the section, “

Estimated Error in the Mean.” Page 53 defines the spread about the mean as Gaussian random. The equation on page 57 is for the total sigma in a weighted mean when the sigmas in the averaged data points are not equal. That is the situation for Case 3a.Your concern is about when ‘sigma_mu = 0 or nearly 0.’ The title of Bevington and Robinson says the book is about the statistics of data reduction “

in the physical sciences.” The book is about applying statistics to experimental data. Sigma_mu should never be zero in real empirical data measured by applying real methods using real instruments.There’s an amusing aside on page 55 that applies to your question. In the middle of this part of their book, B&R include a section titled, “

A Warning About Statistics.” This comes after the derivation of their eq. (4.12), which, for random measurement error, yielded sigma^2_mu = sigma^2/N.They go on to write that, “

Equation (4.12) might suggest that the error in the mean of a set of measurements x_i can be reduced indefinitely by repeated measurements of x_i.”They then discuss the limitations on error reduction as,

time available to the researcher, systematic methodological and instrumental errors, and “non-statistical fluctuations.” They write that, “It is a rare experiment that follows Gaussian distribution beyond 3 or 4 standard deviations.” Non-statistical fluctuations are unexpected outliers due to unknown methodological impacts on the experiment, carelessness, transcription errors, incorrect error modeling, and so on.At the end of this section, they write, “

The moral is, be aware and do not trust statistics in the tails of the distributions.”That applies to your zero sigma_mu, Roman. A sigma_mu of zero is an unlikely outlier. It’s a statistical curiosity but an empirical exile. It needn’t be treated in a book about pragmatic experimental statistics. Your question represents an uncritical application of statistical thinking to an arena of science. That’s actually funny because Steve F.’s, Jeff’s, and Lucia’s insistent and common error is an uncritical application of physical meaning to an arena of statistics.

Bevington and Robinson should be available in any good university library. Or, you can do what I did, and buy a personal copy.

## Pat Frank said

In #19, I agreed with RomanM that T_bar “

is not an estimator of weather noise. Let’s see what Jeff, Lucia, and Steve F. do with that.”In fact, I see that in #14, what Jeff did with it. He wrote, “

As I have written so many times now in #2 Pat has estimated the probability that an individual tau deviates x from the mean not the uncertainty of Tbar.”Jeff, you have incessantly insisted that the the variation of the tau_i about T_bar, (+/-)s, is weather noise. Here is what you wrote about (+/-)s, the magnitude variation about T_bar: “

It does say something about the variance in weather, but that is all it says.”Likewise, Lucia in agreement: “

Precisely. Mind you: The variance in the weather can be interesting….”It has been

this— your untoward assignment of the physical meaning of “weather noise” to Case 2 magnitude variation — that has been at issue for us.Your statement above omits any mention of the core of your objection concerning Case 2.

I’ve always represented (+/-)s the variation of the tau_i about T_bar, and defined that variation as “magnitude uncertainty.” Our entire debate in the prior post was about your insistent equation of (+/-)s to weather noise.

You also claimed that weather noise constituted the variances I reported in paper 1, here, for example, and here and here and here.

So now, finally, we all know that (+/-)s is not weather noise, and that none of the variances I presented in paper 1 represent weather noise.

My unvarying definition of (+/-)s as the variation about T_bar is identical to what you are now claiming you always maintained. Well, you maintained something different. Your reformulated position in #14 is a fine example of re-written history. How “cute” is that?

## lucia said

Pat–

The reason I didn’t explain how weather noise appears in the tables and figures in

paper 1is that I don’t believe I’ve claimed they do.Also, you seem to be decreeing I’ve diagnosed things as “weather noise” based on quotes by SteveF or Jeff and attributing a number of arguments to me I have not advanced. You might be wise to assemble some “collective” argument by SteveF, JeffId and me believing I actually wrote what SteveF or JeffId wrote, or they wrote what I wrote.

My criticism of case 2 has always been that it’s wrong

for signal averaging.One of your numerous mistakes is to

alsoinclude what you call “magnitude uncertainty” in “the total uncertainty in the mean temperature”:From your paper 1:

That is: in case 2, you are including ±s in

the total uncertainty in the mean temperature. Whether you call ‘s’, ‘magnitude uncertainty’ or ‘the standard deviation’ this does not belong in “the total uncertainty in the mean temperature.” This is conceptually wrong– and wrongfor signal averaging.Honestly, you can’t defend yourself against people pointing out that you

areincluding the standard deviation about the mean in “the total uncertainty in the mean temperature”, by showing us that you wrote that “The magnitudeuncertainty, ±s, is a measure of how well a mean represents the state of the system” because, you

alsosay this magnitude uncertainty must be included in the “total uncertainty in the mean temperature”. But no matter what word you apply to s (which most call the standard deviation about the mean temperature)– this s must not be included inthe total uncertainty in the mean temperature. (And you can’t include it if you substitute ‘intensity’ for ‘temperature’.)You also can’t get around this by complaining that people use the words “error bounds” and “uncertainty intervals” interchangeably. They do. Big whip. Get a grip.

## lucia said

Correction:

You might be wise to not assemble some “collective” argument by SteveF, JeffId and me believing I actually wrote what SteveF or JeffId wrote, or they wrote what I wrote.

## Jeff Id said

Pat.

To my knowledge, I never made any claims regarding tables or the figures to the effect of weather noise. Why should I attempt to demonstrate something I didn’t claim? You have ignored my replies on this to date so I wish you would notice it here.

Right there Pat:

And when you look at ‘actual’ temperature magnitude changes TAU from daily variance. Your paper 2 admits it is full of weather noise – which may drop off as sqrt(N) – so I am repeating your words. Whether you say it or not in paper 1, it is what the equation represents when it is applied to actual surface station data.

Now the way I took Roman’s comment was -who cares if it is weather noise or not – I have a question, you wrote above:

See, as it is defined in the quote above, I agree with you, yet in the image grabbed from the paper below we see that you write: +/- s MUST enter into the total uncertainty to the mean temperature as…

How is the “uncertainty in the mean temperature” rectified with “Magnitude uncertainty is not a measure of non-knowledge of the mean value”??

These are the things that drive me crazy about this paper. You’ve written it two different ways, I’ve repeatedly said you’ve calculated the probability of an individual value deviating X from the mean – which is what you just got finished repetitively saying to Roman and Roman said to you if you read it carefully, I think Steve and Lucia may also have made the point, so on this it seems to me that everyone is in agreement. Then you see the equation right there with s stating explicitly that ‘it must enter into the uncertainty of the mean temperature AS’….. the equation after the AS is a problem which is the subject of my next post but it is difficult to see how these two claims exist in the same universe.

Maybe you can clarify it but please don’t write a book attacking everyone’s different opinions, just look at the few I have here and my question. I am hopeful your answer will clarify this apparent contradiction.

## RomanM said

#20 Pat Frank

I am currently traveling away from home and don’t have time right now to address all of your points, however I suggest that you go wrong right from the start.

You will notice that the stationary “noise” variables, n_i, are random. Otherwise, there would be no statistical inference possible. I did indeed assume that the tau_i are deterministic values. In an elementary statistics course, you would learn that the sum of a deterministic variable and a random variable is always a random variable whether you “state” that fact or not. For simplicity , probabilists generally also may treat deterministic variables as degenerate random variables (i.e. r.v’s that take a single value with probability one), but that is not relevant to the matter at hand.

What you wrote here does not make any statistical sense. The reference to “discussing signal averaging” is irrelevant to any mathematical issues. You derive no general properties for t_bar and no context for understanding the relevance of the various standard deviations you throw out later.

Maybe you

shouldtell us exactly what you thinkcanbe estimated by t_bar. If it is a value other than tau_bar, then you will have a bias whose size cannot be estimated from the data without further assumptions on the taus. These assumptions need to be specified in the model.Perhaps you could point out to me where you got the false impression that I even considered “weather noise” or any of the discussion of that ilk in my comment. What I said was that this context obscured the underlying mathematics which I discussed in my comment.

In the same statistics course, you would learn that the sum and the average of independent random variables are also random variables whose properties depend on the summands. The mean (i.e expected value) of the average of the t_i is indeed equal to the average of the tau’s and the variance is the same as that in case 1 – that is not an opinion, but the result of a simple mathematical derivation. The taus do NOT play any role in the variability of the t-bar. You have assumed them to be fixed (non-random, i.e. without noise variance) values. If the taus were to be changed then the mean of the variable t_bar would also change, but the variance of t_bar would still remain the same.

Finally, what “mean” are you referring to in the sentence “A complete description of the variance of T_bar and tau_bar includes the inherent variation of the tau_i about the mean.” I don’t see anything specified in your model that would fit the description. My suspicion is that this omission could explain why you have the notion that the “magnitude” of the tau’s plays some role in the uncertainty considerations for t_bar.

Enough for now. I expect to be home in several days and I will have more time to address some of the remaining issues.

## Steve Fitzpatrick said

Pat Frank,

You seem terribly surprised that people (not just me) assume your paper is about the temperature of the Earth. I find it odd that this so surprises you when the title of the paper that got this these two threads started includes the words: “Uncertainty in the Global Average Surface Air Temperature Index”.

Based on that title, I am not sure how any reasonable person could think that your discussion of magnitude variation (and how that must increase uncertainty in the calculated mean) is not supposed to be related to temperature variation on Earth. But that said, your carrying on about people discussing actual measurements (of anything!) misses the point entirely.

You completely misunderstand what I (and others) have said. Your error

IS NOTa specific error associated with measuring temperatures. It is an error that would apply to the statistical treatment ofanymeasured variable inanyheterogeneous system. Divorcing the math treatment from any specific measurement (eg temperature) does not make it any more correct.You continue to convolute variation in individual measured values with uncertaintly in the calculated mean. That is not correct for Earths temperature stations. It is not correct for any measurement of any heterogeneous system.

You should listen to Roman, Jeff, and Lucia.

## Pat Frank said

#16 Lucia, you wrote, “

What’s important to recognize is Pat’s discussion in case 2 is wrong independent of any application to climate science. The standard deviation in the population describes something, it just doesn’t describe “the magnitude uncertainty in the mean”.”Let’s recall that I defined “magnitude uncertainty” as, “

The magnitude uncertainty, (+/-)s, is a measure of how well a mean represents the state of the system.,” when the system is defined to consist of inherently different intensities.Now you describe your alternative correct meaning of the standard deviation of the variation of inherently different magnitudes about the mean magnitude.

## Pat Frank said

#22 Lucia, you wrote, “

The reason I didn’t explain how weather noise appears in the tables and figures in paper 1 is that I don’t believe I’ve claimed they do.”Your claim of a weather noise error is right here in spades, Lucia. And that claim is at the center of your original assertion of a mistake in my paper.

Jeff purports the same error here and says he got that idea from you.

You specifically referenced your claim of error to paper 1 here. It’s interesting to note in that post you further suggested that, “

You went astray by never defining what the “true” value corresponding to what CRU is trying to report (or alternatively, what you think is the “true” value that might be meaningful.),” when in fact I do that both specifically and contextually in Section 3.As examples: under 3.1, “

It is now possible to evaluate the ±0.2 C uncertainty estimate of Folland, et al. [12],…in any single daily [land air-surface temperature]observation.” Under 3.2.1, “the noise uncertainty in an annual anomaly is now stepwise calculated..” Under 3.2.2 “The systematic measurement errors originating from the field exposure of [temperature sensors]…”What CRU is reporting: Land surface temperature … annual anomaly … air temperature measurement.

You wrote, “

My criticism of case 2 has always been that it’s wrongfor signal averaging.”Not a single one of your posts in the prior thread specifically mentions signal averaging, nor that Case 2 is wrong for

signal averaging. Here is your own posted summary of your objection to Case 2.You were objecting there to (+/-)s as though my paper represented it as a measurement error, which it does not. You repeated that assertion as one of your two main objections throughout the prior post. Your other main objection, as linked above, was that the variances I reported were merely weather noise.

You wrote, “

That is: in case 2, you are including (+/-)s inthe total uncertainty in the mean temperature.Whether you call ‘s’, ‘magnitude uncertainty’ or ‘the standard deviation’ this does not belong in “the total uncertainty in the mean temperature.” This is conceptually wrong– and wrongfor signal averaging.”and “

Honestly, you can’t defend yourself against people pointing out that you are including the standard deviation about the mean in “the total uncertainty in the mean temperature”, by showing us that you wrote that “The magnitude uncertainty, ±s, is a measure of how well a mean represents the state of the system” because, youalsosay this magnitude uncertainty must be included in the “total uncertainty in the mean temperature”.”We know from your post in the prior thread that you have always required that in my paper, “total uncertainty”

mustmean the total ‘“error attributed to the mean”.Mark T called you on that, pointing out that you were in error (Mark T’s less flattering description was, “

This comment by you, lucia, is utter nonsense.“). Yet here you are averring the same thing all over again.It now appears from your further comments, however, that the difference between us has come down to mere rhetorical niceties. You require that “total uncertainty” be canonically restricted to mean

measurement error.I prefer, in contrast, to take more cognizance of the word “

total” in “total uncertainty,” as includingallthe relevant forms of uncertainty. That’s the significance of an unqualified “total,” isn’t it.So, the mean of a heterogeneous system carries with it an uncertainty transmitting the stability of the system about that mean.

You went on to suggest that including magnitude uncertainty as I defined it, is “

conceptually wrong– and wrongfor signal averaging.”I can only wonder how taking explicit cognizance of the inherent variation of a signal about its mean is “

conceptually wrong”You say magnitude variation, as I defined it, is also wrong for signal averaging. But in Case 2, the signal is defined to have an inherently variable magnitude. If one averages the observations of those signals, how is it wrong to notice that there is an inherent variance about the signal mean? And how is it wrong to present the simple math that expresses the standard deviation stemming from that variance?

Your entire case rests on your two claims: that

total uncertaintyis necessarily restricted to meanmeasurement error, and that in my paper magnitude uncertainty has been equated with measurement error.The first claim is now down to merely a dispute about nomenclature that you have converted into a purported mistake. The second claim is not supported in the text of my paper.

You wrote, “

You also can’t get around this by complaining that people use the words “error bounds” and “uncertainty intervals” interchangeably. They do. Big whip. Get a grip.”Widely used or not, it’s still imprecise and sloppy usage. In your hands and in application to my paper, it’s a fundamental mistake.

## Pat Frank said

#25 Roman, you wrote, “

In an elementary statistics course, you would learn that the sum of a deterministic variable and a random variable is always a random variable whether you “state” that fact or not.”Roman, in an elementary science or (very likely) engineering class you’d learn that the sum of a deterministic variable and a random variable represents an imprecise measurement.

In Case 2, the distribution of the plotted summed variable would not be even visually Gaussian, unless the deterministic part is negligible.

The context of my paper is science, not mathematics. The signal averaging statistics are employed in service to a scientific context, Roman. That means the variables take their basic meaning from science, not statistics. The t_i in that context are defined by their deterministic properties, not by the random properties of the associated noise. The statistical meaning you want to apply is scientifically nonsensical.

You wrote, “

What you wrote here does not make any statistical sense. The reference to “discussing signal averaging” is irrelevant to any mathematical issues.”I apologize for being unclear. I did write that, “

I define what T_bar is: the mean of the t_i. In a strictly statistical discussion, that’s all that’s necessary. So, you’re right: there’s no description of what it’s an estimator of.”I shouldn’t have written, “

In a strictly statistical discussion…,” because that phrase implies a mathematical context. I didn’t mean to do that, sorry. The context isn’t mathematics. The context is physical science. My intended meaning would have been more clear if I’d written something like, ‘In a discussion about the statistics of physical measurements, …‘Apologies again.

But what I wrote makes physical sense within a physical context, Roman. Discussing signal averaging is not irrelevant to the mathematical issues because this context puts a bound on the meaning that can be applied to the variables under study.

The physical context includes not applying the meaning of “random” to a deterministic variable, even when that variable is associated with a random noise component.

You wrote, “

Maybe youshouldtell us exactly what you thinkcanbe estimated by t_bar.”I did tell you that — in post #20, where I agreed with you. I quoted you, “

“T_bar is therefore an unbiased estimator for the parameter tau_bar.” and agreed: “Right.”I also agreed with you in post #289 of the previous thread: “

Roman’s conclusion about T_bar and tau_bar is correct. …”You wrote, “

Perhaps you could point out to me where you got the false impression that I even considered “weather noise” or any of the discussion of that ilk in my comment.”I didn’t suggest you considered weather noise, Roman. On the contrary, I observed that you

did not. This is in strict contrast to Lucia, Jeff, and Steve Fitzpatrick, whohaveinsisted that Case 2 is about weather noise. Your point about T_bar being strictly an unbiased estimator of tau_bar fully refuted their assertion. My comment merely noted that fact.You wrote, in the next paragraph, that, “

The mean (i.e expected value) of the average of the t_i is indeed equal to the average of the tau’s and the variance is the same as that in case 1 – that is not an opinion, but the result of a simple mathematical derivation.”Am I to take it that you define “variance” as strictly due only to the noise component of Case 2, (sigma_noise)^2, in other words?

You wrote, “

In the same statistics course, you would learn that the sum and the average of independent random variables are also random variables whose properties depend on the summands.”I’m sure. And in the same science course I noted above, you’d learn that the sum and average of physical measurements are statistics with physical meaning. That physical meaning would include “random” only if a falsifiable physical theory was available to provide that meaning. The meaning of the sum and average of measurements in science are not taken from the content of a statistics class.

Likewise, the scientific context of my paper provides that the meaning of the signals defined in my Cases is not imposed by statistics. The meaning is provided by science, and the statistics follow from that meaning.

You wrote, “

Finally, what “mean” are you referring to in the sentence “A complete description of the variance of T_bar and tau_bar includes the inherent variation of the tau_i about the mean.” I don’t see anything specified in your model that would fit the description.”Case 2 specified the definition: tau_1 not equal to tau_2, …. , not equal to tau_n. That specification fits the description.

I separated out a variance due to the inherent variation in magnitudes of the tau_i. That separate variance is justified on the basis of the different physical meanings assignable to random measurement noise as opposed to inherent variation in intrinsic signal magnitude. Scientific meanings, Roman.

How would you describe the specific variance relevant to the unequal magnitudes of the tau_i?

You wrote, “

My suspicion is that this omission could explain why you have the notion that the “magnitude” of the tau’s plays some role in the uncertainty considerations for t_bar.”Rather, the magnitude uncertainty about T_bar plays a role in the uncertainty considerations inherent in the mean of a physically heterogeneous system.

Notice, from the paper: “

The magnitude uncertainty, (+/-)s, is a measure of how well a mean represents the state of the system.”The magnitude uncertainty is represented as part of the total uncertainty in the mean temperature,

when the mean temperature represents an inherently variable physical state.The prevailing context here is physical science. If this conversation is to proceed constructively, you’re going to have to allow mathematical purity to be sullied by physical meanings, Roman.

## Pat Frank said

#26 Steve, you wrote, “

You seem terribly surprised that people (not just me) assume your paper is about the temperature of the Earth.”That’s not the issue in dispute, Steve. The issue in dispute includes your claims here and here, for example, that the variances in my paper #1 were mere weather noise, that Case 2 is about weather noise, here for example and here, and here (Mark T followed that last post with a heads-up, that you immediately blew off) and that (+/-)s represents the uncertainties in my Figures and Tables, here, for example and here.

None of your error claims are correct. All of them, at best, reflect careless reading.

You wrote, “

… your discussion of magnitude variation (and how that must increase uncertainty in the calculated mean)…”I never wrote that magnitude uncertainty increased uncertainty in the calculated mean. I wrote that magnitude uncertainty is part of the total uncertainty of a mean that reflects a heterogeneous state.

You’ve never appeared able to make the necessary conceptual distinction between magnitude uncertainty due to inherent intensity variation and measurement error (neither has Lucia, for that matter). Hence Mark T’s warning and your off-blowing of it.

You wrote, “

You continue to convolute variation in individual measured values with uncertaintly in the calculated mean.”The variances I report in paper #1 represent real, measured systematic sensor error uncertainties in a temperature mean, Steve. They’re referenced to the peer-reviewed published literature.

The variances I discuss in Case 2 represent a generalized presentation of sigma_random_noise and a generalized variance due to inherent magnitude variability. Pace Roman, these follow from the defined properties of the variables in that Case 2.

You wrote, “

That is not correct for Earths temperature stations.”The specific sensor systematic error variances I reported in paper #1 were derived from explicitly calibration measurements carried out exactly at a carefully controlled Earth temperature station.

You’ve never given any evidence of having carefully read my paper, Steve. All of your critical comments show an equivalent disconnect with the actual content of the paper.

## Jeff Id said

“Jeff purports the same error here and says he got that idea from you.”

Pat,

I have my own thoughts Lucia merely started me on my path. You are feeling beat up a bit but that isn’t my intent, this is just math and I certainly have a lot less than you tied up in it. I have pointed out a very specific issue in 24 which lead to the arise of my opinion, would you mind answering?

The statements pointed out appear contradictory, if you could parse it for me perhaps I could understand you better.

## steve fitzpatrick said

Pat #30,

In your paper, you suggest that the historical trend of average temperatures is so small compared to the “natural variation” that we can’t be reasonably sure there has in fact been a statistically significant increase. This is nonsense. The fact that you will listen to nobody (not scientists, not engineers, not statisticians) about this subject does not make it any less nonsense.

WRT to a careful read of your paper(s): I have read quite enough to understand you are mistaken. If someone proposes the reality of a perpetual motion machine, I don’t bother to dig too deeply into those arguments or equations either. The natural variation that is present in a heterogeneous system does not preclude accurate measurement of a trend in the mean, nor does that natural variation in any way impact our ability to discern that trend. When you say things like:

“It’s very clear that the entire anomaly trend never emerges from the range of natural variability. Even the intense 1998 El Nino anomaly of +0.53 C does exceed the 95% level of natural variability – (+/-)0.56 C about zero.

For 160 years, the global air temperature has been wandering about well within its self-defined natural limits.”

it shows the underlying problem with your entire analysis. You are implicitly assuming that any trend in the mean is “wandering about well within its self-defined natural limits”, and so can’t be due to a causal factor (eg, radiative forcing from GHG’s). You do not know that (it is pure conjecture on your part), because we do not have a representative instrumental record from a period which does not include substantial changes in GHG (and many other!) man-made forcings. We do know the trend in the temperature mean with reasonable accuracy; it does not have anything like the level of uncertainty that you suggest in your graphs. What has actually caused that measured trend to “wander” is a completely different and legitimate question, but is one that your treatment does not in any meaningful way address, even if you imagine that it does.

You have here at Jeff’s blog readers who are about as skeptical of the many pronouncements of climate science as you are going to find anywhere, and many who have a lot of personal experience in science and engineering. Most all recognize what you are saying is simply wrong. I find your argument less credible than Eric Steig’s argument about temperatures in Antarctica. People here didn’t believe Steig; they don’t believe you either. You ought to think about that. But I suspect you won’t.

## steve fitzpatrick said

Jeff,

“The statements pointed out appear contradictory”

They don’t appear contradictory, they

arecontradictory. Let me make a prediction: Pat will never accept any critique of his papers. Carrick was smart enough to see that from the first. I wish you luck if you bother continue with this, but I am pretty sure you are wasting your time.## lucia said

Pat Frank said (in comment 27)

Let’s recall that I defined “magnitude uncertainty” as, “The magnitude uncertainty, (+/-)s, is a measure of how well a mean represents the state of the system.,” when the system is defined to consist of inherently different intensities.

Now you describe your alternative correct meaning of the standard deviation of the variation of inherently different magnitudes about the mean magnitude.

Any alternative definition of “magnitude uncertainty” is utterly irrelevant to my criticism. Please reread my comment 22 , where, among other things, I wrote:

That is: in case 2, you are including ±s in the total uncertainty in the mean temperature. Whether you call ‘s’, ‘magnitude uncertainty’ or ‘the standard deviation’ this does not belong in “the total uncertainty in the mean temperature.” This is conceptually wrong– and wrong for signal averaging.

Uhhnmmm… anyone who clicks the link can see I say nothing about

paper 1and certainly not the “tables and figures in paper 1″. The follow link says you made a mistake in paper 1 (which you did) but does not say that weather noise appears in the tables and figures in paper 1.I’m not going to waste my time with more of this. It”s pointless. You want to think whatever you came up with are uncertainty intervals for… something. And you think the fact that whatever those are supposed to mean, you can make conclusions about something.

## Pat Frank said

#31 Jeff, here’s a reply to your #24: In answer to my question, “

But where in Case 2, do I ever state that the t_i or the tau_i represent physically real temperature observables?”You wrote, “

Right there Pat:” and quoted this: “This case may reflect a series of daily temperatures from any well-sited and maintained surface station sensor.”You know Jeff, suddenly I perhaps see the problem, and in a way it’s my fault. Bear with me a minute. The

veryafter the one you quoted starts this way: “nextsentenceIn this case, the ‘mean temperature (+/-) mean noise’ ….”So, those two sentences together read this way, “

This case may reflect a series of daily temperatures from any well-sited and maintained surface station sensor. In this case the ‘mean temperature (+/-) mean noise’ will again be,…”I can see that the subject in this

nextsentence, “In this case…” could be taken to refer back to the “This case…” subject of the previous sentence; i.e., the sentence you quoted.If the text were read that way, then the “mean temperature (+/-) mean noise” could be taken to refer back to the “

daily temperatures” of the previous sentence, rather than back to the definitional conditions of Case 2 (as I meant it to be).I suddenly see this ambiguity of interpretation in the text, and it’s my fault for writing it that way. This second way of reading the text never occurred to me until just now.

To make what I meant to convey totally clear, the next sentence would have better begun: ‘

Under Case 2 conditions, the ‘mean temperature (+/-) mean noise’ …”That would have made it completely clear that the equations for ‘mean temperature (+/-) mean noise’ — i.e., eq. 5 and eq. 6 — refer to the axiomatic properties of the tau_i and the sigma_i and do not refer to daily temperatures.

Making this change isolates the sentence you quoted. ‘Isolated’ is how I meant it to be. The sentence you quoted was meant to be a kind of throw-away line to suggest that the Case 2 conditions might have some real-world connection.

So, to get a clear picture of what I actually meant in this section, the sentence you quoted can be ignored. Just go from “

… with the ith measurement.” of the previous sentence right over to “In this case, the ‘mean temperature (+/-) mean noise….” and capitalize “case” to ‘Case’ in your minds eye. :-)When reading for my intended meaning, please skip over the sentence you quoted. What I meant to convey should now be apparent.

I’m hoping and guessing this will clarify things.

In writing the paper, I always applied the constant logic that the equations should strictly follow only from the definitional properties of the “signals,” and I thought my readers would understand this derivational logic and do the same. But I can now see how the text as I wrote it, in that spot, could be interpreted differently than the way I meant it, and lead to the presumption that Case 2 was about daily temperatures.

In my own defense in this matter, I did write the sentence you quoted: “

This case(bold added)”. That is, I didmayreflect a series…notwrite,does reflect, oris meant to reflect, or even justreflects.That “

” was meant to convey possibility; a possible application as opposed to a fixed meaning. The sentence you quoted was not meant to state what Case 2 was about. It was only meant to illustrate a possibility.mayIn my further defense, the definitional properties of Case 2 are restricted to”

tau_1, …, not equal to tau_i, …, not equal to tau_nandsigma_noise_1 equal to, …, equal to sigma_noise_n.That’s what Case 2

. I just assumed that my readers would understand this, understand that the equations followed from this, and understand that the subsequent derivations were meant to show the how the defined properties of each Case expressed themselves in the statistical equations.isBut that assumption was wrong, and I see that the text as written is a little ambiguous. And I can see how my interposed throw-away line could have tripped people up.

Please forgive me. :-)

Just to reiterate, “…

mayreflect …” was meant to communicate only possibility; the possibility that some well-maintained sensor might have constant noise. The conditions of Case 2 might then apply, and might be applied to daily temperatures. But this would have to be a judicious application. I didn’t elaborate on how to be judicious, but at the time of writing, and re-writing, and re-writing, …, I never realized the text could be alternatively interpreted as actually advocating the application of Case 2 to daily temperature.So, I hope this clears things up.

By the way, under Case 3, the sentence, “

The latter situation is closest to a real-world spatial average, in which temperature measurements from numerous stations are combined.” was also meant express an illustrative possibility in the same way, and was not meant to define the meaning of Case 3a or Case 3b.## Pat Frank said

#24 and #31 Jeff, you also asked this, “

How is the “uncertainty in the mean temperature” rectified with “Magnitude uncertainty is not a measure of non-knowledge of the mean value”??Please note that I wrote, “

… (+/-)s must enter into the(bold added)”totaluncertainty to the mean temperature as …That word,

total, is critical. Please include it in your thoughts. “Total” uncertainty includes more than just the uncertainty due to measurement error. In principle, it includes all the sorts of uncertainty. In this particular case,totalincludes the uncertainty that is attached to the mean state by virtue of the fact that the system is heterogeneous.One must notice the uncertainty due to the heterogeneity of the system in order to be faithful to the full meaning of the mean state of a heterogeneous system.

For example, it would be misleading to refer to the mean as just mu(+/-)sigma_measurement_noise, when describing the mean of a heterogeneous state. This is true even if one knows the mean precisely (sigma_noise is small).

It would be misleading because the mean state(+/-)sigma_noise, alone, is a poor representative of the heterogeneity of the system, due to the variability of the sub-states.

This heterogeneic uncertainty in the mean could indicate, for example, that the next time one calculates a systemic mean state, even if the mean has the same measurement precision as before, the new mean itself could have a different value (express a different mean magnitude) than the previous mean. This behavior depends on the type of heterogeneity exhibited by the system.

So, to be faithful to the full meaning of the mean state of a heterogeneous system — to impart the full meaning of the mean state — the magnitude uncertainty, (+/-)s

mustbe attached to the mean magnitude, mu.Thus, the magnitude uncertainty expresses some knowledge of the heterogeneity of the system for which that mu is the mean state. And that magnitude uncertainty must be attached to the mean magnitude in order to convey the heterogeneity of the system, and express something about its possible future behavior.

So, (+/-)s is knowledge, not error. It expresses knowledge of system heterogeneity. It expresses the uncertainty expected for the magnitude of the next mean one might calculate, if one subsequently gathered a new and independent set of observational measurements.

The Case 2 system is a heterogeneous system by definition, because the tau_i magnitudes are made to differ inherently. So Case 2 has a magnitude uncertainty, and that magnitude uncertainty is a necessary addendum to the Case 2 mean state, as a description of the heterogeneity of the Case 2 system.

I don’t mean to drive you crazy, Jeff. And I hope this post is a restorative. :-)

I’m going away for a few days, and so my posts may be few or non-existent until the middle of next week.

## Jeff Id said

Pat,

Total is not the problem. The problem is that you have stated that magnitude uncertainty is not a measure of non-knowledge of the mean. As you have defined it, I find the wording strange but this sentence is understood.

Then you added it “+” directly onto the value of non-knowledge of the mean – Total uncertainty. Uncertainty has a very specific statistical meaning.

How do you justify this?

## Pat Frank said

#37, Jeff, I justify it by noting that “uncertainty” with respect to measurement has a larger meaning in science than it does in statistics.

In statistics, “uncertainty” refers to a lack of knowledge concerning the precise value of a number (or a magnitude). In science, “uncertainty” includes this meaning, but can further transmit the lack of availability of a single meaning to describe the significance of a given quantity. Call that uncertainty a kind of empirical fuzziness.

If the magnitude of a mean is its meaning, (+/-)s transmits that there can be more than one possible magnitude. Obviously, a set of variable means must reflect a dynamical system.

If the magnitude of the mean of a dynamical system represents a mean state, then (+/-)s transmits that the total state is not symmetric about its (instantaneous) mean state and the set of substates going into the mean follow multiple, non-random, trajectories. E.g., the overall state evolves in a non-random, non-ergodic way.

This kind of uncertainty transmits knowledge of physical states that includes a meaning distinct from a purely statistical meaning that concerns knowledge of the variance of a set of numbers.

In Case 2, tau_1 not equal to tau_2, not equal,… not equal to tau_n implies a dynamical state. That state was not defined to evolve randomly or ergodically.

The total uncertainty about a mean, under Case 2, p. 972, is written as (+/-)sigma(+/-)s.

The (+/-)sigma transmits the lack of knowledge concerning the statistical value of the magnitude of any given mean; i.e., its precision.

The (+/-)s transmits the message that the magnitude of the mean itself is inherently not constant. Because of inherent variations, the magnitude of any subsequently calculated mean is likely to be different from that of the prior mean. (+/-)s transmits this lack of singular meaning in any given mean.

From a scientific perspective, (+/-)s transmits the uncertainty in the significance of the mean magnitude. That uncertainty refers to the fact that the overall state is heterogeneous, that the intensities of the sub-states vary continually and intrinsically, and that the mean magnitude does not, will not, and, under Case 2 can not, converge to a single value.

My papers, after all, are ultimately within science and are to be read in that context. It’s true they include small statistical derivations in Section 2. But the full significance of the findings from Section 2 cannot be restricted to statistical meanings.

Let me add here, Jeff, that you’ve been outstanding in your civility, patience, intellectual honesty, and personal integrity during the debate, and I’ve really appreciated it.

## Pat Frank said

#34, Lucia, you wrote, “

Uhhnmmm… anyone who clicks the link can see I say nothing about paper 1 and certainly not the “tables and figures in paper 1″”Let’s review this. From the prior thread:

Jeff: “

Pat’s paper implies that this [weather noise] variance, which exists in nature, is in fact error in accurate knowledge of temperature when really it is the variance in the correct knowledge of temperature.Lucia in reply to Jeff: “

Precisely.”So, Lucia you first claimed that the variances I calculated in paper 1 were mere weather noise, and now in #34 claim that your prior claim says, “

nothing aboutpaper 1and certainly not the “tables and figures in paper 1″.”So, let’s see: Jeff was referring to paper 1 and, granting that you were on topic, so were you. If we juxtapose your previous and current claims, they suggest that you’re asserting I made a calculational mistake in my paper that appeared nowhere in my paper. Forgive me for observing that this logic is hard to understand.

You italicized “

paper 1” in your #34 reply, as though your prior claim might have referred to my paper 2. But weather noise is not central to paper 2, either, and was neither miscalculated there nor misrepresented.You wrote, “

The follow link says you made a mistake in paper 1 (which you did)…”You’ve asserted two main errors. The first was that the variances I calculated in paper 1 were merely weather noise. By now, everyone following the conversation should understand that this claim of error was itself erroneous.

Your second claim of error was that Section 2 Case 2 was about weather noise. By now, everyone following the conversation should understand that this claim is not correct, either.

You also continued to insist, also erroneously, that “uncertainty” as used in paper 1 always meant ‘

measurement error,’ until Mark T called you out on it.Thanks, Mark. That

really,helped. :-)reallySo, Lucia, I don’t see that you have any case left at all.

## Jeff Id said

Pat,

Everything you have written to date simply confirms my initial points. This is not an accurate representation of ‘uncertainty’ in anything. Again, your context is one of statistics, whether you define it that way or not. You cannot use this value ‘s’ to determine anything about ‘knowledge of the mean’. You have represented different true values as uncertainty. Similar mistakes are made later and projected all the way to global mean temperature and then to top it off, the papers incorrectly concluded on knowledge of trend. Which you still haven’t admitted you have not even analyzed in this work.

I’m sorry but the work is badly flawed.

You should listen to what Lucia, I and others have written on this, I’m sorry for the tough reception this time around too. It was an interesting discussion anyway.

## lucia said

Pat–

Let’s just quote the full two quotes:

So, in the bit you link I

a) Do not say “paper 1″ and

b) Certainly am not discussing the “tables or figures”.

c) I am not discussing any numerical results whatsoever.

I have not said anything about tables and figures in paper 1.

It is a discussion of the abstract concept you are introducing. I am not discussing the numerical results in tables and figures but your general claims in uncertainty in a mean value.

As I understood Jeff then, and still understand him on rereading, Jeff was discussing a general concept of what you are doing, not the specific numerical values in tables and figures in

anypaper: not 1 and not 2. Jeff wasn’t discussing tables and figures in any paper and I wasn’t discussing tables or figures in any paper. The discussion is aboutconcepts.not tables and figures.In the bit you quote I don’t say “weather noise”. Sheesh.

Look Pat, I don’t take your paper seriously. I think it’s meaningless exercise in sophistry. If you or someone want to take the thing seriously, fine with me. I’ve had my say. I’ve pretty much moved on to other things.

## Pat Frank said

#40, Jeff, I defined (+/-)s to represent a measure of system heterogeneity. This is a simple and straight-forward concept, which you seem either unable or unwilling to grasp.

I called that heterogeneity, “magnitude uncertainty,” because it represented the fact that, under Case 2 conditions, the parent mean could not be a constant.

Perhaps if I’d called it ‘the ambiguity of the mean,’ rather than an uncertainty, you’d have had less trouble with it. Honestly, your problem seems to me rooted in mere confusion brought on by nomenclature.

In my paper, I defined “magnitude uncertainty” to convey the variability in the mean due to inherent system heterogeneity. You have every right to complain that I made a poor choice in nomenclature. But you and Lucia have no business redefining the term to mean something else, and then claiming that I made an error because

yourimposed meaning is different from what I wrote and clearly meant.Within my paper, “magnitude uncertainty” means only what I wrote it means — a measure of system heterogeneity — and nothing else.

You wrote, “

You have represented different true values as uncertainty.” No. I have represented the inherent spread of values in a heterogeneous system as a variability in subsequent calculations of a mean, and called that variability in the mean, “magnitude uncertainty.” You (and Lucia) have consistently and repeatedly misstated the meaning of that term.It’s as though you’re blind to the very direct meaning given repeatedly in the text. Others have clearly understood what I meant. You have clearly failed in this.

You wrote, “

Similar mistakes are made later and projected all the way to global mean temperature…” In paper 1, the uncertainty bars in Figure 3 have nothing to do with (+/-)s or magnitude uncertainty. In paper 2, Figure 4, the natural anomaly variation I included is calculated identically to the method used by Jim Hansen in his 1988 paper with Lebedeff (GRL 15(4) 323-326), and represented by me to have the same meaning he gave it.So I fail to see how any (+/-)s mistake has been projected anywhere. There is no such mistake.

You wrote,

then to top it off, the papers incorrectly concluded on knowledge of trend.” In this, you’re partly right. I did no analysis of the effect of the trend slope on the systematic error reported in the two papers. My result, admittedly.But the simplest statistical approaches to the estimation of uncertainty in trends assume random, identically distributed, error. Tests for the significance of a trend typically assume that the error is given by the y_i minus y_i_hat residuals from an OLS fit, where y_i_hat is the fit-predicted value of y_i at each x_i. But the fit residual is not what paper 1, Figure 3 is about. Figure 3 is about systematic sensor error, which is independent of the fit and is not a fit residual.

The final error bars in paper 1 represents systematic sensor error, which is also not known to be random — a major point of the paper, the significance of which apparently continues to escape you.

In fact, the distribution of systematic error in global air temperatures is entirely unknown. It’s never been measured and its distribution has never been investigated on anything like even a regional scale. That means there is not any way to choose an appropriate statistical error model with which to even

proposeanalyzing the significance of the trend, relative to the uncertainty width.Systematic error is not, in any case, the residual difference between a computed linear trend and the measurements. It’s a fuzziness in the magnitudes of the measurements themselves. This means one can have a series of points that display a statistically significant trend,

even though the values of the underlying data points themselves are highly inaccurate. Such a trend is physically meaningless.A better way to understand the plot in paper 1, Figure 3, would be to see it as a uniform grey band, 0.46 C wide, without any central line of data points. This wide grey band represents what we know about the values of the surface air temperature anomalies. That is, the surface temperature values themselves are not known to better than somewhere within a lower limit fuzziness width that is 0.92 C wide.

Michael Limberg is a Ph.D. candidate at Leipzig University who has made an exhaustive study of uncertainties and errors in the surface air temperature record. After paper 1 came out he emailed me and, among other things, suggested I should have plotted Figure 3 that way — as a point-free grey band. He’s probably right, as doing so would have done a better job of visually communicating the message of physical inaccuracy in the surface air temperature anomaly record.

Michael wrote that he had come to a conclusion very similar to mine after his much more intensive and complete study. We’ve remained in contact since, and he continues in agreement.

So here’s a conundrum for you to consider, Jeff. With systematic error it’s possible to show that a serial trend in the data points themselves is statistically significant, i.e., passes the Student t-test, or, e.g., the Mann-Kendall test, all the while the trend itself remains physically meaningless

because of the poor accuracy of the underlying data points themselves.Your point about trend analysis is implicitly about precision. The systematic error is about accuracy. If the data points are fundamentally of low accuracy, it doesn’t matter what statistics says about their trend. If the inaccuracy in the data is larger than the trend, the trend is physically meaningless.

And that’s the meaning of paper 1, Figure 3. And, really, paper 2 Figure 4 has pretty much the same message, as it’s just the (+/-)0.46 C systematic error (anomaly inaccuracy) convolved with Jim Hansen’s natural variability carried over to the CRU data set.

So, there’s really no point in analyzing the trend as you suggest, Jeff. For one thing, the distribution of systematic error is unknown and so there’s no way to choose among statistical models. For two things, the underlying data are so inaccurate that any trend with a net excursion from the zero line of less than about 0.9 C can’t be said to have any proper physical existence. No matter what the trend statistics say.

Accuracy is not precision, and statistics can say nothing physically useful about trends in terminally inaccurate data.

Finally, you wrote, “

Again, your context is one of statistics, whether you define it that way or not.” No. The papers are about real physical observables. The context is science. Statistics is imported as a tool. It does not govern ultimate meaning.Statistics only illuminates where physical meaning may be derived, and then only under conditions where the accuracy of the data warrants the use of statistical methods.

You and Lucia have no case. Neither of you have shown yourselves capable of allowing my paper to mean what it clearly says.

## steve fitzpatrick said

Stop wasting your time Pat. You still are confusing internal variability with uncertainty about the true state of the system. The true state is what you measure, there is no added uncertainty in that measured state beyond normal measurement uncertainty (the sources of which have been discussed in dozens of papers and on endless climate blog threads).

I once had a guy who worked for me who argued endlessly over nonsense (and he had a PhD in chemical engineering from a reputable university). After about of year, I asked him to look for another job; he was wasting too many people’s time, including mine.

## Pat said

#41 Lucia, after quoting yourself and Jeff, you wrote, “

So, in the bit you link I“a) Do not say “paper 1″ and

“b) Certainly am not discussing the “tables or figures”.

“c) I am not discussing any numerical results whatsoever.

“I have not said anything about tables and figures in paper 1.”I went step-wise through all your posts on the prior thread, Lucia. Your own posts refute you.

Consider: in #50, Jeff wrote that, “

Lucia pointed this out to me so I don’t get credit for finding it, but the problem I have with the method is that monthly weather variance is treated as uncertainty …(bold was italics in the original)”Constructing error bars based on this expected variance tells us little about the error in our knowledge of the actual earth temperature at that month.So, Jeff says that

youpointed out to him that I treated weather noise as an error variance, and constructed error bars on that basis. The errors bars I constructed were taken from the Tables and appear in the Figures of paper 1. Jeff’s comments only make sense if the errors you and he purported actually appear in my paper.So, either:

1) your original claim is sensible in that it supposed that the purported errors actually appeared in the Tables and Figures of my paper and therefore your present rendition of your case is inaccurate,

or;

2) your present rendition is accurate, you originally were

notclaiming that your purported errors actually appeared in my paper and your original proposition was senseless.In the event of 2) being true, of course, Jeff will have totally misunderstood what you had originally pointed out to him, because

hewas clearly under the impression that you were talking about mistaken error bars that actually appeared in my paper.Second, in post #79, Carrick reproduced what he considered your “punchline” comment from your the algebra critique: “

Contrary to Pat Frank’s discussion in section 2, the spread in the temperatures over the “M” days of the month makes absolutely no contribution to the uncertainty in CRU’s ability to estimate the mean value of the M temperatures”This quote shows beyond a doubt that your central objection was based in the idea that I had represented weather noise as an error variance in my paper 1.

Anyone actually reading my paper 1 for content, and critically looking at the Figures, would know your claim is false. Actually, the fact that Carrick credited your claim at all is evidence of his own uncritical approach here: he could not have accepted your claim had he ever read the paper himself. Carrick uncritical. Who’d a thunk … :-)

Third: in post #100, Jeff wrote that, “

Pat’s paper implies that this [weather noise] variance, which exists in nature, is in fact error in accurate knowledge of temperature…”To which you replied, in #101, “

Precisely. Mind you: The variance in the weather can be interesting…”This shows, again beyond a shadow of a doubt, that you were claiming I represented weather noise as an error variance in paper 1. As paper 1 presented the calculated errors in its Tables and Figures, it’s ludicrous of you to now suggest that you never implied that the Figures and Tables of paper 1 should include your claimed mistake of ‘

weather-noise-as-error.’You went on to write that, “

It is a discussion of the abstract concept you are introducing. … The discussion is about”concepts. not tables and figures (bold was italics in the original).In post #132, you go step-wise though your error claims. Apart from revealing that you have entirely misunderstood both of my papers, none of the steps you yourself listed show that you objected merely to a concept. Everything is about what I actually computed. And what I actually computed appeared in the Tables and Figures of the paper.

In post 139 you wrote, “

If all your uncertainty intervals mean is that spread of temperatures in batch A and B overlap– well… all righty.” showing once again your claim that “weather-noise-as-error” should appear in the Tables and Figures of paper 1. And once again all about what I calculated and, nothing about limiting your objections to concepts.You also repeatedly misinterpreted “uncertainty” to be identical to “error,” which it is not, and thanks a lot again, Mark. :-)

That, again, is about calculations and not about concepts.

The first mention of your problem with Case 2 mentions nothing about an objection to the concept.

Your first ever objection was in #71, where you wrote that, “

Your error occurs in paper 1. You went astray by never defining what the “true” value corresponding to what CRU is trying to report…” Whatever else one might suppose, your objection is to an error calculation and not to a concept.Your #41 post here has all the earmarks of you reformulating your argument away from what it actually was. It looks exactly like an attempt to back-pedal away from your original argument — a factually insupportable attempt at face-saving.

Not one of your original objections has survived scrutiny. There are no mistaken weather noise error-bars in paper 1, Case 2 is not about weather noise, and magnitude uncertainty does not mean measurement error. And so having failed in all that, you’ve taken refuge in a non-sequitur.

You finished with, “

Look Pat, I don’t take your paper seriously. I think it’s meaningless exercise in sophistry.”That’s no real problem, Lucia. I no longer take your objections seriously in any objective sense. I’m just sorry that so much dust was raised. Naïve or uncritical folks could suppose that the cloud of dirt you and Jeff kicked up is really smoke, and get misled by your vehement mistakes.

As to sophistry, you have provided so much solid evidence of being unable to read my paper for its actual meaning that your views on its message are objectively meritless.

Jeff finished up #40 with, “

I’m sorry but the work is badly flawed. … You should listen to what Lucia, I and others have written on this…”In fact, though, it’s you and Jeff who should have listened to what I and others have written — others folks being Jstults, Mark T, and even Roman who succinctly repudiated your take on Case 2 and weather noise.

It’s Jeff’s perception, and yours, that is badly flawed. At the end you were left with no case.

## Pat said

#43 you’ve never given any evidence of ever having read my paper, Steve. All you’ve done is echo Lucia’s and Jeff’s mistaken views about weather noise and (+/-)s.

As such your posts have been, and continue to be, empty of any critical content. Jeff and Lucia have been thoroughly refuted. So therefore have you.

## DeWitt Payne said

Pat,

Take an arbitrary function, say the Mexican Hat, on a two dimensional plane. Now take 1,000 randomly placed measurements on the plane with an uncertainty of small fraction of the range of the data. What’s the uncertainty of the average relative to the uncertainty of the measurement? Take a different 1,000 points, what’s the expected value of the difference of the two averages? Now take 1,000 sets of measurements with 100 points, 1,000 points and 10,000 points. Does the uncertainty in the average decrease with the number of points?

We can make the experiment even more complicated by adding time variation of the function with a known period and taking measurements at equally spaced time intervals over the period and averaging over both time and space. But it won’t make a difference.

I have read your papers, they are wrong and Jeff and Lucia have not been refuted at all, much less thoroughly.

## Mark T said

Dewitt… Your example is only relevant for distributions that have a defined mean that is also finite. Your point is true, but it does not address Pat’s assertion.

Mark

## Jeff Id said

Jeff and Lucia have been thoroughly refuted.

Ouch, that makes me want to put more time in but I haven’t gotten anywhere. I am certain of my correctness as you have stated the uncertainty of the mean is +/-s in the quotes above. You have also stated it is not the uncertainty of the mean. While each individual statement is understandable, these are completely contradictory and make the rest indecipherable. It has absolutely nothing to do with the uncertainty of the mean which is in fact your sigma and your sigma alone no ‘s’ required.

## Mark T said

Going back to the Cauchy distribution, which I mentioned previously, it would be intersting to see how it behaves if extreme values are truncated or even ignored. Realistically, any distribution that has infinite tails is impossible in physical systems so any manually generated values will likely result in cases that simply cannot happen, e.g., we will never see a 1000 F jump in temperature in one day (barring some anomalous external event.)

Oh, and btw, Cauchy actually “looks” Gaussian minus the extreme outliers. Many others do, too (heavier tails.)

Mark

## Mark T said

I remember once somebody commenting that “none of you”, skeptics and warmists alike, knows how to deal with heavy tailed distributions. I wish I remembered the context.

Mark

## John Pittman said

MarkT , IIRC it was M Tobis at Lucia from such fun as this http://rankexploits.com/musings/2011/blah-blah-blah-mt-and-communicating-climate/ .

## Mark T said

Damn, figures my phone is dying and I cannot check. That and I’m fairly fitshaced. Last day of “vacation” before the new job. Thanks… I will look later.

Mark

## Jeff Id said

Mark,

I know you’re not particularly interested in this work, spend a few minutes reading the paper #1 and its claims and see if they hold up to your scrutiny. My guess is 5 minutes of serious reading and you will figure this out.

Focus on the uncertainty of Tbar in case 2. After that, the rest follows a similar path.

Pat,

Roman’s comments were not supportive. He can speak for himself though.

I am sorrry, don’t forget that. It isn’t a “mean” thing, it is just incorrect and unconvincing.

## Pat Frank said

#46 DeWitt, you say that you read my papers, the results delimited in which concern subjective estimates of error and systematic sensor error, and then presume to criticize those results in terms of the 1/sqrtN decline in random error. Your criticism is a rational non-sequitur.

Mark T is entirely correct. Thanks yet again, Mark.

Let me summarize again, and I hope this is not too tedious for those who have actually followed the conversation here. I asked Jeff to point to his claimed

weather-noise-as-error-variancemistake in any of my Tables or Figures. He couldn’t do so, because there is no such mistake. Jeff admits as much in his head post. That dispensed with his — and Lucia’s — central claim against the uncertainty variances presented in paper 1.That left Jeff’s second claim, about “weather noise,” strictly isolated within Case 2, with no further ramifications anywhere else in that paper. Jeff’s second claim was that Case 2 was about weather noise. But his claim runs completely against the logic of that derivation, or any derivation, which is to develop the ramifications of the axioms. The definitional axioms of Case 2 are very general and do not necessarily concern weather noise. Jeff’s understanding is wrong. I’ve explained that it’s wrong in as many ways as I could do, here, here, here, here and here, for example, even including pointing out that the derivation in Case 2 retains its identical meaning without any reference at all to weather noise or even to temperature. All to no avail.

Further, whatever else one may think, Roman’s comment agreed with my explanations, pointing out in no uncertain terms that, “

All of the discussion of weather and signals is a red herring in the context of the cases presented by Pat in his paper.”Jeff’s claim that Case 2 is about weather noise is wrong.

That leaves only his third claim that “magnitude uncertainty,” (+/-)s, as it is described in paper 1, represents an error in the mean. Jeff specifically makes that claim here, writing, “

Currently Pat is claiming that his ‘magnitude error’ is not the error in mean. His paper says it is the error in mean,…”Nowhere in my paper does the term “

magnitude error” appear. Jeff has invented this term, has applied it to me, and has then represented it asmymistake. In fact, it’s his mistake.Further, nowhere in my paper is (+/-)s represented as an “error,” and nowhere is “magnitude uncertainty” ever described as an error in the mean. It’s always described as a measure of system heterogeneity; as the inherent variability of the sequentially derived means of an inherently heterogeneous system. I’ve explained this, too, umpteen times, and again to no avail.

The paper can be searched. Jeff’s invented term is not present anywhere. His purported meaning doesn’t exist in my papers.

Jeff and Lucia’s case has been entirely lost. Their #1 and #3 claims are not present and their #2 claim is wrong on its face.

## Pat Frank said

#48 Jeff, you wrote, “

am certain of my correctness as you have stated the uncertainty of the mean is +/-s in the quotes above. You have also stated it is not the uncertainty of the mean.”You’ve shifted your ground. Your claim previously and in your head post was that I had represented (+/-)s as an

errorof the mean.Now you’ve switched to “

uncertainty.”It’s true that (+/-)s is described as an “

uncertainty” in my paper, but that uncertainty is not related to an error. It’s related to the variability of the mean due to system heterogeneity.You wrote that, “

you have stated the uncertainty of the mean is +/-s [and] have also stated it is not the uncertainty of the mean.”This is not true. I have never, ever stated that (+/-)s is an

errorin the mean. In all my posts I have argued against this meaning, which you and Lucia have insistently and unfairly attempted to impose on me. I havealwaysused (+/-)s = “magnitude uncertainty,” strictly as a measure of inherent variability.In order to sustain your claim of a mistake — or in this case a self-contradiction — you have invariably insisted that

error=uncertainty, in terms of analytical meaning.However, that equation is false. It is your invention. It is not supported by anything in my paper.

You’ve got no grounds to stand on, Jeff. You’re just arguing your own invention.

## Jeff Id said

Pat,

You are wrong and I have almost given up. Replace magnitude error with magnitude uncertainty in my previous post please.

“It’s related to the variability of the mean due to system heterogeneity.”

It is bovine scatology and has no relationship to statistical uncertainty it is sophistry because nobody disagrees with your statement as said. But you HAVE contradicted yourself on this exact point several times. It does relate to the true variance of a measurement. Cap for emphasis, not yelling.

AGAIN, I HAVE MADE NO CLAIMS ABOUT YOUR TABLES OR FIGURES ALTHOUGH MY FIRST COMMENTS DID INADVERTENTLY IMPLY IT, I HADN’T EVEN READ THE TABLE AT THAT POINT. MY DISCUSSION AND FOCUS HAS BEEN VERY SPECIFIC AND YOU HAVE FAILED OT REPLY TO MY SPECIFIC POINTS. WHEN YOU HAVE ATTEMPTED TO REPLY, THE CONTRADICTION WAS REPEATED.

If we could resolve the OBVIOUS contradictions in your case 2 text, we could take the next step to the tables, because there are serious and incontrovertible problems there too. Until you reply to my case 2 issues specifically, without writing a book, without discussing the weather noise silliness (on which I am also correct in case 2), we cannot move forward.

You have stated your case two ways —- s is the “uncertainty in the mean temperature” AND “Magnitude uncertainty is not a measure of non-knowledge of the mean value”?? —I have given examples above of your exact quotes.

One is correct, the other is not. The paper has the incorrect version both in the equation and in the text. Resolve this serious problem, and we can move on to the next problem.

In my opinion, you are the one without grounds and we haven’t even gotten past the most basic statistics.

## Anonymous said

It’s not that I’m not interested in the work itself, it’s that I’m not interested in the time sink I know it will become. This is a food fight at best (looking at the bulk of the comments between the several threads.) It’s much easier (and more fun) to simply point out when food is being flung at the wrong target. Citing instances of the LLN and/or CLT while using i.i.d. data, all with stationary statistics, is hardly a convincing argument when Pat has made it clear his assumption is that nothing is i.i.d. or stationary. Whether or not his assertion is true is another matter, as is the construction he has presented (which is really where the time sink comes in… the former point is argued best by Zeke’s demonstration, maybe, though not alltogether convincingly without more investigation.)

The easy stuff is the stuff on the top of my head… same reason it took till tonight before I implemented any of Bart’s code over at CA (only peripheral, btw, I’ve done similar stuff quite often, however, and it all passed the first cut smell test when I read it) and never got too deep into any answers. Damn Nick can be annoying.

Mark

## Jeff Id said

Mark,

Pat’s assertion is not backed up in the text or the equations, and that is where the five minutes comes in. The next step is the time sink as you have pointed out. I was going to put the time in but have lost some interest in that.

## Pat Frank said

#56 Jeff, in your head post you explain your primary objection as “

Equation 6 takes the spread of individual temperature stations and incorporates them into an error calculation s….”However, “s” is not an “error.” It is never defined as an error, nor is “s” ever expressed as an “error” and nor is it ever used to represent an “error.”

That sentence expresses your primary mistake of interpretation. In order to support your argument about “s,” you (and Lucia) have invariably claimed an identity of meaning between “uncertainty” and “error” and you’re just as invariably wrong. The meaning of “s” is not “error” and your use of the term “error” when discussing “s” makes your entire argument no more than merely tendentious.

In your head post you claim that I have, “

incorrectly defined ‘s’ throughout the paper.” But in fact throughout the paper I define “s” in terms of system heterogeneity. That’s how it’s defined when it first appears, and how it’s consistently defined and used throughout the paper.It is you (and Lucia) who redefine “s” to mean measurement error and then enforce your mistaken interpretation throughout my paper, thereby fomenting

yourmisconstrual to bemymistake.You wrote that in my paper I expressed “s” as, “…

” However, that is not true, either. When “s” isthe“uncertainty in the mean temperature”…first introducedit’s described as, “.”a furthersource of uncertaintyNote the use of “further.” That implies something else in addition to measurement error.

Let’s also notice that the introductory sentence, “

However, a further source of uncertainty…”,does notsay, ‘a further source of uncertainty.’ Instead, Case 2 goes on to state that this uncertainty arises from the “in the mean… scatter of the inherently different temperature magnitudes(my bold)” And what does this scatter mean?aboutthe mean…Nothing about measurement error. According to the text under Case 2, (+/-)s “

is a measure of how well a mean represents the state of the system.” Notice that: (+/-)s is not, and is never, represented as a measurement error. It’s also not represented there, nor anywhere else, as an error or uncertainty in the value of the mean itself. Instead, it is everywhere represented as an uncertainty associated with the mean of a heterogeneous state, with that uncertainty due to the inherent variability of the sub-states. I’ve explained this repeatedly, but to no avail.In my paper, the uncertainty in the mean due to random measurement error is (+/-)sigma, and (+/-)sigma was already introduced in Case 1. So by Case 2, the reader is already aware that random measurement error is given by sigma. Then in Case 2, (+/-)s, described as a “further” uncertainty — clearly implying something other than random measurement error. You (and Lucia) have invariably missed this very obvious step of progressional logic.

There’s another point to be made about you describing “s” as, “

.” Your use of the definite article in your phraseology makes it sound as though I represent “s” asthe“uncertainty in the mean temperature”…theuncertainty in the mean, i.e, the primary source of error: the measurement error. That is inferentially wrong, and your definite article phraseology is implicately wrong. You complain about sophistry, Jeff — well, your usage of “the” is sophistry; innocent though it may be. And that sophistry turns your interpretation into “bovine scatology.”At the end, there is no contradiction in my text. There is only your insistent imposition of an inordinate import and a mistaken meaning onto Case 2, and then smearing your mistake all over my paper.

And you may call references to weather noise “silliness,” but in fact

you’rethe one who raised that issue. From your head post: “In other words, Pat is accidentally making the claim that there is an uncertainty s created by the delta tau from the mean of the TRUE temperature magnitudes. This is(my bold)”weather noise, and this is false.In fact, weather noise has no necessary relation at all to Case 2, which I’ve endlessly pointed out and explained the why. But you have invariably brushed it all aside in an apparent determination to stick with your own mistaken interpretation.

## Frank said

Jeff, Pat, and others: It seems to this observer that your disagreements would be resolved by creating artificial temperature data with known statistical properties and then analyzing it (or by thinking through this process). What signals might be put in our artificial temperature data (mean daily temperature anomalies for as many stations as you wish covering one to several centuries)?

Changing Global Climate (processes typically lasting longer than 10 years and effecting all or most stations simultaneously):

1) Anthropogenic Change: a linear increase of 0.5-5 degC/century or a more complicated function (immediate or lagged) that depends GHG measurements or scenarios).

2) The PDO and possibly related oscillations in the form of a sine function with noise and/or chaotic function which tends to change on a multi-decade scale.

3) The 11-year solar cycle as a sine function with noise.

4) ???Other suspected extraterrestrial influences???

Changing Weather (processes typically lasting less than 10 years and often effecting only some stations):

5) ENSO (trig function with noise and changing 2-3 year period)

6) MJO (trig function with noise and 1-2 month period)

7) Chaotic functions chaotic on a time scale of several days and several weeks and several months.

8) Gaussian noise? (I wouldn’t.)

Measurement Issues:

9) Instrument noise (Gaussian)

10) ???Siting issues??? (growing UHI bias as some sites, step function to mimic station changes or moves).

Now plot the annual (or quarterly or monthly) global mean temperature anomalies vs time. What kind of error bars do you want on the data? (You choose, but two standard deviations of the mean for me, since we are plotting means).

At the end of this exercise, my hypothesis is that we will find that: 1) Jeff, Lucia and company are correct about the uncertainty associated with each point and 2) Pat’s analysis is directed towards characterizing the variability (which he may call “magnitude uncertainty”) obscuring the signal(s) associated with changing global climate. Even if my hypothesis is ridiculous, this thought experiment or creating and analyzing data could lead to a simple resolution of your disagreement.