Monthly Averages, Anomalies, and Uncertainties

Jones, CRU: “The global and hemispheric averages are now given to a precision of three decimal places” http://cdiac.ornl.gov/trends/temp/jonescru/jones.html details here http://www.metoffice.gov.uk/hadobs/hadcrut3/HadCRUT3_accepted.pdf
He assumes that more data increases accuracy. On this basis one could calculate very accurate global temperature by getting people around the world to hold their fingers in the air and take estimates.
Even recording accuracy was only +/- 0.5 deg C until recently.

Willis,
Interesting post, made me read their “Robert Rohde, Richard A. Muller, et al. (2013) Berkeley Earth Temperature Averaging Process.” which is kind of disturbing in how they treated the data. After reading that I am pretty much going to ignore BEST from here on out. But relevant to your post is this note buried in their paper, which by the way is supposed to make the data more accurate through a mathematical process. Not sure you can do that without putting your own biases into the data but its what they have done.
“Note, however, that the uncertainty associated with the absolute temperature value is larger than the uncertainty associated with the changes, i.e. the anomalies. The increased error results from the large range of variations in bi from roughly 30°C at the tropics to about -50°C in Antarctica, as well as the rapid spatial changes associated with variations in surface elevation. For temperature differences, the C(x) term cancels (it doesn’t depend on time) and that leads to much smaller uncertainties for anomaly estimates than for the absolute temperatures.”
I hope this helps.
v/r,
David Riser

Willis,
I’m not sure how they are reporting monthly error, but there is a serious logical problem in the published Berkley error reporting which uses an incorrectly modified version of the jackknife method for calculation.
http://noconsensus.wordpress.com/2011/11/20/problems-with-berkeley-weighted-jackknife-method/
http://noconsensus.wordpress.com/2011/11/01/more-best-confidence-interval-discussion/
http://noconsensus.wordpress.com/2011/10/30/overconfidence-error-in-best/
The authors have failed to either attempt a correction or even respond with any kind of discussion on the matter.
The error you report, is exactly the type of thing which could happen by their broken jackknife method.

The shaded regions are the one- and two-standard deviation uncertainties calculated including both statistical and spatial sampling errors.
It’s always bothered me that they can quantify spatial sampling uncertainty.
Tom Karl on the subject,
Long-term (50 to 100 years) and short-term (10 to 30 years) global and hemispheric trends of temperature have an inherent unknown error due to incomplete and nonrandom spatial sampling.
http://journals.ametsoc.org/doi/abs/10.1175/1520-0442(1994)007%3C1144%3AGAHTTU%3E2.0.CO%3B2

It’s rather small, since it is the error in a 30 year mean being added to the error of a single month, so the extra error is prima facie (in that case) about 1/30 of the total.

Let us say I have been driving an automobile for 30 years, & carefully recording the mileage & the amount of fuel used (but nothing else).
What is the formula to derive the error of my odometer from the mileage & fuel used?
If I fuel up twice as often (& thus record twice as many mileage & fuel entries), do my figures get more or less accurate? By what percentage?
How can I calculate my neighbour’s milage from this?

Nick Stokes- you say
“It’s rather small, since it is the error in a 30 year mean being added to the error of a single month, so the extra error is prima facie (in that case) about 1/30 of the total.”
How’s that again? I think you mean 1/sqrt(30).
I hope.

So, if I understand this correctly, the anomaly is a comparison between the monthly value for each station and the average of all the stations for that month over whatever number of years?
Or is it the monthly value for each station compared to the average of that station for that month over whatever number of years?
In the latter case systematic error is present and cancels out (being equal in both sets of figures) in both the monthly and the monthly ‘average over years’ values, whereas in the former case systematic errors common to any one station would be random over all the stations (and thus cancelling with increasing number of samples), but still present for the data from any one station.

I agree with Nick …. I don’t know if he (or Willis) are right or wrong – but I do know denigrating comments absent supporting evidence do zero towards understanding the issue or finding answers.
Science should be about collaborative effort. And anytime you have knowledgeable folks willing to engage in discussion you should take advantage of it.

A. Scott says:
August 17, 2013 at 5:10 pm
Science should be about collaborative effort. And anytime you have knowledgeable folks willing to engage in discussion you should take advantage of it.
>>>>>
That is what WUWT is all about.
Welcome aboard.

Nick Stokes says:
August 17, 2013 at 5:13 pm
An irony here is that skeptics have been nagging climate scientists to get the help of statisticians. So when a group of statisticians (at BEST) do get involved, what we hear here is “Climate science gets it wrong again”.
================
Ironing, not to mention nagging, has what to do with statistics ?

Nick,
Neither Muller or Rhode are statisticians, and based on reading their methodology I am certain that they designed best with the intention to prove CAGW. The methodology they are employing is sketchy at best. Please read http://www.scitechnol.com/GIGS/GIGS-1-103.php
and you will see what I mean.
v/r,
David Riser

Steven Mosher says:
August 17, 2013 at 6:34 pm
“Its not ironic. its typical”
================
Care to enlighten us ?
We’re all ears, that is why we are here.

“If you want to call this something different than temperatures then humpty dumpty has a place on the wall next to him.”
Nice to know where to find you Moshpup!! 8>)

In essence, an anomaly is just creating an offset from some (somewhat) arbitrary base line. No different than setting an offset on an oscilloscope. The question I have is related to the multiple baselines used. The data from each month’s average is converted into an anomaly relative to that months baseline. Each month’s baseline will have a different associated error, hence won’t there be be a potential misalignment between different months? Even worse, if the range used for a particular month is biased relative to the entire dataset for that month (say a cold series of winters), wouldn’t it induce a bias in the anomaly for that month relative to the other months?

Willis,
I join you in wondering about uncertainties. However, I don’t think the uncertainty in the baseline monthly average is a source of error here.
Here’s why: Since we are looking for trends — mostly secular trends using linear regression — we are only concerned with the errors that affect the confidence interval of the slope.
But the baseline monthly average has an effect only on the confidence interval of the intercept of our anomaly graph. So that might affect some sensationalist headlines (“Temperatures are up 2.0 deg C since 1850”, when the true value might be 1.8 deg C), but it will not affect most of them, for the simple reason that most of the headlines focus on the purported effects of forcing, expressed in terms of a slope: “We expect a 0.2 deg C rise per decade for the next two decades.” Those remain unchanged by an epsilon in the monthly baseline.
In other words, we can safely accept the baseline monthlies as givens, and look for trends from there.

From the NIST Engineering Statistics Handbook “Accuracy is a qualitative term referring to whether there is agreement between a measurement made on an object and its true (target or reference) value.” Since we do not have a reference value, and the true value is what we are trying to determine, accuracy statements are essentially meaningless in this case. Individual measurements can have uncertainty associated with them from the precision (repeatability/reproducibility) and the uncertainty of aggregate quantities such as averages calculated. But since the “true” or “reference” value is essentially unknowable, accuracy as such is not a useful term.

Is it not the case that the time series used to calculate the average typically is older by a good margin than the data set under analysis? This is why we see modern anomalies compared to 1979-2000 data, for example.

Hello,
A very good discussion. Lots to learn and digest.
Yes. I asked a question earlier about the probability of ‘a’ particular temperature ‘occurring’ at ‘a’ particular thermometer, whether on — not actually ‘on’ but usually about 1 meter to 2 meter above the ground or in — within a meter below the ocean surface or to due to storminess was more near 5 meter below the ocean surface because of ocean wave dynamics on sea-going mercury thermometer.
I do hope that in the future, yes I really do, those who are ‘reading’ the mercury thermometer — such as “on” land — do employ an accurate chronometer in order to know the ‘real’ time, and longitude a very important number, of the “reading” of the mercury thermometer and do awake in the early morning hours without effects of lack of sleep and ‘associated’ affects of alcohol over consumption — sometime long ago referred to as “consumption” as the papers of the day would read “death by consumption” said papers having been written in the US ‘Prohibition Era.’
Cheers

Whatta bout the the handling of the 1/4 day/year shift of the months vs the solar input. There should be some 4 year period signal in the data that needs to be handled correctly.

“The raw data represents itself as temperatures recorded in C
Using that data we estimate the field in C
If you want to call this something different than temperatures then humpty dumpty has a place on the wall next to him.”
The raw data is the problem , you don’t have enough, neither in quantity, location and time(history) for the small differences.

Lubos, is this true only in cases where the measurements are of the same thing?
“So while the error of the sum grows like sqrt(N) if you correctly add (let us assume) comparable errors in quadrature, the (statistical) error of the average goes down like 1/sqrt(N).”
I think there is confusion about the nature of the raw data. I think the above applies when each recording station has made measurements at all stations and the data sets are averaged. I don’t think it applies to a set of individual data sets, one from each locality. No calculated result made from imprecise data, where each has measured a different thing, can have a greater precision than the raw data.

Willis and all, shurely the main problem is not the error of the average. This must be small due to the large sampling. The main problem is that the statistics only catch random errors and we have no reason whatsoever to think that they are random. Many things have changed that is not random but systematic. Sea routes, built and paved environment, the increase and then decrease of land thermometers, cultivation…and on and on. These are not random errors! McKittrick showed a correlation to industrialization, pielke showed a correlation to land use. Correlations that shouldn’t be there if errors were random!

Mosher, I’m still waiting for your list of 39,000 stations for 1880 – or any particular year of your choosing for that matter.

As surely as you need to draw another breath to survive, Statisticians will always make the assumptions upon which their enterprise is theoretically founded.
The data are useful for exploratory purposes, but the whole enterprise of climate stat inference rests upon contextually fundamentally indefensible assumptions, notably random iid. The confidence intervals cannot sensibly be interpreted as meaning what pure theory would suggest they should mean.
Our understanding of climate is at best in the exploratory stage. It is only by understanding natural climate at a profoundly deeper level that bad assumptions can be weeded out of fundamentally corrupted attempts at climate stat inference.

Willis, I’m with you. But I think the problem is even more basic. This is a link to a lengthy comment from Aug. 12, 2012 that identifies three sources of error in the calculation of the anomalies:

The process has been to first convert the Tmin and Tmax into a daily Tave, having no uncertainty associated with it. Everything from then on uses the Tave and the only uncertainty that is use past that point is what derives from the difference of Tave – Tbase to get an anomaly.
But if we go back to what we actually measured, the Tmin and Tmax, the uncertainty in the system really is large which must be factored into estimates of uncertainty in the slope of the calculated temperature trends.

We do not measure Tave. Tave is a calculation of two components we do measure: Tmin and Tmax. If Tmin and Tmax are separated by just 10 deg C, then the mean std error of a month’s Tave, 30 readings of Tmin and Tmax, must be at least (10/2)/sqrt(30) = or about 0.9 deg C. This is the error of the monthly anomaly against a known base.
But the base, too, has uncertainty, not from 30 years, of 30 Tave for each month, but 30*30 Tmins and Tmaxs. The 30 year average Base temperature each month should have a mean std error of at least 10/2/sqrt(30*30) or 5/30 or 0.16 deg C.
To get the anomally, you subtract Tave – Tbase, but Tave has an uncertainty of 0.9 and Tbase has an uncertainty of 0.16. Uncertainties add via root sum of squares, so the uncertainty of the anomally is 0.914. So the uncertainty of the Tbase can be neglected because the uncertainty in each month’s Tave dominates the result.
BEST compounds this error by slicing long temperature records into shorter ones, ignoring the absolute temperature readings and working only with short slopes. I have repeatedly objected to this from a Fourier Analysis, Information Theory approach, with its loss of low frequency content. The scalpel may be removing the removing the essential recalibration and keeping the instrument drift as real data.
But let’s just look at the implication of BEST slicing records from an uncertainty analysis point of view. The uncertainty of the slope of a linear regression is ( I think) proportional to 1/(square of time length of the series). Cut a 400 month time series into two 200 month series, and the uncertainty in the 200 month slope will be 4 times the uncertainty of the 400 month series. Sure, you now have two records and optimistically you might reduce uncertainty by sqrt(2). Slicing the record was a loosing game for the uncertainty in slope is at best increased by 2*sqrt(2) or almost a factor of 3.

There is something that has been bothering me about ‘error’ in statistics. in particular when standard deviations are used on ‘average’ calculations. I dont mean the average of a dataset, I mean an average of the averages of a number of datasets.
So you have a group of averages that are then averaged out again, for example the Artic Sea Ice Extent – On one of the charts you see the +/- 2 Standard Deviations. But what is the SD calculated from. Is it the total individual data points or is it the average of the data points ? Or daily average temperatures (h+l)/2
If I give an analogy of why this is not right it may help understand my question.
If each year the height of each schoolchild on their 3rd birthday is taken at each school in a county and the average is calculated (which would be around 37 inches) .. This would include those that were say 33 inches to 42 inches which would (probably) be in the normal deviation … However, when the averages are sent to a central state center and the standard deviation is calculated on the averages, then the distribution will be much MUCH smaller and those at the extremes of the distribution would be considered abnormally short or tall. This is probably not how height statistics are calculated, but for Artic Sea Ice and temperature I can not see how they would discern between a (daily high + daily low)/2 is not an average. Also, do they phisically go out and measure every single ice flow in the Arctic ?
I think what would be better in temperatures would be to take the daily highs and lows separately then do the statistics on those, but to collate them all together come up with an average and SD would not be truly representative of the statistics.
If I am wrong in these assumption, then i have leant something, but if i am right I hope someone else learns something.

SIGHhhh….
This seems to be a discussion where the same word is used to mean two entirely different things (So what else is new in the Post Normal World.)
To those of us in industry the word ‘Error’ has a very specific meaning. It is the deviation from the ‘correct’ or ‘true’ value.

Definition:
A statistical error is the (unknown) difference between the retained value and the true value.
Context:
It is immediately associated with accuracy since accuracy is used to mean “the inverse of the total error, including bias and variance” (Kish, Survey Sampling, 1965). The larger the error, the lower the accuracy.
http://stats.oecd.org/glossary/detail.asp?ID=5058

Mosher on the other hand is talking about a different definition of error. Kriging is a method of Interpolation when there is not enough data. Within the concept of Kriging there is the nugget effect.

..The nugget effect can be attributed to measurement errors or spatial sources of variation at distances smaller than the sampling interval or both. Measurement error occurs because of the error inherent in measuring devices…
http://help.arcgis.com/en/arcgisdesktop/10.0/help/index.html#//0031000000mq000000

That is the ‘Error” Mosher is talking about.
Kriging comes out of gold mining and other geological work where the geologist is trying to make the best guess with insufficient data. The ‘Error” in Kriging is the error of that guess, and really has nothing to do with the error of the data which Pat Frank and I and others are talking about.
It also seems to be ‘Controversial” to say the least. According to J W Merks ( Vice President, Quality Control Services, with the SGS Organization, a worldwide network of inspection companies that acts as referee between international trading partners) It is a fraud for producing data when you do not have any. see Geostatisics: From Human Error to Scientific Fraud http://www.geostatscam.com/
Also see the wikipedia talk section on Kriging: http://en.wikipedia.org/wiki/Talk:Kriging#Further_revision_proposal_by_Scheidtm
Hope that helps Willis.

So you average a month then do a difference between that average and the data to show the anomaly. An anomaly of what? The average is 0ne month not the average temperature of the same month in a climate cycle, roughly 30 years. Meaningless really. You are assuming that your month of choice is truly average but none of them are since every month of a cycle will be slightly different.
This is not a method I would use since it assumes too much and gets the guessing index up.

http://iopscience.iop.org/1748-9326/8/3/034018/article
Another global warming doom….

I’m not qualified to speak on the actual data/topic but wanted to say I am glad there are people always examining the data.
thanks willis.

Pat Frank,
“I carried that argument, Steve. If you don’t understand that after all that was written, then your view at best reflects incompetence.”
Without reopening a sore spot, I don’t remember it that way.
My opinion is that the BEST CI method is likely to be close but has the potential for big errors depending on the distribution of values.

It is a certainty that the measurement errors are systemic, and that is a can of worms, varying by instruments, by location changes, by nations, by mumber of stations, by station locations. It is also a certainty that the recorded official T is still being changed. The recorded data sets have been changed many times, and are still changing, so from which data set do you get your meausremnts and anomalies.
A far more accurate way of getting the measurement uncertainty is to observe it, instead of teasing it out through numeric sophistery. That is, observe the land based differences in anomaly trends, verses the satelite trend over the same land area. Otherwise it is fair to ask why the vast majority of the changes to measured T warm the present, and cool the past.
Of course, this is all academic, as all the reported disasters of CAGW are not happening. (Even NH sea ice is now strongly rebonding with changes in ocean currents and jet streams, besides which, no one ever gave observational evidence of world calamity with melting NH sea ice) ) So the3 disaster are not happening, and neither is the warming. In CAGW neither the “C” or the “W” (for the past 15 years) is happening. The entire SH never warmed much at all so the “G” is also missing. Very Shakeperian, “Much ado about nothing”

Nick Stokes says:
August 18, 2013 at 5:56 am
It’s probably best to just think of a Monte Carlo
I was thinking of a much more basic thought experiment.
Start with lots of devices that measure twice a day.
Each device has an assumed accuracy of x
Human reading is to the nearest 0.5 degree which introduces error y
Tmax and Tmin are averaged to give daily average which does zz to the error
Daily average is averaged again to give monthly accuracy
Interpolations to cover missing values
Outliers are removed etc etc
Give all these errors names so that everyone is clear they are talking about the same thing, then show how they are accounted for and whether or not they are significant

The statement:
“If you want to know whether Aprils are getting warmer, say the BEST proponents according to my understanding, it doesn’t much matter what the error is in the average-April number you subtract from the individual-April numbers to get the April anomaly.”
seem debated. My focus is that on top of that, if some today measure april average as the average taken every second during the month, and earlier year was measured as the average of three readings per day, then there is a huge error added. Similarly, if different stations use different methods for their averaging, and then stations are used to adjust each other, errors are introduced.
— Mats —

Repost with correct words: There is an obvious problem with daily high/low averages, that is this: if the day time high is 72F for three hours vs 72 for 5 minutes, or the low is -15F for 5 hours vs 1 hour. See the problem with leaving duration out of the picture?

Thanks Nick

@Nick Stokes says:
August 18, 2013 at 5:56 am
It’s probably best to just think of a Monte Carlo
You need however to remember a basic problem with that … is climate deterministic? I think everyone agrees it is slightly chaotic and is probably the biggest source of error you are trying to work out how to handle.
Climate scientists actually need to do a little discussion with engineers who are well versed in slightly chaotic signals and signal processing. Brad above actually gave you one possibility by using an Unscented Kalman Filter (UKF … because it’s not linear) and I see a few in climate science have played with trying to adjust Lorenz (1996).
Until you people all realize you can’t solve these sorts of problems with statistics you are going to continue bang your head against the wall.
Willis you know I respect you and your tenacity but do you see the problem and that there is no way into this by statistics solely you need to deal with the slight chaos something statistics can’t do by itself. Willis spend some time talking to someone at a local university about signal processing chaotic signals. The other choice is someone involved in deep space radio comms such as flicker noise spectroscopy and the like.

Willis Eschenbach:

I understand all of that, Steven, and thanks for the explanation. The part I don’t understand is, how can the error in your published climatology be GREATER than the error in your published anomaly. That’s the part that you and Nick haven’t explained (or if so, I sure missed it).

If I’m following your question, you can have larger errors on an absolute scale than on a relative scale. If there is an overall offset error that is constant across measurements, using one (or a subset) of measurements as a new base line that you subtract all measurements from, reduces this offset error.
There’s a bit of discussion in this long thread on Lucia’s blog that may be helpful.

@Lubos and the all other wise contributors
“yes, 1/sqrt(N) is the behavior of the statistical error, e.g. a part of the error from individual errors that are independent for each measurement.
“Then there can be a systematic error that is “shared” by the quantities that are being averaged. The systematic error of the average of similar pieces of data doesn’t decrease with N but it is the same as it is for the individual entries – it can never be greater.”
+++++++++
Thanks. I almost entirely agree. There could be systematic errors in the way the averaging is done so I hold open that caveat. I think the problem with temperatures is the instruments have different precisions and accuracies but there are ways to account for that. That does not contradict your point at all, however.
My second worries on this relate to the kinda loose manner in which ‘accuracy’ and ‘precision’ are being used in peer reviewed works. When one gets a lot more data points within a territory, extending the shooting range metaphor I sued above, it means getting a lot more shooters to use a different rifle each to take shots at their respective targets. None of that increase makes the shooters more accurate or precise and again, the precision of each rifle is not improved by having more of them.
My refinement of the perspective is this: Imaging you did not see any of the shooting take place. You do not know where the bullseye was for any shooter. All you have is a blank target with a large number of holes shot into it. The task at hand is to work out where the bullseye really was – the actual average temperature. The calculations demonstrated do not decrease the size of the error bars which is rooted in the precision of the rifle and the shooter and the accuracy of the shooters.
What is increased is the precision of the location of that point where we can confidently place the center of the error bar. We can have more confidence that it lies at exactly a certain point, to several significant digits, but in no way does this certain knowledge reduce the vertical height of the error bars. For temperatures it is still ±0.5°C for most of the land record.
This point about confidence as opposed to knowledge is not being stated clearly. Claims for increased precision in our knowledge of the position of the center of the error bar is being claimed to be an increase in the precision of the calculated answer. They are two different things entirely. One is like the GPS coordinates of a car, the other is like the size of the car.
Here is another example in the form of a question:
If I measure the same mass piece with 1000 different calibrated scales each having a resolution of 20 g, how precisely and now accurately can I state that single piece’s true mass?
If I measure 1000 different mass pieces, one mass piece weighed once on each scale, with what precision and accuracy can I claim to know the average mass of all the pieces?
These are conceptually different problems. The latter question is the one that applies to temperature measurements at different stations. The final answer cannot be better than the best instrument but might be worse than the worst instrument because of various systematic errors in the processing of data, or experimental errors in acquiring, transcribing or archiving it.
Regards
Crispin

@Wilis: Super resolution microscopy uses techniques where total error (measuring the position of a light emitter) is smaller than the measurement error (the diffraction limit of light).
See: http://en.wikipedia.org/wiki/Super_resolution_microscopy
The PALM & STORM methods use the aggregate measurement and infer true position from assumptions about the point spread function and the probability of having more than one emitter in the field.
I’m not sure how that affects your argument, but, these techniques have ben validated in subcellular structures in many ways and they are examples of what you were asking for.

Pat Frank says:
August 18, 2013 at 11:05 am
Great work, Pat Frank. Too bad that those whom you intend to help just cannot imagine that empirical matters are relevant to their statistical claims. Please keep up your good work because many benefit from it, as does empirical science generally.
To those who have no respect for empirical matters in the surface temperature record, especially BEST, I have one question. When are you going to accept Anthony’s five-fold classification of measurement stations, work your statistical magic on his numbers, and then address his claims about bias in the measurement records?

To allow future improvement of measuring technology would use of absolute degree hours be very useful to calculate monthly average. K x h summed for each month. More frequent reading will show up as a more exact value. Old values will still be usable. Add then the monthly variance when presenting the resulting trends to give the reader a sanity check to see if the trend is within normal climate variance.

Since it’s been brought up a few times, there was a bit of discussion on Jeff’s blog of Pat Frank’s E&E paper. This is a good place to start.
Lucia had a follow-up here.
This is another place the basic issues surrounding measurement uncertainty in absolute versus relative temperature are explored.

Pat,
My critical position hasn’t changed. One thing you did do, that a lot of your critics don’t, is put your work out there and I hope people understand that is no small thing. It would be good to see more effort put into the data but we are not government paid climate citizens.

“Is there a way to transfer the raw data range into the anomaly?”
,
Bob Tisdale calculates weekly anomalies all the time. If you had daily data (and lots of it), you could calculate daily anomalies. If you had hourly data, say 10 years worth, or 87600 points of data, then yeah sure, you could do hourly anomalies as well.
I always find it fascinating that with just about all climate data anomalies there is an underlying cycle /trend (diurnal, seasonal) that is removed so that we can see the departures from what is considered ‘normal’. Sea Ice is an interesting anomaly recently because now even the anomalies themselves have a cycle embedded in them (recurring loss of summer sea ice). Maybe removing this new ‘recurring loss’ sea ice from the data might highlight something interesting (i.e. an anomaly of an anomaly!!!).
As for all this stuff about errors its given me a head ache! I will only add that when removing an recurring underlying cycle in the data, such as calculating anomalies, the standard deviation for each month will be exactly the same whether it an anomaly or not. So in essence the (monthly) data are unchanged.

In science (sorry to mention that largely taboo subject) you model known information and show that your treatment/theory/etc. can explain known observation. When you have convinced yourself (and others) of the validity of your treatment, you then make predictions. Applying this ‘scientific’ approach to the discussion here, one would have various options. For example, you could generate some synthetic temperature data for a set of geographically distributed measurement sites with known variations, trends, noise, etc. based on random numbers (aka Monte Carlo), and show that the error estimates and trends recovered by your analysis match those used in creating the synthetic input. Of you could could divide your input information in half and use half your input to show that it never falls outside the projected estimates obtained from the data input to your procedure, etc. (This might in fact turn out to be reasonable use of a GCM – see if the land temperature analysis can recover the inputs to the GCM…).
From what I’ve read here, the true believers (and funding recipients) are (as usual) trying to convince the unwashed masses by shouting rather than showing how it is that their analysis can magically reduce the errors in inputs. How about providing a demonstration? (And no – I don’t have to provide a demonstration – the climatologists are trying to ‘prove’ something – not me!).

@Doug P
Final paragraph is quite right. I see it all the time in crazily correct climate claims (C^4).
@matsibengtsson
“All rifles tend to go a little high to the right. You do not have a better clue of where the bullseye where, as long as there is a systematic error.”
That is an accuracy problem – the shooter should have compensated for a known issue – it should have shown up in calibration. That begs the question about who is doing the calibrations and how well are they doing it. Think about the usefulness of calibrating really well and getting a really precise measurement of a temperature that is strongly affected by UHI.
Measurement systems require system-level, rational analysis. I see great opportunities for mathematical sleight-of-hand when passing casually back and forth between anomalies. Willis is really poking the right dog here.
When I see someone in the lab measure something with a thermocouple that is precise to 0.02 degrees and then [black box, hum-haw, kerfuffle, kersproing] Presto! Out comes a reading to 6.5 digit precision! Then I know there is a carpet bag in the cloak room.

@Pamela Gray says:
August 18, 2013 at 12:51 pm
Any kind of filter at all, removes the most important part of the data series in my opinion.
The problem is then you miss a very important science fact that a signals as opposed to noise can not be removed by any filter it needs a “specific filter” because a signal is not at all random.
This problem has a very important historic overtone .. look at the Holmdel horn antenna.
http://en.wikipedia.org/wiki/Holmdel_Horn_Antenna
What Arno Penzias and Robert Wilson were trying to work out is why they couldn’t silence and filter the noise out of the Holmdel horn antenna. They realized it was impossible because it was a signal and a signal is not random and you can not filter it out without understanding how the signal is distorting your data. It lead to one of the most important signals of all time the cosmic background radiation and the signal was everywhere.
To deal with a signal and filter it out you need to understand the signal and you can construct a filter to remove it because it isn’t a random effect it’s not 0.5 here and -0.5 there, it has a pattern and that pattern is not random.
The only way you could ignore the effect is if you were absolutely sure that the interfering signals were very small and you could even assign an error value to them so you would say that the background chaotic signals had a value of +- ? degree. For example in a normal radio scenario the CMBR is unimportant because the signal is so large compared to it but as you saw you start looking for signals from deep space and it became a problem. The CMBR level varies over frequency range and that itself also became important but that’s another story.
The same is probably true of the climate signal there will be different chaotic signals at different frequencies and so you will need to understand the background to remove them.
Eventually Pamela if climate science is right, and I suspect it is at least somewhat right, the climate signal will grow and you will be obviously able to separate it from the background (the radio equivalent here on earth) at the moment however you are in the deep space signal mode looking for a very small signal with other signals possibly of equal size mixed into the back of it.

OK, so I want to retract my statement above that the errors in monthly averages would affect only the intercept. This was a clear case of having the wrong mental picture: since the errors in monthly averages are all different, they add a 12-month periodic error sequence to the anomalies.
However, I’ve done some experiments that convince me that any reasonable-sized error in the monthly averages will have negligible effect on the trends in anomalies.
Here are the experiments. Take the BEST Anchorage, AK data from 1900 on (no missing values from then on). Compute the trendline.
Now, generate a random set of 12 monthly errors. I used a normal distribution mu = 0, sigma = 0.1 on the theory that this is worst-case: since there is a 30-day sample period per month, even one year’s worth of data should have an error in the mean of less than 0.1 per month.
Now systematically add those random errors to the monthly anomalies and recompute trends. I did this for the following time periods:
1900 – 1902
1900 – 1910
1900 – 1920
1900 – 1930
for 100 runs for each time period.
Results:
1900 – 1902. Baseline trend: -0.0253. Average of trends with error: -0.0256. Std dev: 0.0020
1900 – 1910. Baseline trend: -0.0026. Average of trends with error: -0.0026. Std dev 8E-5
1900 – 1920. Baseline trend: 0.002859. Average of trends with error: 0.002861. Std dev 2E-5
1900 – 1930. Baseline trend: 0.00406. Average of trends with error: 0.00406. Std dev 9E-6.
I conclude from this that an error of size 0.1 in the monthly averages — and this is large given a 30-day sample period, I would imagine — is negligible in terms of error in trends.
Thoughts?

@all: The time intervals above are “off-by-one.” They should read
1900 – 1901 (inclusive)
1900 – 1909
1900 – 1919
1900 – 1929
@LdB: I don’t know much about shooters and wind. However, a normal distribution of errors *for the monthly averages* is not a necessary assumption. The reason is simple, and has to do with why my experiment gives such consistent results.
Take the BEST data from 1900 to present and run a linear regression with your favorite stats package. I happen to be using JMP. Here is the output:
T(t)=β_1 t+β_0+ε_t
Results:
β_1=0.0013 95% CI:[0.0009,0.0017]
r^2=0.029 σ=2.8609
Notice two things about this. First, the variability is huge — 2.86 even with 1359 observations. This means that although the confidence interval for the mean is very tight, the confidence interval for individual estimates is very wide.
The second thing to notice is an obvious corollary: r^2 is tiny, laughably so.
The accompanying scatter plot tells the tale. The variability is humongous (and is approximately normally distributed).
In light of that large variability, a small error in the monthly averages is meaningless. That’s really the story here. If we add in quadrature the natural variability of the anomalies together with the error in the monthly averages, we essentially get the natural variability.
So I could change my distribution to be anything at all and get pretty much the same result. I bet you 10 quatloos.

@Jeff Cagle
@LdB: I don’t know much about shooters and wind. However, a normal distribution of errors *for the monthly averages* is not a necessary assumption. The reason is simple, and has to do with why my experiment gives such consistent results.
But you are missing the point you are assuming a controlled enviroment and this is real world data you may be only getting consistancy based on a fallacy.
Lets extend the problem initially the shooters only shot on fine days because windy days makes there sights wobble so they avoid shooting on windy days. Later sights are improved and they shoot on windy days and sunny days suddenly your neat assumption goes to pieces.
Anthony’s own urban heat island argument is a classic in this sort of problem you need to understand the problem properly you can’t just assume you can average it away.
This is why particle physics, radio communications and telescope observatories study there backgrounds in detail they actually study it almost as much as they study the signals.
Some interesting stories on background noise controls recently and how they deal with them
Ethan Segal on why Earth telescopes fire lasers into space
http://scienceblogs.com/startswithabang/2013/07/24/why-observatories-shoot-lasers-at-the-universe/
Tommaso Doringo on which is more sensitive to the Higgs ATLAS or CMS
http://www.science20.com/quantum_diaries_survivor/atlas_vs_cms_higgs_results_which_experiment_has_more_sensitivity-113044

Averaging a time series is bad statistical practice. What the anomaly is the average of an average of a time series. A double whammy.
It is impossible to get the true temperature of any object until that object is at thermodynamic equilibrium. The earth never is at this impossible state.

richardscourtney says:
August 19, 2013 at 4:55 am
No wonder climate science is in a mess if they think like you and I may sound like a Pompous Git but perhaps just stop and listen.
You are talking about ENERGY not some nebulous concept … ENERGY is invariant it can not be created and destroyed and the particular question you are asking is the energy on the earth increasing or decreasing because of the addition of CO2 into the atmosphere.
I don’t care how you define your point to measure ENERGY the datum exists because ENERGY is defined clearly and precisely.
Your problem in climate science is people want to choose a different condition you are calling the metric of global temperature.
The answer is simple make them define that particular definition (we call it a frame of reference) to that of the global energy balance after all that is the question you are seeking.
Look at invariant mass and guess how we make them define it
http://en.wikipedia.org/wiki/Invariant_mass
See what we make scientists do:
The invariant mass, rest mass, intrinsic mass, proper mass, or (in the case of bound systems or objects observed in their center of momentum frame) simply mass, is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference related by Lorentz transformations.
Pick whatever frame of reference you like make up your own definitions, make up your own units do what you like because you MUST tie it to the total energy balance of the earth there by definition has to be a transform between any two frames of reference.
If you make every climate science group as part of there choosing whatever reference point they like to measure tie it to global energy balance you can create a translation of results GUARANTEED because energy can not be created or destroyed.
If the scientists in climate science haven’t worked that out by now they have serious issues and need to get some real physicists involved ENERGY is not a toy you can redefine.

richardscourtney says:
August 19, 2013 at 5:34 am
BOLLOCKS!
We are talking about temperature and NOT energy.
Temperature is NOT energy
OMG Richard please don’t say another word
Here is google for you:
http://en.wikipedia.org/wiki/Temperature
Please read it before you make any more statements I am not trying to pick on you simply explaining and you obviously need some things explained.

LdB said:
“ENERGY is invariant it can not be created and destroyed and the particular question you are asking is the energy on the earth increasing or decreasing because of the addition of CO2 into the atmosphere.”
More energy need not affect temperature.
Potential Energy is not registered as heat by thermometers.
If a gas molecule rises more of its energy converts from KE to PE and it cools.
Any molecule that absorbs energy so as to become warmer than its surroundings then becomes part of an expanded, less dense and lighter air parcel which rises against gravity until it cools once more so the net effect on temperature of radiative characteristics is zero but PE increases instead.
No need for a rise in surface temperature to push the atmosphere higher. One only needs the presence above the surface of molecules capable of carrying more energy in PE form.

Stephen Wilde says:
August 19, 2013 at 6:47 am
Potential Energy is not registered as heat by thermometers.
If a gas molecule rises more of its energy converts from KE to PE and it cools.
You are dealing with earths energy balance Stephen cycling energy like that is irrelevant as you already worked out the energy is the same so it doesn’t matter.
PE or KE is irrelevant energy is energy if earth is retaining it is a problem because it can easily become temperature as you have already shown above.
The energy can’t hide that is why you make every group reconcile the full energy balance.
What happens at the moment yes like you are trying to do here you can slide the energy into somewhere you aren’t measuring your system is open … you have to close the system and everyone must close there system you enforce it .. no closure = no publish.
What then usual happens is several groups will pick certain frames of reference and others will do just small bits an tie back to these group positions rather than having to do a full balance themselves.
So yes you can change the energy from KE to PE but it matters not it will get accounted.
No more playing hide the heat or energy anymore 🙂

@LdB: You wrote,
But you are missing the point you are assuming a controlled enviroment and this is real world data you may be only getting consistancy based on a fallacy.
I think you’re misunderstanding the purpose of the experiment. I am not trying to use a linear model to predict trends in monthly anomalies. That would be futile: the r^2 value is 3%, which is ludicrous.
All I’m doing is to try to answer the question that Willis posed: Does an error in the monthly averages have an effect on secular trends?
The answer is, No.
Then you wrote, Lets extend the problem initially the shooters only shot on fine days because windy days makes there sights wobble so they avoid shooting on windy days. Later sights are improved and they shoot on windy days and sunny days suddenly your neat assumption goes to pieces.
Anthony’s own urban heat island argument is a classic in this sort of problem you need to understand the problem properly you can’t just assume you can average it away.
In other words, you suggest that improvements in thermometers and UHI are two potentially confounding variables that affect trends? I agree. And if I were trying to predict trends, your objection would need to be fully accounted for.
However, I cannot think of a reason for either of those variables to alter the answer to Willis’ question. Can you? If you can, then construct a multivariate experiment with all three variables, and show the effect when you control for each.
It will significantly advance the discussion if you can quantify your objections rather than arguing by analogy. As it stands, your shooter analogy does not match the state of the data very well. Instead of a marksman (tight variance on shots) we have a novice (only hits the target — anywhere! — about 68% of the time, corresponding to a temp variance of 2.86 deg C). Instead of a wind (random large variance), we have a kind of “target-gremlin” (small variance with a pattern repeating every twelve shots, corresponding to an error in monthly averages of 0.1 deg C). Arguments by analogy are potentially misleading or confusing; straight math would be better.
So for example, if you think my estimate of 0.1C is too small, tell me why.

Man Bearpig says:
August 18, 2013 at 1:50 am
If each year the height of each schoolchild on their 3rd birthday is taken at each school in a county and the average is calculated (which would be around 37 inches) .. This would include those that were say 33 inches to 42 inches which would (probably) be in the normal deviation … However, when the averages are sent to a central state center and the standard deviation is calculated on the averages, then the distribution will be much MUCH smaller and those at the extremes of the distribution would be considered abnormally short or tall.

You are wrong Man Berarpig, because you mix two notions, one is “Standard deviation” and the other is “Standard error of the mean”. The first one will not be smaller for the central state statistics than for the school. However, the standard error of the mean will decrease toward zero as the sample size increases. Standard deviation is a measure of the spread and Standard error of the mean is a measure of the accuracy of the mean in I finite sample size.

Joseph Murphy says:
August 19, 2013 at 7:17 am
I need to learn how to quote. But anyways, energy = temp – work, if I am not mistaken so no energy and temp are not the same thing. Honest question, why would we consider the earth a closed system
Sure so you just subtracting two things that are not the same and you got something else so an apple – orange = banana 🙂 They are all the same thing or you couldn’t do what you just did.
Put “change of” in front of energy in your equation and you are right and yes work is also energy 🙂
Earth isn’t closed however earths energy balance IS CLOSED BY DEFINITION
Earth’s Energy Balance extended definition:
Earth’s Energy balance describes how the incoming energy from the sun is used and returned to space. If incoming and outgoing energy are in balance, the earth’s temperature remains constant.

@Richardscourtney says:
August 19, 2013 at 7:45 am
NO! Not by a temperature measurement it won’t.
Since you failed to understand the point made by Stephen Wilde, I offer you another example.
No you failed to understand my answer .. I understand what you think you are doing.
Temperature = energy … it is actually roughly the Kinetic energy of the molecules its a slight approximation but we can ignore for now.
PE = energy … as Stephen says.
You can argue whether PE is climate change or not that’s up to each groups view. So in Stephens example the excess energy is going into PE but at least we know that and account for it because he has to balance the full budget.
The likelyhood of that energy coming back out to haunt you the groups can argue about that but you aren’t arguing with “hiding energy” everything is out on the table.
You also get large amounts of energy disappear into chemical processes predominately photosynthesis by plants and that can come back to haunt you in the same way.
Now some groups may add those different energies into climate change some may not at least everyone has the same total energy and there is no missing energy.
Wherever you want to put the energy is up to each groups framework but anyone can easily follow what energy is where and that is how you start to build reference frames.

richardscourtney says:
August 19, 2013 at 8:00 am
NO!
If incoming and outgoing energy are in balance, the earth’s EFFECTIVE RADIATIVE TEMPERATURE remains constant.
Woot you got that bit that …. we can do a full closure if you simply now measure the suns output because that varies.
So now we drill each of the earth energies apart.
Full balance …. so you are monitoring energy in temperature change
Full balance ….. so you would be monitoring Stephens PE
Full balance ….. so you would be monitoring chemical uptakes ocean/plant etc
Full balance ….. so you monitor any thermal energy coming out of earth
I am not a climate scientist there are probably a lot more like winds etc
Different groups may argue which things are climate change some may want to stick Stephens PE in some wont they argue they are only interested in temperature.
It matters not you can easily work the views between those two reference frames.
So we have group A who says temperature is the only Climate change. Group B says no climate change is temperature and PE gain.
Group B publishes a report saying PE is increasing we have terrible climate change. Group A says no look the temperature is constant there is no climate change. See you can reconcile the two views and eventually groups will start talking about energy which is the reality of what you are trying to do.