Guest Post by Willis Eschenbach (@weschenbach on Ex-Twitter)
The iconic “hockeystick” simply refuses to die. It was first created by Mann, Bradley and Hughes in their 1998 paper Global-scale temperature patterns and climate forcing over the past six centuries (hereinafter “MBH98”).
Figure 1. Original hockeystick graph
MBH98 claimed to show that after a long period with very little change, suddenly the world started warming, and warming fast.
Back a couple of decades ago, Steve McIntyre over at Climate Audit did yeoman work in discovering a host of errors in MBH98. And somewere in that time, someone, likely Steve but perhaps not, noted that the curious (and mathematically incorrect) procedure used in MBH98 could actively mine hockeysticks out of red noise.
[UPDATE]: The unstoppable Rud Istvan noted in the comments that McIntyre and McKitrick published Hockey sticks, principal components, and spurious significance in 2005..
I also find Mann, Bradley and Hughes reply to that study, Reply to McIntyre and McKitrick: Proxy-based temperature reconstructions are robust, which says in part:
McIntyre and McKitrick’s claim that the common procedure (6) of screening proxy data (used in some of our reconstructions) generates “hockey sticks” is unsupported in peer-reviewed literature and reflects an unfamiliarity with the concept of screening regression/validation.
This post will show that statement by MBH is incorrect. Read on …
Despite all of that, MBH was succeeded by various of what I call “hockalikes”, studies that purported to independently find a hockeystick in the historical record and thus were claimed to support and validate the original MBH98 hockeystick.
Of course, these repeated many of the same errors as had been exposed by McIntyre and others. Here is the money graphic from my post Kill It With Fire, which analyzed the Mann 2008 attempt to rehabilitate the hockeystick (M2008).
Figure 2. Cluster dendrogram showing similar groups in the proxies of the M2008 hockalike
Note that the hockeystick shape depends on only a few groups of proxies.
Now, what I realized a few days ago was that although I’d believed that the MBH98 incorrect math could mine hockeysticks out of red noise, I’d never tried it myself. And more to the point, I’d never tried it with simpler math, straight averages instead of the uncentered principal components method of MBH98. So this is basically my lab notebook from that investigation.
The most expansive of these hockalikes involve the PAGES dataset, which has had three incarnations—PAGES2017, PAGES2019, and PAGES2K. PAGES2K starts in the year 1AD and contains 600+ proxy records. Here are several temperature reconstructions using PAGES2K data done by different groups of investigators, from a Nature article promoting the claim that there is “Consistent multidecadal variability in global temperature reconstructions and simulations over the Common Era“
Figure 3. Several historical reconstructions using the PAGES2K dataset.
Now, as Figure 3 shows, it’s true that several different investigations done by different teams have yielded very similar hockeystick shapes. While this seems to greatly impress the scientists, this post will show why that is both true and meaningless.
To do that, first we need to understand the steps in the process of creating proxy-based historical temperature reconstructions. A “proxy” is some measurement of differences in some measurable variable that changes with the temperature. For example, in general when it is warmer, both trees and coral grow faster. Thus, we can analyze the widths of their annual rings as a proxy for the surrounding temperature. Other temperature proxies are isotopes in ice cores, sediment rates in lakes, speleothems, magnesium/calcium ratios in seashells, and the like.
The process of creating a proxy-based historical dataset goes like this:
- Gather a bunch of proxies.
- Discard the ones that are not “temperature sensitive”. Temperature-sensitive proxies can be identified by seeing if they vary in general lockstep (or anti-lockstep) with historical temperature observations (high correlation).
- They might be positively correlated (both temperature and the proxy go up/down together) or negatively correlated (when one goes up the other goes down). Either one is sensitive to the temperature and thus, is useful. So we need to simply flip over the proxies with negative correlation.
- Use some mathematical method, simple or complex, to average all or some subset of the individual proxies.
- Declare success.
Seems like a reasonable idea. Find temperature-sensitive proxies, and average them in some fashion to reconstruct the past. So … what’s not to like?
To start with, here’s the description from the paper announcing the PAGES2K dataset, entitled A global multiproxy database for temperature reconstructions of the Common Era.
Reproducible climate reconstructions of the Common Era (1 CE to present) are key to placing industrial-era warming into the context of natural climatic variability.
Here we present a community-sourced database of temperature-sensitive proxy records from the PAGES2k initiative. The database gathers 692 records from 648 locations, including all continental regions and major ocean basins. The records are from trees, ice, sediment, corals, speleothems, documentary evidence, and other archives. They range in length from 50 to 2000 years, with a median of 547 years, while temporal resolution ranges from biweekly to centennial. Nearly half of the proxy time series are significantly correlated with HadCRUT4.2 surface temperature over the period 1850–2014.
So PAGES2K has completed the first step of creating a proxy-based temperature reconstruction. They’ve gathered a host of proxies, and they’ve noted that about half of them are “temperature sensitive” based on their agreement with the HadCRUT surface temperature.
Again … what’s not to like?
To demonstrate what’s not to like, I created groups of 692 “pseudoproxies” to match the size of the PAGES2K dataset. These are randomly generated imitation “time series” starting in the Year 1, to match the length of the PAGES2K. I created them so their autocorrelation roughly matched the autocorrelation of the temperature records, which is quite high. That way they are “lifelike”, a good match for actual temperature records. Here are the first ten of a random batch.
Figure 4. Randomly generated pseudoproxies with high autocorrelation, also called “red noise”.
As you can see, all of them could reasonably represent the two-millennia temperature history of some imaginary planet. How good is their correlation with post-1850 temperature observations? Figure 4 shows that data.
Figure 5. Correlations of 692 random pseudoproxies with the Berkeley Earth modern temperature observations.
This is about what we’d expect, with approximately half of the pseudoproxies having a positive correlation with the observational temperature data, the other half with a negative correlation, and most of the proxies not having a strong correlation with the temperature.
And here’s the average of all of the pseudoproxies.
Figure 6. Average, 692 pseudoproxies. The red line shows the start of the Berkeley Earth instrumental record. Note that there is no hockeystick—to the contrary, in this case, to avoid biasing my results, I’ve chosen a batch of pseudoproxies whose average goes down at the recent end. Nor is there any significant trend in the overall data.
OK, so we have the proxies, and we’ve calculated the correlation of each one with the instrumental record. Then, following Step 3 in the procedure outlined above, I flipped over (inverted) those proxies that had a negative correlation to the instrumental record. That meant all the proxies were positively correlated with the Berkeley Earth data.
At this point, I was going to see what an average would look like if I selected only the pseudoproxies with a high correlation with the instrumental record, say 0.5 or more … but before that, for no particular reason, I thought I’d look at a bozo-simple average of the whole dataset after inverting the negatively correlated pseudoproxies. Color me gobsmacked.
Figure 7. Average of all of the pseudoproxies after simply flipping over (inverting) those with a negative correlation with the instrumental data.
YOICKS!
Here, we can see why all the different averaging methods yield the same “historical record” … because the procedure listed above actively mines for hockeysticks in random red noise.
Please note that it’s not necessary to flip (invert) those pseudoproxies which have a negative correlation to the temperature. We can get the same hockeystick result by simply discarding all the negatively correlated proxies.
One interesting detail of Figure 7 is that there is a sharp drop in the average before the start of the period used for the correlation. I assume this is because to get that large an increase, you need to first go down to a low point.
And this drop in the average starting around 1775 is of interest because you can see it in both Panel A and Panel B of the PAGES2K reconstructions shown in Figure 3 above. The same post-1775 drop is also visible in the MBH hockeystick in Figure 1, although it’s stretched horizontally by the different time scales of the MBH and PAGES2K graphs.
Another item of note is that the procedure has introduced a slight downward trend from the beginning to a sharp drop around 1775. I ascribe that to the procedure favoring “U” shaped datasets, but hey, that’s just me.
In any case, the slight downward trend is a real effect of the procedure. We know that because there’s no downward trend in the full dataset. We also know it’s a real effect for a second reason—we see the same slight downward trend in the original MBH Hockeystick in Fig.1, and also in Panel “a” of Figure 2.
Finally, why is there so little variation in the “handle” of the hockeystick? Are the temperatures of the past really that stable?
Nope. It’s another artifact. The handle of the hockeystick is just an average of some presumably large number of random red noise datasets. When you average a bunch of random red noise datasets, you get a straight line.
Moving along, my next thought was, how much do I have to disturb the pseudoproxies in order to produce a visible hockeystick?
To investigate that, I took the same original dataset. In this case, however, I inverted only 40 proxies, the ones with the greatest negative correlation. So I was flipping only the strongest negative signals, and leaving the rest of the proxies that had negative correlation as untouched red noise. Here’s that result.
Figure 8. Average of all of the pseudoproxies after flipping over those with the top forty negative correlation with the instrumental data.
Note that less than six percent (forty) of the pseudoproxies were flipped, and all four hockeystick characteristics are already visible—a straight handle, a slight downward trend to 1775, a sharp drop to 1850, and a nearly vertical hockeystick “blade” from 1850 on.
How about at the other end, where we select only the ones with the strongest correlation? Here’s the average of only the top quarter of the data (176 pseudoproxies) as measured by their correlation with the observational temperature.
Figure 9. Average of only the top quarter of the data, those with the best correlation with Berkeley Earth data.
Same thing. Straight handle on the hockeystick. Slow decline to 1775. Sharp drop. Vertical hockeystick blade after that.
Finally, after sleeping on it, I realized I’d looked at the best-case scenarios … but what about the worst-case? So here’s the half of the pseudoproxies with the worst correlation with the observational temperature.
Figure 10. Average of only the bottom half of the data, those with the worst correlation with Berkeley Earth data.
Despite using only the half of the pseudoproxies with the poorest correlation with temperatures, those with a correlation of 0.22 or less, we get the same story as before—same straight hockeystick handle, same slight drop to 1775, same sharp drop to 1850, and the same vertical hockeystick blade after 1850.
Now, there’s an interesting and easily missed point in the graphics above. While the shape stays the same, the greater the correlation, the taller the blade of the hockeystick. The different procedures changed the tip of the blade from ~0.1 with only 40 flipped, to ~1.5 using the worst-correlated pseudoproxies, to ~0.3 with all pseudoproxies flipped, to around ~0.7 using only the best correlated. So all of them showed the identical “hockeystick” form, and they only varied in the size of the blade. Curious.
Now I stated up above that this post would show why it is both true and meaningless that various studies all come up with hockeysticks. And I said above that I’d show the MBH claim wrong, where they said that the idea that the procedure “generates hockeysticks” is “unsupported”.
The reason is quite evident in the figures above—no matter what the investigators do, since they are all using some variation of the standard procedure I listed at the top of the post, they are guaranteed to get a hockeystick. Can’t escape it. That procedure definitely and very effectively mines hockeysticks out of random red noise.
Here, it’s an irenic long summer evening, with bursts of children’s laughter radiating out of the open windows. I have the great joy of living with my gorgeous ex-fiance, our daughter and her husband, a granddaughter who is “almost five, Papa!” and a grandson heading towards three.
Is there a lovelier sound than their laughter?
Best to all,
w.
The Usual: When you comment please quote the exact words you are discussing. I can defend my words, but I can’t defend your interpretation of my words. And if you want to show I’m wrong, see How To Show Willis Is Wrong.
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Could “temperature sensitive” trees be better described as woke trees and temperature
insensitive trees climate denier trees?
But how do we know that woke trees have been “temperature sensitive” for their entire millennium-long lives, maybe they have only recently acquired their temperature sensitivity, maybe the climate denier trees were sensitive early in their lives but only adopted an insensitive attitude in the last century.
me thinks the temp sensitive trees might be gay! 🙂
A happy tree is a healthy tree.
Not sure if this is already been mentioned, but what if you assumed from year one through the year 1624? Would you still get a hockey stick – I’m expecting you would – in the last hundred or so years starting around 1500?
The selection of proxies as temperature sensitive or not is flawed.
One can get no pattern in the data because either (a) the biological system indeed does not respond to temperature or (b) the temperature change at the location of the tree was zero, so no pattern is generated (though other mechanisms might produce an irrelevant pattern that misleads.)
Geoff S
Discard the ones that are not “temperature sensitive”
=======
In statistics this is an error. Formally it is known as “selecting on the dependent variable’. The results are spurious correlations. Why?
Because the proxies that are not “temperature sensitive” are telling you that the temperature sensitive proxies may simply be due to chance.
If a proxy, for basic physical reasons for physical reasons, has no correlation with temperature, then including that proxy tells you nothing about the proxies that are temperature sensitive.
Discard the ones that are not “temperature sensitive”
=======
so say you have 10 tree ring proxies, only 1 of which is temperature sensitive. Do you throw the other 9 away? Or do the 9 tell you not to trust the 1?
How do you know they are proxies if they are not temperature sensitive?
The assumption is that they are temperature sensitive. The analysis purports to find those that are most sensitive. This is “science” at its worst, supported by loud, bullying advocates with financial and reputational incentives.
That’s the point. You don’t know they are proxies if they are not temperature sensitive.
So, why think that the one that is temperature sensitive is a proxy?
Imagine a billion proxies, not 10, with only 1 showing temperature sensitivity. Would you think that the one was a real proxy, or just coincidence?
The error made by Mann is called the Monty Hall Problem. You can learn about it by a simple Google search.
No trees are temperature sensitive unless you can hold all other variables constant which is impossible. So the idea of using trees as temperature proxies is terrible science.
I don’t believe they are deciding which proxies are temperature sensitive or not by examining the results. They should be looking at the physics behind each proxy and from that determining which are temperature sensitive and which are not.
Here is a ChatGPT explanation of the hockeystick error.
https://chatgpt.com/share/b25aefcf-93a4-4ca8-a78f-6ad3d01faf1d
Am I supposed to submit it?
Do you expect me to get the results you got?
What results did you get?
Just read the chat at that link.
Here’s the text so you dont have to follow the link
Anonymous
Don’t consider my history in the the following chat
ChatGPT
Sure, I won’t consider your chat history. How can I help you today?
Anonymous
Is ‘selecting on the dependent variable’ mathematically sound
ChatGPT
“Selecting on the dependent variable” refers to a practice in statistical analysis where the sample is selected based on the values of the dependent variable. This approach can introduce bias and lead to incorrect inferences because it ignores variation in the independent variables that might be crucial for understanding the relationship being studied.
Here’s why it’s problematic:
To ensure sound statistical analysis, it’s important to select samples that represent the full range of both dependent and independent variables. This allows for a more accurate understanding of the relationships and effects within the data.
Here is a more in depth analysis with ChatGPT
https://chatgpt.com/share/01a8461d-3ec6-4423-9b25-53e8d1e86683
Here is a chatgpt critique of the methodology:
https://chatgpt.com/share/91c6fffa-e54f-42ae-bb91-6a3e0e70f6ed
“Of course, these repeated many of the same errors as had been exposed by McIntyre and others.”
That’s a generous take. There were no errors involved. The subsequent stonewalling and name-calling proves they weren’t errors, but purposeful manipulations. Yes, numbers can lie.
Also, I’m curious. How do local proxies correlate with a non-existent “global average temperature”? That’s just completely nonsensical.
Given the nature of the “error” it is unlikely anyone with a math background would make it accidentally.
This error is however all too common in the social Sciences.
The problem is that “common sense” says you should “clean” the data. But the math behind statistics says your sample is random. If you clean the data it is no longer random.
Interesting question. Generally, proxies are compared to local rather than global temperatures.
In any case, to begin to answer your question, I looked at the correlations of historical individual 1° latitude x 1° longitude gridcell temperatures with the global average temperature. Here’s a histogram of the results.
w.
Sorry, Willis. I don’t know what that means.
It’s how well correlated the temperature record of each individual 1°x1° gridcell is with the global average temperature.
w.
Willis, you have two “Figure 8″s; maybe you should change them to “8a” and “8b.”
Thanks, Clyde, fixed. I hate typos.
w.
It will be interesting to see if the Gavin and the gang at Real Climate (are they still around?) have anything to say about this. This would probably be over the heads of the audience for Skeptical Science.
I don’t understand why they don’t change the name of their club to Unskeptical Science.
Do they even try to answer skeptics anymore? I thought that ended the day Gavin the coward walked off stage to avoid Dr. Roy Spencer.
Wouldn’t something like this be worthy of a published paper?
Probably, but it’s shouting into the void, and has small odds of seeing publication.
w.
Perhaps the real issue is that an error first found years ago (from reading the previous posts) remains uncorrected by retraction of the paper.
Either that or the fact that it was not discovered in “peer review”.
AND not noticed by the huge number of “scientific experts” in the field of climate crisis.
Ironically its neither novel, nor new.
Tree ring proxies for global temperature are only (maybe) valid if there are samples all over the world. If you select a few trees, it’s valid only for that locale. In addition, many non-tree-ring proxies have a quite different filter than annual.
Trees CANNOT be proxies for temperature due to all the other variables that effect the size of the rings. So you could get millions of them across the planet- and it’ll make no difference.
Willis, from your startling finding that only a few percent of highly correlated data need be selected to generate a hockey stick, I think you have discovered Mann’s real ‘trick’. He used principal components analysis as a smoke screen to make his selection and all the other data for camouflage.
With the high profile mathematical criticisms of his work, he spent the next 25 years largely defending himself, writing about the climate wars, op eds, suing people, ruining other people’s careers… the hockeystick was basically his life’s work.
Very interesting. Flipping the negative correlation is clearly a bigger source of hocksticks than removing the low correlations.
But why flip the data? The negative correlation doesn’t mean tree rings are negative thermometers.
Tree ring widths are (sometimes) positively correlated with temperature. Other proxies are (sometimes) negatively correlated with temperature.
If the correlation whether negative or positive is strong and significant, they might be useful as a temperature proxy.
Lots of “ifs” in that, however.
w.
And IMO if you have a physical basis for the proxy, its unnecessary to flip the data as it should have been framed in such a way as to give a positive correlation in the first place. Finding a negative correlation and flipping is simply fishing.
“If the correlation whether negative or positive is strong and significant”
which will never be shown to be true regarding tree rings and temperature since there are so many other variables effecting the size of the rings- you’d have to grow trees in a lab for many decades and hold all other variables constant- at least then you’d actually know the temperature and size of rings- otherwise, there is no way to know what the correlation is
I’ll admit to knowing next to nothing about statistics but I do know trees.
Not all of the proxies used were trees.
The discussion is about whether flipping a negative correlation prior to processing is a valid or invalid technique.
Real data would be better than any kind of proxie.
True. However, prior to the invention of accurate thermometers, real data is noticeable in it’s absence.
Some things increase with temperature, those are the positive correlations.
Some things decrease with temperature, those are the negative correlations.
If you just add these things together, you might come up with the conclusion that nothing changes due to temperature, which is incorrect.
Flipping the negative correlations is the same as taking the absolute value prior to processing.
I’ve added the following update to the head post.
[UPDATE]: The unstoppable Rud Istvan noted in the comments that McIntyre and McKitrick published Hockey sticks, principal components, and spurious significance in 2005.
I also find Mann, Bradley and Hughes reply to that study, Reply to McIntyre and McKitrick: Proxy-based temperature reconstructions are robust, which says in part:
This post will show that statement by MBH is incorrect. Read on …
Best to all as always,
w.
Willis
“YOICKS”
What you have done bears no resemblance to what climate scientists do, particularly your device of just inverting the data with negative correlation. The only connection was that you used correlation with BEST T, but you could have used the Dow, population of Paris, or whatever. I modelled 692 similar sequences, treated the same way but just looking at correlation with time, ie regression slope >0:
Same result. And there is a simple arithmetic reason. What strongly causes a rising trend 1850-2000? A dip near 1850 and a peak near 200. What causes a negative slope? A peak near 1850 and a dip near 2000. Normally these cancel. But after your trick, there is a super dip near 1850 and a super peak near 2000. Nothing to do with climate science. Jusr arithmetic of random numbers.
Nick Stokes June 29, 2024 11:05 pm
Inverting the negatively correlated proxies was what MBH98 did, it’s hardly “my device”. They defend it in the link above. Other than by inverting them, how do you propose to use them in a large-scale average?
But that makes no difference.
Why? Because we can get the same hockeystick result with the same characteristics if we just discard all the proxies with negative values.
I used correlation with temperature in order to replicate the procedure for determining temperature sensitivity described by the PAGES folks in the head post.
Then you say:
Again, it’s neither my “trick” nor my “device”. You get the exact same result by simply choosing those proxies with the best positive correlation to temperature. That has nothing to do with negatively correlated proxies, inverted or not.
You close by saying:
Couldn’t agree more.
Best to you,
w.
Willis,
“Inverting the negatively correlated proxies was what MBH98 did, it’s hardly “my device”. They defend it in the link above.”
Firstly, the link refers to a Mann 2008 paper, not MBH98. But it does not refer to inverting the data, which you have done, and climate scientists never do. It refers to the sign of the principal component, which is indeed not significant. Normally, you can invert it with no effect on the results. In the particular case of the Tiljander data of M2008, this may not be true. But this has nothing to do with MBH98.
Your process leaves out important other things that climate scientists do. They don’t just select the ones that correlate, over a calibration period (1902-1980 in MBH98). They then test over a verification period whether the resulting model works (verification 1850-1902 in MBH98). That tests whether the correlation has predictive value. Most of your random proxies would have failed verification.
Do you really believe splitting the testing data into two and then effectively testing twice makes the proxy any more a valid measure than testing once over the whole period?
Nick believes in whatever method best supports his conclusion.
TTTM,
Not sure if this is what Nick is on about, but there is a case to be made for not using all of the available data to estimate a model, and then using the ‘held out’ data to check the model’s efficacy. If all is ‘well’, the excluded data would then be added back to re-estimate the model’s parameters. Of course, if the model blows-up with the excluded data, it’s time to get out of Dodge. I should also say that leaving out 50% of the data seems excessive given the relative scarcity of ‘robust’ proxy data.
And how does that help? At most it simply reduces the amount of data available but can’t give any more certainty the data is actually a proxy for temperature.
‘And how does that help?’
I was simply pointing out that ‘best practices’ for estimating statistical models include holding back some of the available data so that the model can be checked against these as a preliminary safeguard against accepting spurious results as valid.
Yes but it doesn’t actually help does it. It’s a strategy designed to make the process sound robust when in fact its just a smokescreen
Nick, your whole question of how we treat the negatively correlated proxies is a total side track and distraction. If we just discard the negative ones and only use the positively correlated proxies, we still get the exact same hockeystick result.
So please, stop with questions of how to handle negative correlation. This phenomenon occurs with purely positively correlated proxies. Your issue is irrelevant to that.
Next, yes, you can choose proxies based on a two-part calibration/verification test rather than a simple correlation test like I used. And yes, fewer pseudoproxies would pass that test. Just like only a quarter of the pseudoproxies passed the test to be chosen for Figure 9 above.
But that doesn’t touch the deeper issue—whether you use your specially chosen proxies, or my specially chosen proxies in Figure 9, or impose some other test involving AIC or something more complicated, you’ll still get a hockeystick out of red noise.
Best to you,
w.
Aww, you might have a look at the Cape Ghir (offshore Morocco) series published by McGregor as discussed here by McIntyre.
https://climateaudit.org/2021/08/11/the-ipcc-ar6-hockeystick/
Not only does it not resemble local temperature, but according to McIntyre in that example it was also the only longer proxy used by Neukom et al. to reconstruct the first figure 1a for the IPCC AR6 summary for policy makers.
They somehow fail to address how this proxy represents global temperature while it does not represent local temperature even remotely.
IMHO it should not have been statistically overweight as they have done it there and if already the very first figure in the report is questionable, I expect little more for the rest of it..
And once more Kudos to McIntyre for pointing it out
Oh and of course you are wrong, it seems common practice over the last decades to invert proxy signals if the climate scientist feels it helps their believe.
And I write a 3rd time here, that any climate proxy reconstruction not including an analysis of the relevance and validity of the proxy for the global temperature is completely worthless.
Wasn´t it you claiming to be a mathematician? You should be able to understand McShane and Wyners points on this!?
Climate Audit went over the math.
https://climateaudit.org/2006/02/06/mbh-calibration-estimation-procedure/
Nick Writes
And yet
— Michael Mann
As I said to Willis, that comment by Mann is about the sign of predictors (principal components), not data. And it was not about MBH98.
Well I’m no expert in PCA but it seems to me that PC1 represents some quantity about the data which must itself have an underlying physical cause – for temperature sensitivity in this case.
Why is arbitrarily inverting PC1 valid when inverting the data its derived from, not?
A PC is just an eigenvector. An eigenvector of matrix A is just a solution of the equation
Av=λv, where λ is the eigenvalue. And obviously
A(-v)=λ(-v) is also a solution. In fact there is no basis for saying one is right and the other wrong.
I think you’re avoiding the question. From what I do know, PC1 has a physical meaning in the case of a proxy analysis, its not just an abstracted mathematical representation of the data and arbitrarily flipping it could reverse the implication of that meaning.
If it didn’t then PCA would be invalid as a tool for proxy analysis.
The statement “Multivariate regression methods are insensitive to the sign of predictors.” is from a mathematical viewpoint, as is yours.
The PCs are intermediates. Their sign is indeterminate, but has to be used consistently. The ambiguity then disappears in the subsequent reconstruction.
If you want the full gory math, the eigenvalue decomposition of symmetric matrix A is
A*L=L*Λ where L is an orthonormal matrix and Λ the diagonal matrix of eigenvalues. Or solved,
A=L*Λ *t(L) t=transpose.
For PCA you just select a subset of largest Λ values and corresponding eigenvectors.
What if you replace L by L*J, where J is any diagonal matrix with 1 or -1 on the diag. That has the effect or arbitrarily changing the signs of the eivecs. Then L*J*Λ *t(L*J)=L*J*Λ* J*t(L)=L*Λ* J²*t(L), since diagonal matrices commute. And of course, J²=I.
This isn’t an answer either. You need to clearly state that the eigenvector has no physical meaning so that flipping it is valid. Are you willing to do that?
That would be an interesting statement from Nick,
given that the proxies this particular discussion is about seem to include an artifact according to Tijander et. al. “This recent increase in thickness is due to the clay-rich varves caused by intensive cultivation in the late 20th century.”
( https://climateaudit.org/2009/10/29/upside-down-proxies-baffle-the-team/ )
and even Mann´s Co-author Kaufman acknowledges that they used the Tiljander proxies upside down!
And obviously
A(-v)=λ(-v) is also a solution.
====$=$
No! If 2=sqrt(4) does that mean -2=sqrt(-4)?
In math the solution has a domain in which it is valid and thus a domain in which it is not. You cant simply assume the solution for positive temperatures is also a solution for negative temperature.
sqrt is not a matrix. Or more generally, a linear operator. And what I wrote is just an expression of linearity. It fact you could multiply v by any scalar (except 0) and it would still be an eigenvector. But it is conventional to require the eigenvectors to be unit vectors. You can’t make a general requirement about the sign.
And in case you think Mann et all adequately addressed the concerns they’re here in all their glory
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2644169/
And it includes this quote (below) which has no references (well why would they find reason to support the argument used against them) and amounts to a simple denial.
Meanwhile many, including Willis now, have proved them wrong.
No, there is a fundamental misunderstanding here of what the hockey stick is. It has a blade, known from thermometric data, which is much better than proxy. And it has a shaft, known from proxy data. We do not need proxy data to tell is about the time of AGW. It tells us about the time before.
“It tells us NOTHING about the time before.”
It is all part of the fakery.
It’s your petard that you hoist yourself on.
There is so much wrong with that post.
The blade is correlated to the data from the time of AGW (as per Willis’ post) and is selected to increase.
The thermometric data isn’t comparable to the proxy data despite what you want to believe.
The shaft tends to average out (also as per Willis’ post) and rewrites archaeological history by obliterating variability.
And finally to address your argument. No they likely didn’t choose proxies that gave an unreasonable dip, they likely chose the proxies that fitted the temperature record much better than that.
It seems more likely to me, based on looking at individual proxies, that they grossly overweighted those that gave the “proper” result. We know this happened in MBH98, with Bristlecones being the star of the show. Without them there was no HS. Almost none of the other proxies even remotely resemble a hockey stick.
Yes, agreed. Weighting is another strategy for getting the results they’re after.
“Just arithmetic of random numbers.”
Ah well done….. Nick exposes the Mickey Mann methodology !!
PAGES 2k 2019 weighted proxies by their correlation with the instrumental target. In other words, negatively correlated proxies were inverted (and scaled).
Using proxies with an inverse relationship to temperature is not the same as arbitrarily disregarding the sign of the correlation, as Willis does.
PAGES 2k inverted negatively correlated data. Nick Stokes said they didn’t.
Nothing about the passage you quoted suggests that the authors invert proxy series based on correlation to the instrumental data.
Read it again.
What do you suppose happens to negatively correlated data? It gets inverted.
Negatively correlated series would be assigned a low weight, not flipped and weighted highly. You’re just making things up here.
That would be true if they weighted by correlation squared, but that’s not what they did. They weighted by correlation. Negative correlation means negative weight.
Can you demonstrate this? It seems central to your thesis.
Okay. Here is the part where they explain that they weight by correlation, not by absolute value of correlation, or correlation squared, or whatever you imagine they did.
1) How low?
2) What is the effect of that weighting?
Those weren’t meant to be rhetorical questions, but c’est la vie.
PAGES 2k never invert data. The weights you refer to are always positive.
You’re wrong, Nick. Some correlations are negative, which means data gets inverted in PAGES 2k’s CPS method.
Steve did all the hard work on this one. From here
https://climateaudit.org/2018/10/07/pages2k-2017-south-america-revisited/
we have a number of inverted proxies in the Pages 2K network
Oops. Lol.
https://climateaudit.org/2013/04/21/pages2k-reconstructions/#comment-415197
You are conflating a proxy having a negative relation to temperature with a proxy being arbitrarily inverted, as Willis does. If the proxy goes down as temperature goes up, then you need to flip the proxy record in a CPS reconstruction, that isn’t an arbitrary inversion based on correlation to the instrumental series, it’s based on the underlying physical relationship between the proxy and temperature. You would flip the proxy whether the flip yields a positive correlation or not.
d18O of ocean sediments has an inverse relationship to temperature, for instance, so to convert from the measured value to a temperature estimate, you need to flip the series (this only matters when you’re using some kind of “simple” averaging approach to reconstructing temperature, such as CPS).
Steve McIntyre wants to insist there can be no different relationships among trees between growth and temperature, but that is because he is ignorant of the field and can only make broad, uninformed generalizations (it’s the hallmark of his auditing process).
The problem is how do you determine that the absolute temperatures have gone up (or down) to cause the ΔT. You all follow the climate science prescription that ΔT’s are a temperature when they are actually only a rate of change.
I am no tree expert but I have spent some time trying to find data showing that certain tree species always have a confirmed negative growth rate in relation to temperature. I have found none. Most relate higher temps with less rainfall which confuses the issue.
Unless there is a confirmed natural lack of growth caused by higher temperature in a given species, one cannot assume a negative growth rate means higher temps which would allow inverting the data.
No we’re not. Nick stated there were no negative correlations in pages 2k but there provably are.
And so for two Pages 2k tree ring series a few kms apart you think the correlations reverse? The underlying physical relationship is being declared ex post. Can I prove it? No. But its very obvious that’s what’s happened and that’s a big no no.
Steve McIntyre is an expert in the field after years of work there and he has a statistical background. He has forgotten more about proxy reconstructions than you’ll ever know. Scientists hate him because he knows more than they do and holds them to account publicly.
He did not say anything of the kind, Nick said PAGES2k didn’t invert any proxy series based on negative correlation, that doesn’t mean no proxies exist that are inversely correlated to temperature. See my comment above noting the relationship between d18O of ocean sediments to temperatures to understand the distinction.
Nick and Eli’s comments from 2013 already address this, and you quoted them in the link above, so I’m not sure why you are falsely claiming that the relationship is being declared ex post, except that you are just parroting McIntyre’s take, which is a baseless borderline conspiracy theory that he refuses to own outright but is happy to let his readers fill in for him.
Scientists don’t hate Steve McIntyre, they think he’s a pest, which he is. Despite decades of time spent nitpicking the work of other people, Steve has yet to make a single forward contribution to the field. His entire body of work amounts to a handful of minor corrections to a couple of papers.
The negative correlated series are listed right there a few posts up. Inversion is what happens when the series is correlated against the temperature record for proxy selection in a study.
It’s obvious you’ve not spent any time at climate audit.
This is a thing you keep saying, but it’s not a thing you’ve actually shown to be happening.
I’ve unfortunately wasted way too much time reading CA.
It’s negatively correlated against temperature. How do you think the compared them?
I’ve not seen any posts by you.
Right, the correlation is calculated, and that is used to determine the weight. If a proxy series that is known to have a positive correlation to temperature (i.e. goes up when temperature goes up) is found to go down when temperature goes up (negative correlation), the proxy is assigned a low weight, it is not flipped over (inverting the temperature signal of the proxy arbitrarily) and reweighted.
I don’t comment there.
That’s not how it works. A negative correlation doesn’t mean its a bad correlation and should be assigned a low weight. It could be strongly negatively correlated and so be a good measure. But to test it, you need to flip it to compare to the increasing temperatures.
A proxy that responds inversely to temperature is correctly flipped, which you seem to agree with. The claim you are trying to defend here is that proxies are arbitrarily flipped in orientation based on their correlation to the GMST in CPS reconstructions, but you can’t seem to point specifically where it is said that this is being done in PAGES2k.
Like Tiljander you mean?
You should tell Nick that proxy indeed get flipped that was the first question.
As for your question about good or bad proxies look at Willis post up there showing that random data is good enough to produce hockey sticks..
So maybe this comes really down to showing first the proxys are good and relevant, because the hockey stick is a result regardless.
Vinther’s NGRIP1 record is a good example. Positively related to local temperature but negatively correlated with global temperature. Used upside down in PAGES 2k 2019.
This requires a citation or additional elaboration.
PAGES 2k 2019 refers to Neukom et al. 2014. It states the following in the SI:
And from Neukom et al. 2011:
So there you go, they weighted NGRIP1 by its correlation coefficient ρ with the GMST target. Not |ρ|, or ρ², or ρ+1, or e^ρ, or whatever would make it a positive weight. Hence, NGRIP1 is upside down in PAGES 2k 2019.
It’s pretty similar to what Willis did.
To clarify, I’m asking for a reference indicating that NGRIP1 is incorrectly used inversely to its relationship to temperature in PAGES2k 2019, not whether NGRIP1 has a negative weighting owing to the inverse relationship. If a proxy is inversely related to temperature, the correct thing to do in a CPS reconstruction is to flip its orientation.
I agree, but NGRIP1 is listed as positively related to temperature in the PAGES 2k 2017 database. See row 51 in Table S1.
https://static-content.springer.com/esm/art%3A10.1038%2Fsdata.2017.88/MediaObjects/41597_2017_BFsdata201788_MOESM218_ESM.xlsx
Neukom-style CPS weights it by its negative correlation with the GMST target.
Regarding
one author apparently stated
What should happen is you declare a series to have a particular correlation based on an expected physical characteristic at sample collection time.
What happened here is the physical characteristic was reasoned ex post because they’re claimed to be “results” and because of the use of “probably”.
Does this sound like a growth rate change solely due to temperature? Where is the attribution study to separate temperature from rainfall?
You’re misinterpreting the passage. The authors know that it responds inversely to temperature (high temperatures reduce radial growth), the only “probably” is the why. This is a growth pattern in the trees that is being observed in the present day, it isn’t merely a speculative comment on potential past behavior.
McIntyre insists the reasoning is all done post-hoc, but he doesn’t have any evidence that this is the case, he just wants to hint at a narrative of scientists all colluding to make hockey sticks (again, building a conspiracy theory mad-lib that his followers get to fill in with their own fantasies).
You didn’t answer my question.
As ferdberple pointed out how do you know which side of the “U” the data is on when it comes to absolute temperatures?
You don’t understand the problem.
What ferdberple posted about a Π shape for growth makes more sense than trees having a reversed biological function whereby it grows better radially as absolute temperatures fall.
The Π shows that as absolute temperatures warm toward the “best” temperature there is a large positive correlation. While the best growing range is in effect, there would be a very small anomaly. As the temps exceed the best range one would experience large negative correlation in anomalies.
Show us a study where this has been studied for trees that show larger rings as temperature fall below the nominal best temperature.
Nick, PAGES2k is a dataset, not a study or a method. As a result, they just present the data. They don’t either invert or not invert anything.
w.
w – That makes a lot of sense. The question now is: How can you use proxies so that they don’t give an artificial (ie, artefact) hockey-stick? If you have lots of proxies of each type, you could divide them randomly into two sets, each of which covers the same overall period in roughly the same proportions. Then you select subperiods of the measured period with various different overall trends. in one set of proxies, you work out which proxy types correlate negatively with one of the subperiods. You then flip those types of proxy in the other set and plot the result just from that other set. IOW, none of the proxy series used for determining correlation are used in the result. You then do the same with the same division of proxies for the other subperiods.
If you do that with your “pseudoproxies”, they are presumably all of one type (“pseudoproxies”), so each time you either flip them all or you flip none (best if you un-bias them by not getting them to decline a bit in recent years). Your results will all I think be trendless over the whole period.
Now, what result would you get if you do this with the PAGES2K proxies? If they are genuinely good temperature proxies, I predict that for each subperiod you will match the temperature trend over the subperiod, with brief movement the other way at each end of it, and with the chart levelling out as you get further in either direction from the subperiod. But if you just get trendless results, then the proxies were useless as temperature proxies in the first place.
If someone can provide the PAGES2K data in a format that I can read, I would be happy to do the work, though (a) you could do it much faster using R, and (b) I’ll be travelling for a while so it could be some time before I could do it.
Mike,
The Pages2K data is here. I used the V1.0 (2013) version here. It’s XL, not too large and very well organised.
What would you have to do with the data to get an hockey stick where the blade pointed down instead of up?
Start with observations of a time “now” when the temperatures are going down, not up.
Question. I thought there was another problem with the hockey stick that Richard Muller spoke about some years ago where he showed that Mann was very selective in picking which tree ring series to include and that when the series showed that temps should be going down when in fact they were going up, he substituted the actual temps record in place of his proxy data to create the blade and the the climategate emails called this Mike’s nature trick or hide the decline. Do you know if he was he referring to the hockey stick or was it something else, or am I just completely mistaken? I can’t yet find the video of Muller, but will keep looking.
this one? (easy to find googling “richard mueller temperature mann hockey stick”)
Nice work Willis. Should be easily understood by all readers.
You have contrived a way to make hockey sticks, but you have not replicated how scientists create paleoclimate reconstructions from proxies, so I’m not sure what the point of the exercise is.
It’s called science. And it investigates the argument that Mann’s method created a hockey stick regardless of the underlying data.
Willis doesn’t use any method that Mann has used, he just invents a contrived way to generate hockey sticks out of series of random data by picking hockey stick like series out. At least M&M claimed to be replicating MBH methodology in their critiques (although they did not in reality).
It’s clear you don’t understand what’s happened here. You said as much.
No, I understand very well what has happened here. That’s why I said I don’t see the point in it. Your comment offers nothing of value to the conversation. Try to do better.
Mann et al. 2008 famously inverted the Korttajärvi sediment data. They also screened the proxies by their correlation with the target, which is what Willis did for the last few figures.
Mann 2008 contains an error in the composite plus scaling reconstruction approach (the study presents numerous different reconstructions based on different statistical approaches), where a series of lakebed sediments is “inverted” due to calibration issues in the late portion of the series, but this was not an intentional choice by Mann (Mann even calculated the CPS reconstruction without the sediment series in his supplemental figures owing to these very concerns about the calibration of those proxies), and it does not follow any common convention used in other reconstructions. If Willis is basing his “inversion” approach on this issue in Mann 2008, he is sorely mistaken in doing so.
Whether the use of inverted data was an important issue or not, it was still the process Mann used without due care and you never addressed proxy selection by correlation which is what this post is really about.
Both of those are what Mann did and directly refutes your claim “Willis doesn’t use any method that Mann has used”
Perhaps you’d like to explain what Mann did and how it differs from what Willis did?
The issue is entirely Willis’s flipping procedure. Ensuring that all of the red noise series point up at the end will ensure that the average points up at the end, and indeed it could do nothing else. The more Willis selects for pointy-uppiness at the end the more the average will point up at the end (and Willis finds this a novel and compelling result for some reason). So it is a convenient way to make hockey sticks out of red noise series, but it does not replicate how scientists assemble paleoclimate reconstructions, not even using CPS, which is what Willis is attempting to emulate.
Mann’s screening approach picks proxy series by how closely they correlate to the local temperature field over the calibration interval, not by how pointy-uppy they are at the end (I.e. how well they correlate with the global instrumental index).
Nope. The flipped data can be excluded and the same result happens with the data that naturally increases. Willis has made this clear.
This is garbage. Mann screened by correlation to the recently increasing temperatures in the instrumental record.
Yes, it should be no surprise that you can also omit all the pointy-downy series and still get a pointy-uppy result. This is, again, a contrivance, not an insight into how paleoclimate reconstructions work.
Please read Mann 2008:
“For the CPS approach, we employ only those proxies estimated to reflect local temperature variations (based on a screening analysis over the calibration interval using instrumental grid box surface temperature data).”
Alan, AS I POINTED OUT IN THE HEAD POST, it doesn’t matter whether you invert the negatively correlated proxies or you simply throw them away.
You get a hockeystick either way, which clearly proves that the issue is NOT the inversion of the negative proxies as you falsely claim.
I encourage you to give it a try. Make some pseudoproxies, and just select those with high correlation with modern temperatures. Ignore the negative ones. Tell us what you get.
Finally, you say:
However, since we only have the local temperature field during the last 150 years and we never have that field at the start of paleoproxies, and since the overwhelming majority of local temperatures are “pointy-uppy at the end”, the result is the same—a hockeystick.
w.
I’m not questioning the results that you obtained, I’m expressing skepticism that the results are important. Any actions you take to increase the hocky-stickishness of your red noise series will increase the hockey-stickishness of the aggregate. You can flip the negative ones or omit them entirely.
The local temperature fields do a lot more than just “point up”, and some of them point down. And yes, using the instrumental period (part of it, at least) as the calibration period is necessary and appropriate, precisely because we know what the aggregate result ought to look like if the proxies yield a valid reconstruction.
He gets similar results when there is no inversion, so the process still mines hockey sticks in that case.
Alan, the point of the exercise is to show that regardless of the method used to combine the data (PCA, simple average, etc.), the selection or discarding of the proxies on the basis of their “temperature sensitivity” creates a hockeystick.
That IS how scientists create paleoclimate reconstructions from proxies, by selecting the “temperature sensitive” proxies based on, as the PAGES folks say, whether or not they are “significantly correlated with HadCRUT4.2 surface temperature over the period 1850–2014.”
And since you seem to think differently, please show us the right way to do it, one that does NOT create a hockeystick out of red noise.
Regards,
w.
Well, yes, this is exactly correct, but not in the way you want it to be. It’s correct because the shape of the temperature evolution of the past 2kyr creates a hockey stick. If you screen or weight proxies sensitive to temperature change then you end up with a reconstruction that looks like the global temperature.
It’s already been shown, by experts who have spent decades carefully developing approaches to paleo-reconstructions and publishing them in the peer reviewed literature.
“It’s already been shown, by experts who have spent decades carefully developing approaches to paleo-reconstructions and publishing them in the peer reviewed literature.”
That’s the funniest thing I’ve read all day.
What you don’t seem to be able to grasp the significance of is that…
If you screen or weight
proxies sensitive to temperature changered noise data then you end up with a reconstruction that looks likethe global temperaturea hockey stick.What does that tell you about proxy reconstructions?
Well, no, only if you screen or weight them based on how hockey-stick like they are (and make them hockeystick-like if they aren’t), as Willis does. This exercise, as noted in my initial comment, tells you nothing about proxy reconstructions, because it has nothing to do with how proxy reconstructions are compiled.
Neukom (PAGES 2k 2019) tested his CPS method on red noise. It produced hockey sticks.
https://www.ncei.noaa.gov/pub/data/paleo/pages2k/neukom2019temp/input/recons.PCP.ARnoise.RData
The file seems to contain the 1000 member ensembles Neukom et al. generated for each reconstruction type they considered:
This isn’t a “test” of the CPS method using red noise, it is the CPS method as used in the study, using ensembles to allow for probabilistic uncertainty estimates.
You are mistaken. They are the results from using Neukom’s CPS method (and two other methods) on red noise pseudoproxies. The CPS simulations are very similar to Willis’ results.
Thanks for pointing this out. The data file seems to align with the description you cite (it contains 1000 realizations each for three different methods), but it isn’t clear to me how you are certain that these are simply red noise proxies. The paper notes that the red noise reconstructions are incapable of explaining observed 20th century trends, which the proxy-reconstructions capture robustly.
On further reading, I think this is more complex than what Willis has done, although I’m still trying to wrap my head around the details. The hosking.sim() function used to generate red noise series from the proxies will yield long persistence in the synthetic series, and it isn’t clear to me what implications this has for the analysis presented. This is one of the specific issues pointed out with M&M’s analysis of MBH.
Willis probably used a different noise model but his results have the same salient features as Neukom’s CPS and PCR simulations. In contrast, PaiCo doesn’t generate hockey sticks from the noise Neukom used.
Not clear why you think there’s a problem with using red noise (noise with high Hurst exponent). Both the observational data and the PAGES2k proxy data have long-term persistence.
So any pseudoproxies should have the same characteristics.
w.
Alan, I looked at the paper “Consistent multidecadal variability in global temperature reconstructions and simulations over the Common Era”, and I couldn’t find evidence for your claim that they said that “the red noise reconstructions are incapable of explaining observed 20th century trends, which the proxy-reconstructions capture robustly.”
I ask in part because my red noise reconstruction (using only positively correlated pseudoproxies with no weighting) compares very well to the 20th Century trends …
Regards,
w.
Alan, I do NOT make anything “hockeystick-like”. And I’m not screening them based on how “hockeystick-like” they are. I’m screening them based on how “modern-temperature-like” they are.
Please read my post again, you have huge misconceptions about what I’ve done.
w.
Wolfgang Pauli had words of wisdom which apply to that approach.
The reality is that tree rings are not a linear proxy for temperature. They have an optimum temperature, and growth drops off above and below this temperature
Naive statistics however is linear. Hidden behind the scenes the math is trying to fit the data to a straight line.
But in reality the tree ring graph is a curve. Increasing to a maximum as temperatures go up, then decreasing as temperatures continue upwards.
This is why you get positive and negative correlations. Not because the answer is to flip the data. But rather because you are attempting a linear fit of non linear data.
Visualize tree ring growth as an upside down “U”. The apex is the temperature of optimum growth.
Now fit a straight line to the data, with positive slope, that intersects the upside down U. The better this line fits the U, the better the correlation.
If you now take a sample of the data from the left side of the U you will get positive correlation. If you sample data from the right side of the U you will get negative correlation. If you sample the data near the apex you will get near zero correlation.
Nowhere in this is there any justification to flip the data on the right side of the upside down U to “correct” the negative correlation.
Flipping the data would mean that tree rings would get wider no matter how hot it got. If the trees were at 200 degrees F they would have wider rings than trees at 70 F. Which is a nonsense.
I like this. It is what I was trying to say in a couple of my posts. If you don’t know the absolute temperatures, you don’t know which side of the U that the ΔΤ occurs in.
Are we really to believe that tree rings are thermometer? That trees growing at 130F will have wider rings than trees growing at 72F!! How about trees growing at 212F. Will they have even wider rings.