Comments on: More Linear Algebra

By: MBH Calibration-Estimation Procedure « Climate Audit

MBH Calibration-Estimation Procedure « Climate Audit — Mon, 06 Jun 2011 20:58:20 +0000

[...] The final RPC includes a re-scaling step described in the following post on this topic here. [...]

By: Steve McIntyre

Steve McIntyre — Wed, 22 Feb 2006 03:17:10 +0000

Phil – the trouble with MBH is that there are so many statistical horror stories than it makes a rather sick beauty contest. I apologize for not fully going through the temperature PC issues right now, but I will. One aspect that I’m going to write up a little bit more at some point is the massive overfitting in the calibration phase.

If you look at the equations, the calibration period for the 22 15th century proxies is no more than a representation of Neofs series of length 79 against 22 proxies. The representation is not quite a multiple linear regression, but since many of the proxies are fairly uncorrelated, viewing the operation as something like a multiple linear regression is not too far off. Since the target PC1 especially is very autorcorrelated and the “effective” length of the series is much less – certainly no more than 22: so there is overfitting of massive proportions and most of the covariance relationships are insignificant. It’s even worse with 112 proxies in the later steps.

But you also have one classic spurious regression. But MBH doesn’t fail a Durbin-Watson statistic (although the other multiproxy studies all do). The reason why it doesn’t fail a Durbin-Watson statistic is because of all the overfitting, which ruins the DW diagnostic.

By: Phil B.

Phil B. — Wed, 22 Feb 2006 01:58:59 +0000

Steve, I know you’re busy but let me summarize what you probably already understand. During the calibration period, the proxies are regressed against the retained PC(almost) orthogonal vector basis set to obtain a G matrix that will be used for reconstruction during the verification period and further back in time. Normally the PC are orthonormal and it would be possible to exactly reconstruct the retained PC during the calibration period given conditions necessary for the inverse to exist. The point I want to make here was that it is easy to get very good RE and residuals during the calibration period.
MBH98 uses essential your equation 29) à?’€ž= à…⪠* àŽ» to obtain NH temperature both during the calibration and verification periods and back to 1400 AD. The problem with equation 24) is that the lambdas are only valid during the calibration phase. Martin Ringo post is pointing at this fact. SVD decomposition is not additive if one appends a matrix to another matrix to add more records. What I mean by this is that if you one does takes the svd of the standardized gridcell T matrix from 1854 to 1980 (assuming all gridcells available or filled in by a RegEm procedure) than the last 79 elements of that PC1 (even with scaling) will not be equal to the PC1 of mbh98 calibration period, nor will be the lambdas in equation 24).
This really begs the obvious question of why didn’t mbh98 “skillfully resolve” or regress the proxies directly onto the NH or global temperature series, as it would make obvious the gain factor associated with each proxy such as the bristlecones or Gaspe.
Note in my #10 post, instead of “retained U’s” it should be “retained PCs used to form U”

By: kim

kim — Tue, 21 Feb 2006 22:49:52 +0000

I can’t archive my work, and God only knows where the decimal point is, but I think that’ll do it.
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By: kim

kim — Tue, 21 Feb 2006 22:44:55 +0000

I got time, let me try. Hey kids, where’s the abacus?
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By: Steve McIntyre

Steve McIntyre — Tue, 21 Feb 2006 22:34:17 +0000

#10. Phil, thanks for the comment. I’ll have to think about it. I’m pressed on the NAS presentation right now and even the Briffa stuff that I’m posting up now is linked to that; could you refresh this comment in about 14 days if I forget. Cheers, Steve

By: Phil B.

Phil B. — Tue, 21 Feb 2006 21:32:10 +0000

Steve, normally the PCs in a PCA type analysis are columns of a unitary matrix, so the PCs (columns of the unitary matrix) are orthonormal to each. In mbh98, they use monthly data and than average to get yearly PC’s, such that the resulting PC1-16 (from website) are almost orthogonal but not quite. In Zorita et al JC may 2003 which has been referenced by Mann as supporting his reconstruction, they proceed directly to annual PCs by doing a svd on the standardize yearly T matrix resulting in orthonormal PCs. It is interesting to look at your above equations and treat the retained U’s as orthonormal. G becomes G ”€ ’? UT *Y[503:581,] and substituting G into the à…⪬ and dropping [503:581,] to simplify and for clarity. Gives the following
à…⪠”€ ’? Y * YT *U *( UT*Y*YT*U)-1
Now if the inverse exists and keeping in mind U columns are orthonormal, than with modest effort one can show that UT*à…⪠= à…⫔*à…⪠= Identity_matrix and à…⪭U = 0_matrix, so one can say that U=à…⪠or one can get exact reconstruction RPCs during the calibration period. Now in order for the inverse to exists the column space of the proxys have to span the column space of the retained U’s. So any arbritrary proxy set (including women skirt length) that provides an inverse can provide exact reconstruction and small residuals and error bounds. The NH temperature errors during calibration are based only on the number of retained U’s and have nothing to do with the proxies themselves expect for the inverse requirement. So now all mhb98 needs is to manage some statistical significance during the verification period. Hope this analysis sheds some light. Phil

By: Steve McIntyre

Steve McIntyre — Thu, 16 Feb 2006 22:30:51 +0000

#8. I’ve never figured out quite what they did. There’s nothing in their source code released last summer about it. Ammann and Wahl didn’t replicate it either. It’s one more loose end.

By: Phil B.

Phil B. — Thu, 16 Feb 2006 22:12:03 +0000

Steve, I was hoping in this post that you would shed some light on MBH98 “skillfully resolving” methodology in chosing the RPC to keep in their NH temperature reconstruction. Phil

By: Martin Ringo

Martin Ringo — Tue, 14 Feb 2006 18:08:56 +0000

Re: “MBH Calibration-Estimation Procedure” and “More Linear Algebra”

There is a story about Albert Einstein whose subplot might pertain here. Einstein was invited to tea party at Princeton, the hostess of which was naturally thrilled by the honored guest and asking Einstein if we would speak on relativity theory.

Without any hesitation Einstein rose to his feet and told a story. He said he was reminded of a walk he one day had with his blind friend. The day was hot and he turned to the blind friend and said, “I wish I had a glass of milk.”

“Glass,” replied the blind friend, “I know what that is. But what do you mean by milk?”

“Why, milk is a white fluid,” explained Einstein.

“Now fluid, I know what that is,” said the blind man. “but what is white?”

“Oh, white is the color of a swan’s feathers.”

“Feathers, now I know what they are, but what is a swan?”

“A swan is a bird with a crooked neck.”

“Neck, I know what that is, but what do you mean by crooked?”

At this point Einstein said he lost his patience. He seized his blind friend’s arm and pulled it straight. “There, now your arm is straight,” he said. Then he bent the blind friend’s arm at the elbow. “Now it is crooked.”

“Ah,” said the blind friend. “Now I know what milk is.”

Well reconstruction ain’t relativity, but now we know what “milk” is.