Comments on: Inversions from Partial Correlation Coefficients

By: UC

UC — Sun, 22 Apr 2007 12:55:06 +0000

I just rotated the original temperature vector (Jean S can explain this part more clearly [ref: personal communication]).

Background: Juckes et al, Appendix A1, optimal estimate for the coefficients is

in matrix form,

where T is instrumental and P proxy data. To find another T that yields the same betas we can use method that Willis brought up; find a vector that has same sample correlation coefficients with N other vectors. I just took proxy matrix instead of temperature matrix, and got a new temperature series that yields the same betas as in JBB INVR. Low correlations make it possible to rotate that temperature vector quite a lot, as you can see from the differences between new and the original.

By: Steve McIntyre

Steve McIntyre — Sat, 21 Apr 2007 20:19:06 +0000

UC, as usual this looks interesting. Can you make your write up a little (actually a lot less terse) as I can’t figure out what you’ve done, and, if I can’t, this may apply to others as well.

By: UC

UC — Sat, 21 Apr 2007 19:39:53 +0000

Yes, hmm hmm, now for the next question: If I make my own temperature series (better than HadCRUT), and it happens to have exact correlations with some proxy collection, INVR method (as described by Juckes et al) will give exactly the same reconstruction? Works with JBB

By: bender

bender — Thu, 19 Apr 2007 08:47:21 +0000

Re #7 Glad to see we agree.

By: UC

UC — Thu, 19 Apr 2007 06:32:48 +0000

sim1 is obtained using red noise initial guess. It is true that the correlation with the original (r=0.36) is stronger in the HF band (0.5...1) that in the LF band. But note how weak that correlation is, given that these two series have identical correlations with 12 temperature series . Correlations from 0 to 0.5 give quite a lot of freedom for the alignment of that vector, the situation would be completely different if at least some of those correlations would be 0.9..0.99.

Basically, a Pearson correlation coefficient is frequency-independent, and is thus a poor measure of match' between two non-iid process. Most good dendroclimatologists understand this already.

Hmm, if a reconstruction explains X % of the variance, it can be explaining HF part or LF part. In the GW sense the low frequencies are interesting, but for that band we need lots of samples.. Problems in climatology are very difficult, and it seems that due to this some of those scientists take the short cut. And never admit a mistake. Ref RC .

By: bender

bender — Thu, 19 Apr 2007 03:04:17 +0000

It is trivially obvious that significant correlations may arise from high-frequency coherence or low-frequency coherence. I believe that is what is happening in the red and green cases above (each compared to blue). Basically, a Pearson correlation coefficient is frequency-independent, and is thus a poor measure of ‘match’ between two non-iid process. Most good dendroclimatologists understand this already.

Willis’s skepticism about one’s ability to robustly estimate confidence intervals stands, however. That’s precisely why the dendros don’t do it. Not because they haven’t thought of it, but because they don’t know how.

By: Joe Ellebracht

Joe Ellebracht — Wed, 18 Apr 2007 22:22:18 +0000

This discussion is over my head, but let me make a suggestion anyway.

The ring widths are a function of at least 2 things, temperature and precipitation (OK, maybe CO2 concentration but I am ignoring that and nonlinearity and the interaction between temp and precip). Assuming for the purpose of discussion that the effects of temperature and precipitation are additive, then using a one factor temperature model assumes, in the model’s forecasts (backcasts), that precipitation has a unitary value at the average of its value during the calibration period. Starting from the obviously existing 2 factor model sitting there waiting to be calculated allows one to test the reasonableness of the one-factor model (temp) assumptions about the other factor (precip). Also to look for interaction factors.

Additionally, since (I think) some have calculated historical precipitation in some of the same regions from tree ring widths, perhaps these precipitation estimates could be used in a 2 factor model to get temps.

As to the slope of the one factor line, in a linear regression wouldn’t it be obvious from the error statistics whether the error of the estimate of the slope allows for the slope to be negative within the significance test?

By: bernie

bernie — Wed, 18 Apr 2007 21:02:28 +0000

But isn’t a problem with the real data is that the set of linear equations are not independent and, therefore, the solution is over-determined??

By: UC

UC — Wed, 18 Apr 2007 19:21:03 +0000

It is quite interesting problem, you are given 12 vectors (zero mean) . You need to find a vector P that is aligned so that cosines of the angles between P and those 12 vectors are the given 12 values (sample correlation coefficients). I got my solutions using pseudoinverse + iterations, under-determined set of linear equations + constraint on the solution, I’m quite sure there is a better way ..
With simple 3D example it is easy to show that there can be 0, 1, or more solutions. If those given correlations are close to 1 it is harder to find solutions.

By: bernie

bernie — Wed, 18 Apr 2007 16:26:36 +0000

OK:
I have to admit that what you guys are doing statistically is beyond me. Logically, however, I am completely confused as to how it makes sense to talk about 12 separate monthly or 4 quarterly correlations of temperature with a single dependent variable annual ring width. There is clearly considerable correlation among months – winter months are cold and summer months are hot which amounts to a lack of independence among the independent variables. Wouldn’t a better approach be to class each year by the pattern of monthly temperatures against some norm and use those classifications as dummy variables? At moment, I sense a huge effort to construct findings that are more a matter of chance than a menaingful and definable mechanism. Of course once you do this you destroy the precision of being able to use rw as proxies for temperature – but at least you then can build a model that reflects the actual rw growth mechanisms. At the moment, the whole exercise looks serendipitous. I also think that there may be a problem in the degree of freedom in these models given the actual number of tree chronologies. What you have done is made me dig out all my stat books and rework things I haven’t looked at for 25 years.