Is it possible we’re looking at a new branch of linear algebra?

and I completely agree with your response:

#46. Not a chance.

I was trying to point out, clumsily it seems, that the HT (domestic U.S. and now Euro) has asserted the right to invent, and employ prior to testing, extraordinary statistical methods, so perhaps we should expect no better when it comes to linear algebra.

so there must be weaker conditions than positive semidefinite which still generate positive eigenvalues.

Interesting point. For a symmetric matrix M, eigenvalues are positive if and only if the M is positive semidefinite. For a non-symmetric matrix, eigenvalues are positive if the symmetric matrix 0.5*(M+M’) is positive semidefinite.

I’m not sure what theorems might apply to the Mannomatic. Is it possible we’re looking at a new branch of linear algebra?

In fact, there’s an interesting botch of this in MBH. He does PCA on the temperature grid cells by month, stating incorrectly that months were necessary because you need more time periods than gridcells to do PCA. Nature referees were unequal to the task of identifying this faux pas. After calculating monthly PCs, he then annualized them. The resulting series are no longer orthonormal; they are close to being orthogonal but depart from orthogonality.

Then he does one more thing – the annual series were calculated over 1902-1993, but he then truncates to 1902-1980 for matching to proxies, further moving the series from orthogonality.

Wahl and Ammann calculate temperature PCs annually maintaining orthogonality at this stage, but then truncate like Mann, so, once again, they are not orthonormal or orthogonal, thought they are close to it.

In the 15th century step, the proxies which are supposed to have a "signal" are surprisingly close to being orthogonal. So the PLS regression ends up being very similar to an OLS regression. There has really been far too little attention paid to the regression aspects: think about it- he is regressing a series of length 79 against 22 near-orthogonal proxies.

Where it gets really funny is in the no-PC case of Wahl and Ammann. Then they end up doing a PLS regression of a series of length 79 against 95 proxies.

if you calculate the correlation among the parts of series (eg. instru against eigenvector), then changing the sign of the eigenvector changes the sign of that correlation

1. Ok, I may be dense, but give me a little credit here; this is more than a little obvious. (I’ve read Richman & Thurstone & Craddock cited in that 2nd primer.)

2. I’m not talking about instrumental vs. proxy data. I’m talking about just the proxies.

Let’s start simple. Orthogonality among whole eigenvectors (which in this case are temporal) does not imply orthogonality at all time scales. Agreed?