I then compared verification statistics for the different reconstructions as shown below. OLS yielded much the “best” fit in the calibration period, but the worst fit in the verification period.
If OLS is equivalent to ICE, it actually finds the best fit (minimizes calibration residuals), and in proxy-temperature case makes the most obvious overfit. Let me try to get an understanding of the methods applied. As we know, core of MBH9x is the double pseudoinverse, in Matlab
where P is standardized (calibration mean to zero, calibration de-trended std to 1) proxy matrix, TPC target (temperature PCs) and T_select is that odd selector of target PCs. After this, RPCs are variance matched to TPCs, and brought back to temperatures via matrix multiplication. As you can see, I can replicate MBH99 reconstruction almost exactly with this method:
Differences in 1400-1499 are related problems with archived data, and in 1650-1699 they are due to unreported step in MBH procedure. Steve and Jean S have noted these independently, so I’m quite confident that my algorithm is correct.
I’ve considered this method as a variation of classical calibration estimator (CCE) and Steve’s made a point that this is one form of PLS. These statements are not necessarily in conflict. Original CCE is (with standardized target)
where matrices and are ML estimates of and , obtained from the calibration experiment with a model
By setting , I get exactly the same result as with double pinv. Which verifies my observation that MBH9x is CCE with special assumption about proxy noise and with incorrect variance matching step after this classical estimation.
Back to OLS (ICE) estimate, which is obtained by regressing directly X on Y,
this is justified only with a special prior distribution for , which we don’t have. Thus OLS is out of the question. Yet, it is interesting to observe that OLS is actually a matrix weighted average between CCE and zero-matrix (Brown82, Eq 2.21) :
It would be interesting to compute MBH99 results with real CCE, but S does not have inverse for AD1700 and later steps. But results for up to AD1700 are here, CCE:
As you can see, variability of ICE is lower, as it is weighted towards zero. But hey, where did that hockeystick go ?