This next replication exercises shows the results of attempting to reconstruct the NH temperature index from the archived RPCs, with the usual puzzling discrepancies.
The final calculation of the NH temperature index from the reconstructed temperature series would seem to be straightforward. The process is not described in detail in MBH98. The most straightforward interpretation of the procedure would be as follows:
These steps are all linear processes and can be be summarized as follows:
(1) T ”€ ‘? à…⪠* diag(àŽ») * E * diag (à?Æ’) * Ie * 1/n,
where T is the average NH temperature, à…⪠is the matrix of reconstructed temperature PCs, àŽ» is the vector of temperature eigenvalues, E is the matrix of temperature eigenvectors (EOFs), à?Æ’ is the vector of gridcell standard deviations, Ie selects the NH gridcells and 1/n is a vector with the length of the number of NH gridcell locations (707).
This equation can be simplified by first calculating a vector of length 16 as follows:
(2) w ”€ ‘? diag(àŽ») * E * diag (à?Æ’) * Ie * 1/n.
The number of temperature eigenvectors ranges from 11 in the AD1820 step to 1 in the AD1400 step. Thus, in the simplest case, the AD1400 step, one would expect that the NH temperature reconstruction would be a multiple of the RPC1. Because Mann et al. have only archived the spliced reconstruction and not the individual steps, confirmation of most steps is not possible. that much more complicated.
In order to test the possibility of a linear relationship of the archived NH temperature reconstruction to the RPC1 in the AD1400 step, I carried out a linear regression of the NH temperature reconstruction against the RPC1 in the 1400-1449 period. The top left panel of Figure 1 below shows the contribution of the RPC1 to the MBH98 temperature index; the top right panel shows the residual from the actual MBH98 temperature index. There is an exact linear relationship in the 1400-1449 (zero residual), although, as I will discuss in my next post, the regression coefficient does not tie together with the expected weight from Equation (2) above. This suggests some still undisclosed re-scaling procedure. It is evident that the RPC1 accounts for the majority of the total variance. The RPC1 accounts for most of the variance up to 1700 with lower eigenvectors making larger contributions in the later periods.
Then, I regressed the RPC2 against the residual in the 1450-1600 period. The reconstruction of the MBH98 temperature index using these 2 series was exact. The coefficient for the RPC1 was identical to the 1400-1450 step. The effect of the RPC2 is obviously very small on the reconstruction.
The rest of the reconstruction cannot be determined on the present record. The earliest of the remaining 3 archived RPCs is RPC5 beginning in 1650. This has an R2 of 0.50 for the period 1650-1749. The coefficient is 1.63 (calculated above: 1.06). The remaining two archived RPCs commence in 1750 and 1760 respectively. They were regressed against residuals for the period 1760-1980 yielding an R2 of 0.56; the coefficients were 0.54 and -2.45 (calculated above: -0.08 and -1.55).
Figure 1. Left – reconstruction element calculated by regression from archived RPC; Right – residual after deduction of reconstructed contribution.
It is odd that a method which works exactly in the first 2 steps cannot be extended into later steps. There is also something odd about one RPC starting in 1650 – this is not a calculation step and it appears nowhere else. It seems like something is missing.
In the next few days, I’ll post up something about what happens with the emulated RPCs and the differences between the calculated weights from equation (2) and the observed weights from the regressions.