Dummies guide to the latest “Hockey Stick” controversy

As discussed above, a different convention (MM05 suggest one that has zero mean over the whole record) will change the ordering, significance and number of important PCs. In this case, the number of significant PCs increases to 5 (maybe 6) from 2 originally. This is the difference between the blue points (MBH98 convention) and the red crosses (MM05 convention) in the first figure. Also PC1 in the MBH98 convention moves down to PC4 in the MM05 convention. This is illustrated in the figure on the right, the red curve is the original PC1 and the blue curve is MM05 PC4 (adjusted to have same variance and mean). But as we stated above, the underlying data has a hockey stick structure, and so in either case the ‘hockey stick’-like PC explains a significant part of the variance. Therefore, using the MM05 convention, more PCs need to be included to capture the significant information contained in the tree ring network.

This figure shows the difference in the final result whether you use the original convention and 2 PCs (blue) and the MM05 convention with 5 PCs (red). The MM05-based reconstruction is slightly less skilful when judged over the 19th century validation period but is otherwise very similar. In fact any calibration convention will lead to approximately the same answer as long as the PC decomposition is done properly and one determines how many PCs are needed to retain the primary information in the original data.

different conventions

5) What happens if you just use all the data and skip the whole PCA step?

This is a key point. If the PCs being used were inadequate in characterizing the underlying data, then the answer you get using all of the data will be significantly different. If, on the other hand, enough PCs were used, the answer should be essentially unchanged. This is shown in the figure below. The reconstruction using all the data is in yellow (the green line is the same thing but with the ‘St-Anne River’ tree ring chronology taken out). The blue line is the original reconstruction, and as you can see the correspondence between them is high. The validation is slightly worse, illustrating the trade-off mentioned above i.e. when using all of the data, over-fitting during the calibration period (due to the increase number of degrees of freedom) leads to a slight loss of predictability in the validation step.

No PCA comparison

6) So how do MM05 conclude that this small detail changes the answer?

MM05 claim that the reconstruction using only the first 2 PCs with their convention is significantly different to MBH98. Since PC 3,4 and 5 (at least) are also significant they are leaving out good data. It is mathematically wrong to retain the same number of PCs if the convention of standardization is changed. In this case, it causes a loss of information that is very easily demonstrated. Firstly, by showing that any such results do not resemble the results from using all data, and by checking the validation of the reconstruction for the 19th century. The MM version of the reconstruction can be matched by simply removing the N. American tree ring data along with the ‘St Anne River’ Northern treeline series from the reconstruction (shown in yellow below). Compare this curve with the ones shown above.

No N. American tree rings

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