FIGURE 9-2 Five simulated principal components and the corresponding population eigenvector. See text for details.

normalized to a common scale may have widely different variances. Huybers comments on tree ring densities, which have much lower variances than widths, even after conversion to dimensionless “standardized” form. In this case, an argument can be made for using the variables without further normalization. However, the higher-variance variables tend to make correspondingly higher contributions to the principal components, so the decision whether to equalize variances or not should be based on the scientific considerations of the climate information represented in each of the proxies.

Each principal component is a weighted combination of the individual proxy series. When those series consist of a common signal plus incoherent noise, the best estimate of the common signal has weights proportional to the sensitivity of the proxy divided by its noise variance. These weights in general are not the same as the weights in the principal component, as calculated from either raw or standardized proxies, either of which is therefore suboptimal. In any case, the principal components should be constructed to achieve a low-dimensional representation of the entire set of proxy variables that incorporates most of the climate information contained therein.