Climatic Indices, Principal Components, and the Gauss-Markov Theorem

If indices are to be used as the variables predicted by linear statistical models, it is important to be able to recover as much local information as possible from the values forecast for the indices. Here it is shown that the indices that encapsulate the most information about the local climatic state are determined by a generalized (two-matrix) eigenvalue problem that is equivalent to the usual (one-matrix) eigenvalue problem involving the sample correlation matrix. Thus, the best indices in the sense of providing the most location-specific information are familiar principal-component indices. Regarding the indices as predictors in linear statistical models similar to those routinely used for estimating meteorological fields from observations reveals the role of the Gauss-Markov theorem in EOF analyses. From this perspective each index can he characterized by two EOF-like maps: the first illustrating the linear combinations of the data used to define the index, and the second displaying the Gauss-Markov weights for the index to predict local variables, both of which are related to the eigenvectors of the sample correlation matrix. Other maps can be used to display information about sampling errors: one to characterize the uncertainty of the weights; another to display the skill with which the index accounts for the training data; and a third to show how well it explains independent data. Such maps are illustrated within the context of 43 years of North Atlantic seasonal sea surface temperature anomalies. The analysis presented here underlines two additional points. First, any linear combination of the indices would result in an equivalent model yielding exactly the same forecast. Consequently, it may be desirable to use indices that are easier to interpret physically. Second, when indices are regarded as being variables of a linear statistical model, the analysis of sampling error can be formulated in terms of the uncertainty of the Gauss-Markov weights inferred from a limited training set rather than in terms of the sample-to-sample variability of eigenvalues and eigenvectors.