Metric-based principal components: data uncertainties

Seeking an index characterizing the best-determined mode of variability leads to a natural generalization of principal-component analysis with an explicit metric characterizing the uncertainties of the data. This formalism, which distinguishes between state-space patterns and patterns of coefficients defining principal components, allows the more accurate data to exert a greater influence on the definition of the indices than they do in conventional principal-component analysis; in all other aspects, the new formalism is the same as the old. Within the context of the simple example of Bretherton and collaborators, metric-based principal-component analysis is shown to be capable of finding correlated patterns of variability in two different data sets.