Cross-Covariance and the Singular Value Decomposition

Abstract

At the end of the previous chapter we have introduced the concept of the simultaneous analysis of different fields. We have introduced the Combined EOF that, after a suitable scaling, allow us to produce patterns of variability that reflect the covariance properties of different data types. This is an interesting development because it leads to the consideration of the cross-covariance along the same lines we have used for the covariance of a single field. The program we have followed in Chaps. 4 and 5Generalizations: Rotated, Complex, Extended and Combined EOFchapter.5.151 has been inspired by the attempt to analyze the variance of a single field, finding the best way to represent the data, maximizing the variance with the smallest number of patterns. The modes we have found have been identified as “preferred” modes of variations and we have shown that they are linked to the number of degrees of freedom in the data space.