Multiple Linear Regression Methods

Abstract

In the previous chapters we have introduced techniques to find relations between two or more different fields. In this chapter we describe a more general framework that may provide additional insight into previously analyzed methods. In general a relation between fields can be formulated symbolically as

$$\begin{array}{rcl} \mathbf{Z} = f(\mathbf{S})& &\end{array}$$

(8.1)

where, for instance, Z represents the geopotential and S the SST. The exact form of the relation is unknown, but it is probably time dependent and thus includes effects of time lags and so on. In practice, it is really difficult to investigate arbitrary functional forms for f in (8.1); assuming f to be linear represents a simplifying but viable alternative. In this case the function f may be represented by matrices. We have seen in the previous chapters that powerful methods have been devised to identify relations of the form (8.1) assuming that f(S) is a linear function. We have seen linear correlation methods, teleconnection analysis and finally methods that analyze systematically the linear relation between two data sets, such as the Singular Value Decomposition (SVD) or Canonical Correlation Analysis (CCA). We will now define a general framework that includes the latter as special case.