A Caveat Concerning Singular Value Decomposition

Abstract

An assessment is made of the ability of the singular value decomposition (SYD) technique to recover the relationship between two variables x and y from a time series of their observations. It is shown that SVD is rigorously successful only in the special cases when either (i) the transformation linking x and y is orthogonal or (ii) the covariance matrix of either x or y is the identity matrix. The behavior of the method when theSE conditions are not met is also studied in a simple two-dimensional case. That this caveat can be relevant in a meteorological context is demonstrated by performing an SVD analysis of a time series of global upper-tropospheric streamfunction and vorticity fields. Although these fields are linked by the two-dimensional Laplacian operator on the sphere, it is shown that the pairs of singular patterns resulting from the SVD analysis are not so related. The problem is apparent even for the first SVD pair and generally becomes worse for succeeding pairs These results suggest that any physical interpretation of SVD pairs may be unjustified.