Canonical Correlation Analysis for finding patterns in coupled fields

First CCA pattern of Sea Level Pressure (SLP) and Sea Surface Temperature (SST) monthly anomalies for the region between -180 °W to -70 °W and +30 °N to -30 °S.

The following post demonstrates the use of Canonical Correlation Analysis (CCA) for diagnosing coupled patterns in climate fields. The method produces similar results to that of  Maximum Covariance Analysis (MCA), but patterns reflect maximum correlation rather than maximum covariance. Furthermore, the output of the model is a combination of linear models that can be used for field prediction.

This particular method was illustrated by Barnett and Preisendorfer (1997) - it constructs a CCA model based on a truncated subset of EOF coefficients (i.e. "principle components") instead of using the original field (as with MCA). This truncation has several benefits for the fitting of the model - First, one reduces the amount of noise in the problem by eliminating the higher EOF modes, which represent poorly organized, small-scale features of the fields. Second, by using orthogonal functions, the algebra of the problem is simplified (see von Storch and Zweiers 1999 for details). Bretherton etal. (1992) reviewed several techniques for diagnosing coupled patterns and found the Barnett and Preisendorfer method (hereafter "BPCCA") and MCA to be the most robust.

Exponentiation of a matrix (including pseudoinverse)

The following function "exp.mat" allows for the exponentiation of a matrix (i.e. calculation of a matrix to a given power). The function follows three steps:
1) Singular Value Decomposition (SVD) of the matrix
2) Exponentiation of the singular values
3) Re-calculation of the matrix with the new singular values

The most common case where the method is applied is in the calculation of a "Moore–Penrose pseudoinverse", or a matrix raised to the power of -1. In this case, the function returns the same result as the "pseudoinverse" function from the corpcor package (although maybe not as nicely implemented). This should also be analogous to the pinv function in Matlab.

Less common is the need to calculate other powers of matrices. For example, I use this function to calculate the square root of a matrix (and square root of the inverse of a matrix) within Canonical Correlation Analysis (CCA). I hope to write a post shortly on the use of CCA in identifying coupled patterns in climate data.

Below is the code for the exp.mat function as well as some demonstrations of its use in R.