REDUNDANCY ANALYS IS AN ALTERNATIVE FOR CANONICAL CORRELAT ION ANALYS IS

A component method is presented maximiz ing Stewart and Love's redundancy index. Relationships with multiple correlation and principal component analysis are pointed out and a rotational procedure for obtaining bi-orthogonal variates is given. An elaborate example comparing canonical correlation analysis and redundancy analysis on artificial data is presented. Key words: principal components, generalized multiple correlation analysis, cross battery principle component analysis. 1. In t roduct ion In canonical correlation analysis components are extracted from two sets of variables simultaneously in such a way as to maximize the correlation, #, between these components. Mathematically, the criterion to be maximized under restrictions is (1) w'Rxyv- 1.t(w'Rxxw- 1)- v(v 'Ryyv- 1), where u and v are Lagrange multipliers. Elaboration then leads to the follow-ing eigenvalue igenvector equations [Anderson, 1958]: (2) (Rxx-~RxyRyy- lRyx- U2I) w = O, and (3) (Ryy- lRyxRxx- lRxy- v2 I) v = 0 The eigenvalues u s and v 2 are equal, and are also equal to the squared canonical correlation coefficient. After extraction of the first pair o f canonical variates, a second pair can be determined having maximum correlation, with the restric-tion that the variates are uncorrelated with all other canonical variates except with their counterparts in the other set, and so forth. A Fortran lV program for the method of redundancy analysis described in thispaper can be obtained from the author upon request