There are two groups of random variables, x1,x 2, …, X p' and y1,Y2, …, Yq', so we say p ≤ Q. For x, y, we assume that the mean and covariance matrix of the first group of variables are e x1cov x ∑/. The mean and covariance matrix of the second group of variables is e y 2cov y ∑ 22, while the covariance matrix of the first and second groups of variables is cov x y ∑12 ∑ 21X. Therefore, for the matrix z, there is a mean vector Eze (9-1-1). Two variables, 2Y221∑1∑12p× PP× q ∑ 21∑ 22q× pq× q, X 1, X 2, ..., X p,.

First, we make a linear combination of two groups of variables, namely UA1x1a2x2LAPP' xvb1Y2Lbqyqb' yaa1a2LAP. If B b 1 b2 L bq is an arbitrary non-zero constant coefficient vector, you can get'', var ua' cov x aa' ∑11a var VB' cov bb' ∑ 22b Cov ua' cov x bb' ∑12 b to call u and v typical variables. That is, the problem of A' ∑12b ρ ∑1ab' ∑ 22b canonical correlation analysis is how to choose the optimal linear combination of typical variables.

The selection principle is: among all linear combinations U and V that make U1a ′ x11,choose U and V with the largest typical correlation coefficient, that is, choose A and B with the largest correlation coefficient with V1b ′ y (among all U and V)/. Then choose A and B to make the correlation coefficient between U2A ′ 2x and V2B ′ 2y the largest among the combinations U and V which are not related to U 1 and V 1, and continue until all the linear combinations U which are not related to U 1 are U 1 and V1v2lvp, respectively.

V At this time, P is equal to the rank typical variables U 1 and V 1 of the covariance matrix between variables X and Y, and U 2 and V2 ... u P and V p are extracted from large columns and small pairs according to the correlation coefficient of U P and V P until the correlation between the two groups of variables is decomposed.