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Canonical correlation
[A,B] = canoncorr(X,Y)
[A,B,r] = canoncorr(X,Y)
[A,B,r,U,V] = canoncorr(X,Y)
[A,B,r,U,V,stats] = canoncorr(X,Y)
[A,B] = canoncorr(X,Y) computes the sample canonical coefficients for the n-by-d1 and n-by-d2 data matrices X and Y. X and Y must have the same number of observations (rows) but can have different numbers of variables (columns). A and B are d1-by-d and d2-by-d matrices, where d = min(rank(X),rank(Y)). The jth columns of A and B contain the canonical coefficients, i.e., the linear combination of variables making up the jth canonical variable for X and Y, respectively. Columns of A and B are scaled to make the covariance matrices of the canonical variables the identity matrix (see U and V below). If X or Y is less than full rank, canoncorr gives a warning and returns zeros in the rows of A or B corresponding to dependent columns of X or Y.
[A,B,r] = canoncorr(X,Y) also returns a 1-by-d vector containing the sample canonical correlations. The jth element of r is the correlation between the jth columns of U and V (see below).
[A,B,r,U,V] = canoncorr(X,Y) also returns the canonical variables, scores. U and V are n-by-d matrices computed as
U = (X-repmat(mean(X),N,1))*A V = (Y-repmat(mean(Y),N,1))*B
[A,B,r,U,V,stats] = canoncorr(X,Y) also returns a structure stats containing information relating to the sequence of d null hypotheses $${H}_{0}^{(k)}$$, that the (k+1)st through dth correlations are all zero, for k = 0:(d-1). stats contains seven fields, each a 1-by-d vector with elements corresponding to the values of k, as described in the following table:
Field | Description |
---|---|
Wilks | Wilks' lambda (likelihood ratio) statistic |
df1 | Degrees of freedom for the chi-squared statistic, and the numerator degrees of freedom for the F statistic |
df2 | Denominator degrees of freedom for the F statistic |
F | Rao's approximate F statistic for $${H}_{0}^{(k)}$$ |
pF | Right-tail significance level for F |
chisq | Bartlett's approximate chi-squared statistic for $${H}_{0}^{(k)}$$ with Lawley's modification |
pChisq | Right-tail significance level for chisq |
stats has two other fields (dfe and p) which are equal to df1 and pChisq, respectively, and exist for historical reasons.