Canonical Correlation Analysis, Redundancy Analysis and ：典型相关分析，冗余分析.ppt

Canonical Correlation Analysis, Redundancy Analysis and Canonical Correspondence Analysis  Hal Whitehead BIOL4062/5062 Canonical Correlation Analysis Redundancy Analysis Canonical Correspondence Analysis Multivariate Statistics with Two Groups of Variables Look at relationships between two groups of variables species variables vs environment variables (community ecology) genetic variables vs environmental variables (population genetics) Canonical Correlation Analysis Multivariate extension of correlation analysis Looks at relationship between two sets of variables Canonical Correlation Analysis Given a linear combination of X variables: F = f1X1 + f2X2 + ... + fpXp and a linear combination of Y variables: G = g1Y1 + g2Y2 + ... + gqYq The first canonical correlation is: Maximum correlation coefficient between F and G, for all F and G F1={f11,f12,...,f1p} and G1={g11,g12,...,g1q} are corresponding canonical variates Canonical Correlation Analysis Canonical Correlation Analysis The first canonical correlation is: Maximum correlation coefficient between F and G, for all F and G F1={f11,f12,...,f1p} and G1={g11,g12,...,g1q} are corresponding first canonical variates Canonical Correlation Analysis So each canonical correlation is associated with a pair of canonical variates Canonical correlations decrease Canonical correlations are higher than generally found with simple correlations as coefficients are chosen to maximize correlations Canonical Correlation Analysis Correlation Matrix: X1 X2 X3 ... Xp Y1 ... Yq X1 X2 . A (pxp) C (pxq) . Xp Y1 . C (qxp) B (qxq) . Yq Canonical correlations are: Squareroots of Eigenvalues of B-1 C A-1 C Canonical variates for Y variables are Eigenvectors Number of canonical correlations = min(No. X’s, No. Y’s) Can test whether canonical correlations are significantly different from 0 Canonical Correlation Analysis What are the canonical