1、Chapter 10,Canonical Correlation Analysis,10.1 INTRODUCTION,Canonical correlation analysis seeks to identify and quantify the associations between two sets of variables. H. Hotelling (5,6), who initially dev-eloped the technique, provided the example of rel-ating arithmetic speed and arithmetic powe
2、r to re-ading speed and reading power. (See Exercise 10.9) Other examples include relating governme-ntal policy variables with economic goal variables and relating college “performance” variables with precollege “achievement” variables.,Canonical correlation analysis focuses on the correlation betwe
3、en a linear combination of the variables in one set and a linear combination of the variables in another set. The maximization as-pect of the technique represents an attempt to con-centrate a high-dimensional relationship between two sets of variables into a few pairs of canonical variables.,10.2 CA
4、NONICAL VARIATES AND CANONICAL CORRELATIONS,It will be convenient to consider and joi-ntly above. We assume, in the theoretical develop-ment, that represents the smaller set, so that . For the random vectors and , let ; ;,has mean vectorand covariance matrix,The main task of canonical correlation an
5、aly-sis is to summarize the associations between the and sets in terms of a few carefully chosen covariances (or correlations) rather than the co-variances in .,Linear combinations provide simple summary measures of a set of variables. Set for some pair of coefficient vectors a and b. then, using (1
6、0.5) and (2.45), we obtain,We shall seek coefficient vectors a and b such thatis as large as possible. We define the following: The first pair of canonical variables, or first canonical variate pair, is the pair of linear combi-nations , having unit variances, which,maximize the correlation (10.7);
7、The second pair of canonical variables, or second canonical variate pair, is the pair of linear combinations , having unit varian-ces, which maximize the correlation (10.7) among all choices that are uncorrelated with the first pair of canonical variables.At the kth step,Result 10.1. Suppose and let
8、 the random vectors and have , and , where has full rank. then The kth pair of canonical variables, ,maximizesAmong those linear combinations uncorrelated with the preceding canonical variables. Here are the eigenvalues of .,Example 10.1 (Calculating canonical variates and canonical correlations for standardized variables)P.549,10.3 INTERPRETING THE POPULATION CANONICAL VARIABLES,
