SAS学习资料 Canonical Correlation Analysis.pptVIP

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 power 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 between 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 CANONICAL 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 vector and covariance matrix The main task of canonical correlation analy-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 (10.5) and (2.45), we obtain We shall seek coefficient vectors a and b such that is 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 (