RESTRICTED CANONICAL CORRELATIONS

Given two random vectors Y(1) and Y(2) the first canonical correlation between them is defined as: sup{ Correlation(alpha'Y(1), beta'Y(2)) : alpha is an element of R(p), beta is an element of R(q)}. However, in many practical situations (e.g. educational testing problems, neural networks), some natural restrictions on the coefficients alpha and beta may arise which should be incorporated in this maximization procedure. The maximum correlation subject to such constraints is referred to as the restricted canonical correlation. This problem is treated here under the nonnegativity restriction on alpha and beta. The analysis is extended to more general form of inequality constraints, and also when the restrictions are present only on some of the coefficients. Restricted versions of some other related measures are also discussed. This includes principal component analysis and different modifications of canonical correlations. Anomalies with higher order correlation are also described. Some properties of restricted canonical correlation, including its bounds, are studied.