Part and bipartial canonical correlation analysis

Abstract

multivariate analysis,

Generalized Constrained Redundancy Analysis

“…Note that the structure vector and standardized cross loadings reported in Table 5 are correlations between redundancy component and the original predictor (X) and criterion variables (Y ) rather than their constrained counterparts (XH X and Y H Y ). Table 6 provides part redundancies (Timm and Carlson, 1976) for all 153 combinations. The total redundancy is 72.33 for the combination of [9] X and [1] Y .…”

Linear relative canonical analysis of Euclidean random variables, asymptotic study and some applications

“…That is known in the literature as partial canonical correlation analysis (see e.g. Rao (1969), Timm and Carlson (1976)). Then this last analysis appears as a particular case of the general relative canonical analysis of subspaces (see Dauxois and Nkiet (2002), Dauxois et al (2004)), obtained by considering subspaces generated by specific linear functions of the original variables.…”

“…In order to show up this property we prefer to use the terminology linear relative canonical analysis instead of partial canonical analysis. Notice that the part and bipartial canonical correlation analysis developed by Timm and Carlson (1976) can be reobtained from our framework by considering the CA of E1 and E2.3, and E1.3 and E2.4 respectively, where E2.4 is constructed as in equation (2.2) with another Euclidean r.v. )(4.…”

“…One knows that (see, e.g., Timm and Carlson (1976)) when (X1,X2,X3) has a normal distribution this property is equivalent to the conditional independence of X1 and X2, given )(3. Following an approach which has been used for classical LCA (trainer and Nicewander (1979), Lin (1987), CIdroux and Lazraq (1988), Nkiet (1997b), Nkiet (2000)), a class of linear relative association measures have been introduced by Dauxois and Nkiet (2002).…”

“…Fujikoshi and Khatri (1990), Baccini et al (2001)). For canonical analysis, the same approach gave partial (see Rao (1969)) part and bipartial (see Timm and Carlson (1976)) canonical correlation analysis. In this later work, statistical inferences based on the canonical coefficients related to these analyses were proposed.…”