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

Abstract

Abstract. We introduce the Linear Relative Canonical Analysis (LRCA) of Euclidean random variables. Then similar properties than for usual linear Canonical Analysis are obtained. Furthermore, we develop an asymptotic study of LRCA and apply the obtained results to tests for lack of relative linear association, dimensionality and invariance.

On invariance in canonical analysis with applications to covariate discriminant analysis

“…Assertion (ii) of the preceding lemma shows that E 2.3 has the form of subspaces that are involved in relative canonical analysis (see Dauxois et al 2004aDauxois et al , 2004b; that is why we will term the DA of Y and X 2·3 , the relative discriminant analysis (RDA) of Y and X 2 relative to X 3 , and we put…”

Asymptotic theory of multiple-set linear canonical analysis

“…Remark 2.1 1) The constraints sets given in (2) and (3) can be expressed by using covariance operators defined for (k, ℓ) ∈ {1, · · · , K} 2 by:…”

“…(e.g., Muirhead and Waternaux (1980), Anderson (1999), Pousse (1992), Fine (2000), Dauxois et al (2004)). It would be natural to wonder how the obtained results extend to the case of MSLCA but, to the best of our knowledge, such an approach has never been tackled.…”

Functional partial canonical correlation

“…Given three random vectors X 1 , X 2 and X 3 , the partial canonical correlation of X 2 and X 3 relative to X 1 was defined as the ordinary canonical correlation betweenX 2 = X 2 − P X1 X 2 andX 3 = X 3 − P X1 X 3 , where P X1 denotes projection onto the linear space spanned by X 1 . Related work by Dauxois and Nkiet [3] and Dauxois et al [5] comes with the restriction of a closed range for covariance operators which, again, confines statistical applications to the finite dimensional setting that was already treated in Roy's original work. In Section 3, we show how the partial canonical correlation concept can be rigorously extended to infinite dimensions and functional data.…”

“…Dauxois and Pousse [6], Dauxois et al [4], Dauxois and Nkiet [3] and Dauxois et al [5] largely ignore this issue with the consequence that their statistical applications become relevant only for finite dimensional covariance operators whose ranges are necessarily closed. Such results are, of course, already subsumed by the original Hotelling [12] work.…”