Measures of Association for Hilbertian Subspaces and Some Applications

Abstract

Measures of association are introduced for Hilbertian subspaces, that are defined by a few axioms and are shown to be symmetric nondecreasing functions of the canonical coefficients. When particular subspaces are considered, classical measures of association are obtained as particular cases. Moreover, the proposed framework allows one to introduce new approaches for measuring partial noncorrelation, partial independence and linear predictability of a stationary process.

An extension of Fisher's discriminant analysis for stochastic processes

“…(see, also, related work by Dauxois and Nkiet [8], Dauxois et al [9] and He et al [23]) Although beyond the scope of this paper, we note in passing that it is possible to extend the well-known connection between the finite dimensional Fisher's LDA and canonical correlation analysis (see, e.g., [28,Chapter 7]) to the infinite dimensional setting that was considered here. Further, through the use of Parzen's formulation for the density functional of a Gaussian process (e.g., [33]) it is possible to develop a parallel of the classical finite dimensional Bayes' classifier that can be used in an infinite dimensional context.…”

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

“…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.…”

“…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). These measures have the form m/xa (X1, X2) := ~(A), where is a continuous symmetric function from IRm to R+ satisfying ~(x) = 0 r x = 0.…”

“…These measures have the form m/xa (X1, X2) := ~(A), where is a continuous symmetric function from IRm to R+ satisfying ~(x) = 0 r x = 0. These measures can be used for testing for lack of linear relative association since equation (4.1) is equivalent to rn/xa (X1,)(2) = 0 (see Dauxois and Nkiet (2002) PROOF. Since(c~{{)a)l<i<m (resp.~c~ (j)' < 2.3~1<j_m ~ is an orthonormal basis of X1 (resp.…”

Testing conditional independence using maximal nonlinear conditional correlation

“…If the above (f i , g i )'s exist, then one can define ρ i (X, Y |Z) = E(f i (X, Z)g i (Y, Z)|Z) for each i and the ρ i (X, Y |Z)'s can serve as a conditional version of canonical coefficients. A measure of conditional association satisfying (P1)-(P5) can be obtained by taking a proper combination of the ρ i (X, Y |Z)'s, following the approach in [4]. Examples of such combinations include ρ 1 (X, Y |Z) and 1 − exp(− i ρ 2 i (X, Y |Z)).…”