Sylvie Viguier-Pla

Factor-based comparison of groups of curves

A method able to detect a possible common structure between two (or more than two) groups of curves is considered. This technique is part of the recent developments within the field of Functional Data Analysis. It is discussed how the standard approach based on factorial analysis for comparing groups of multivariate data can be used in this infinite-dimensional framework. The potentiality of such a functional method in terms of application is illustrated through a spectrometric functional data set coming from food industry quality control.

Tensor products and statistics

Abstract A dictionary between operator-based and matrix-based languages in multivariate statistical analysis is proposed. Then this formulary is applied to asymptotic factorial analyses, especially for giving asymptotic covariance matrices and operators in an explicit form. Finally, we present the mathematical foundations on which are based the functional tools, i.e. tensor products of linear spaces, of vectors, and of operators.

Gap between orthogonal projectors-Application to stationary processes

In the statistics of processes, the usual convergence of projectors does not provide a good idea of the continuity of some phenomena. In order to address it, we introduce a measure of the gap between two projectors, that we link to another type of convergence of sequences of projectors. This allows to define the gap between two spectral measures and then to develop various properties associated with the latter. In particular, we establish the continuity of the convolution product. A study of the closeness between two continuous random functions and the exhibition of a common filter for them illustrate the possible applications in functional statistics.

On proximity between PCA in the frequency domain and usual PCA

The principal components analysis (PCA) in the frequency domain of a stationary p-dimensional time series (X n ) n∈ℤ leads to a summarizing time series written as a linear combination series X′ n =∑ m C m ° X n−m . Therefore, we observe that, when the coefficients C m , m≠0, are close to 0, this PCA is close to the usual PCA, that is the PCA in the temporal domain. When the coefficients tend to 0, the corresponding limit is said to satisfy a property noted 𝒫, of which we will study the consequences. Finally, we will examine, for any series, the proximity between the two PCAs.

Factor-based comparison of k populations

A generalized canonical analysis is considered in order to build a test statistic for the comparison of factor subspaces of k related populations. This test constitutes a generalization of a test of comparison of two factor subspaces (Dauxois, J., Romain, Y. and Viguier, S. (1993). Comparison of two factor subspaces. J. Multivariate Anal., 44, 160–178). A first stage of the generalization is made in the case of two populations, with the infinite dimension, and without the hypothesis of sphericity. A numerical example is given to compare the test statistics for k = 2, and another example illustrates the k-sample case.

Commuter of operators in a Hilbert space

Commutativity between two stationary functions is a notion that generalizes the concept of stationary correlation. Two stationary functions commute if and only if their associated spectral measures and unitary operators commute. When the operators do not commute, we define the concept of commuter for two operators, and then derive it for two projectors, for a projector and a unitary operator, and for two unitary operators. We establish relations between these commuters and some other tools related to proximity between processes. Finally, we propose a method which retrieves the commuter for two stationary series of finite spectrum, and we study some convergence properties.

Commutator of projectors and of unitary operators

We define and study the concept of commutator for two projectors, for a projector and a unitary operator, and for two unitary operators. Then we state several properties of these commutators. We recall that projectors and unitary operators are linked with the spectral elements of stationary processes. We establish relations between these commutators and some other tools related to the proximity between processes.

Contribution of Functional Approach to the Classification and the Identification of Acoustic Emission Source Mechanisms

In a context of nuclear Reactivity Initiated Accident, we describe acoustic emission signals, for which a problem of classification is open. As classical approaches with a reduced number of variables do not give satisfactory discrimination, we propose to use the envelopes of the received signals. We perform a k-means clustering and discuss the first results of this approach.