Principal Subspaces Research Articles

Self-Consistency and Principal Component Analysis

Abstract I examine the self-consistency of a principal component axis; that is, when a distribution is centered about a principal component axis. A principal component axis of a random vector X is self-consistent if each point on the axis corresponds to the mean of X given that X projects orthogonally onto that point. A large class of symmetric multivariate distributions are examined in terms of self-consistency of principal component subspaces. Elliptical distributions are characterized by the preservation of self-consistency of principal component axes after arbitrary linear transformations. A “lack-of-fit” test is proposed that tests for self-consistency of a principal axis. The test is applied to two real datasets.

Specific analysis of a euclidean cloud: application to the study of questionnaires

In this paper, we propose a method of specific Correspondence Analysis which allows to treat questionnaires when some responses are missing, and thus to free oneself from the yoke of complete disjunctive encoding. The method of specific analysis is presented within the general framework of Geometric Data Analysis for a Euclidean cloud, then particularized to multinumerical protocols and to questionnaires. We show that, in this approach, beweighted Principal Component Analysis (PCA) is privileged and that Multiples Correspondence Analysis (MCA) is equivalent to a biweighted PCA on indicator variables. Finally, we compare the specific analysis to the conventional one by writing inequalities between eigenvalues and studying the rotation of principal subspaces when one goes from the global analysis to the specific one.

Modified Oja's algorithms for principal subspace and minor subspace extraction

Principal component extraction and minor component extraction are well known and powerful techniques in time series. Modified Oja‘s algorithms for extracting principal and minor component subspaces are proposed. The only difference between them is the sign of the first term in the iteration equations. Simulations show that the algorithms are effective.

Combinatorial constructions of modules for infinite-dimensional Lie algebras, I. Principal subspace

This is the first of a series of papers studying combinatorial (with no “subtractions”) bases and characters of standard modules for affine Lie algebras, as well as various subspaces and “coset spaces” of these modules. In part I we consider certain standard modules for the affine Lie algebra g ̂ , g: = sl(n + 1, C) , n ≥ 1, at any positive integral level k and construct bases for their principal subspaces (introduced and studied recently by Feigin and Stoyanovsky (1994)). The bases are given in terms of partitions: a color i, 1 ≤ i ≤ n, and a charge s, 1 ≤ s ≤ k, are assigned to each part of a partition, so that the parts of the same color and charge comply with certain difference conditions. The parts represent “Fourier coefficients” of vertex operators and can be interpreted as “quasi-particles” enjoying (two-particle) statistical interaction related to the Cartan matrix of g. In the particular case of vacuum modules, the character formula associated with our basis is the one announced in Feigin and Stoyanovsky (1994). New combinatorial characters are proposed for the whole standard vacuum ĝ-modules at level one.

Approximation Orders of Principal Shift-Invariant Spaces Generated by Box Splines

We investigate the approximation orders of principal shift-invariant subspaces ofLp(Rd), 1<p<∞, generated by exponential box splinesMassociated to rational matrices. Moreover, under some regularity assumptions onMthe exact approximation orders are determined.

Principal Points and Self-Consistent Points of Symmetrical Multivariate Distributions

The k principal points ξ 1, ..., ξ k of a random vector X are the points that approximate the distribution of X by minimizing the expected squared distance of X to the nearest of the ξ j . A given set of k points y 1, ..., y k partition R p into domains of attraction D 1, ..., D k respectively, where D j ,consists of all points x ∈ R p such that ∥ x − y j ∥ < ∥ x − y l ∥, l ≠ j. If E[ X | X ∈ D j ] = y j for each j, then y 1, ..., y k are k self-consistent points of X (∥·∥ is the Euclidian norm). Principal points are a special case of self-consistent points. Principal points and sell-consistent points are cluster means of a distribution and represent a generalization of the population mean from one to several points. Principal points and self-consistent points are studied for a class of strongly symmetric multivariate distributions. A distribution is strongly symmetric if the distribution of the principal components ( Z 1, ..., Z p )′ is invariant up to sign changes, i.e., ( Z 1, ..., Z p )′ has the same distribution as (± Z 1, ..., ± Z p )′. Elliptical distributions belong to the class of strongly symmetric distributions. Several results are given for principal points and self-consistent points of strongly symmetric multivariate distributions. One result relates self-consistent points to principal component subspaces. Another result provides a sufficient condition for any set of self-consistent points lying on a line to be symmetric to the mean of the distribution.

Principal Points and Self-Consistent Points of Elliptical Distributions

The $k$ principal points of a $p$-variate random vector $\mathbf{X}$ are those points $\xi_1, \ldots, \xi_k \in \mathbb{R}^p$ which approximate the distribution of $\mathbf{X}$ by minimizing the expected squared distance of $\mathbf{X}$ from the nearest of the $\xi_j$. Any set of $k$ points $\mathbf{y}_1, \ldots, \mathbf{y}_k$ partitions $\mathbb{R}^p$ into "domains of attraction" $D_1, \ldots, D_k$ according to minimal distance; following Hastie and Stuetzle we call $\mathbf{y}_1, \ldots, \mathbf{y}_k$ self-consistent if $E\lbrack\mathbf{X}\mid\mathbf{X} \in D_j\rbrack = \mathbf{y}_j$ for $j = 1, \ldots, k$. Principal points are a special case of self-consistent points. In this paper we study principal points and self-consistent points of $p$-variate elliptical distributions. The main results are the following: (1) If $k$ self-consistent points of $\mathbf{X}$ span a subspace of dimension $q < p$, then this subspace is also spanned by $q$ principal components, that is, self-consistent points of elliptical distributions exist only in principal component subspaces. (2) The subspace spanned by $k$ principal points of $\mathbf{X}$ is identical with the subspace spanned by the principal components associated with the largest roots. This proves a conjecture of Flury. We also discuss implications of our results for the computation and estimation of principal points.

An approximate test for common principal component subspaces in two groups

Principal component analysis has made an important contribution to data reduction. In two sample problems, one great interest is whether we can reduce the number of variables to a smaller number in similar fashions for both samples. More precisely, we consider the hypothesisH m that the subspaces spanned by the latent vectors of the population covariance matrices corresponding to the first principal components are the same in two groups. In this paper, we propose a simple and easily interpreted test procedure forH m .

Some Tests for Common Principal Component Subspaces in Several Groups

Some Tests for Common Principal Component Subspaces in Several Groups

Some tests for common principal component subspaces in several groups

SUMMARY An approximate test, based on sample eigenprojections, is obtained for testing the hypothesis that the subspaces spanned by the first m principal components of several different covariance matrices are identical. This procedure can fairly easily be extended to an analogous test of hypothesis concerning correlation matrices. Extensions to a robust principal components analysis are also considered. In recent years principal components analysis has been generalized from the typical single population setting to that of several populations. This paper focuses on an extension that may be useful in those situations in which one needs a variance preserving simultaneous reduction of dimensionality in several groups. This sort of reduction will be possible when the subspace spanned by the most important principal component vectors of a particular group is the same as that of any other group. Formally, suppose the same p variables are being measured on objects in g different groups with the covariance

Elliptic Problems Involving an Indefinite Weight

We consider a selfadjoint elliptic eigenvalue problem, which is derived formally from a variational problem, of the form $Lu = \lambda \omega (x)u$ in $\Omega$, ${B_j}u = 0$ on $\Gamma$, $j = 1, \ldots ,m$, where $L$ is a linear elliptic operator of order $2m$ defined in a bounded open set $\Omega \subset {{\mathbf {R}}^n}\quad (n \geq 2)$ with boundary $\Gamma$, the ${B_j}$ are linear differential operators defined on $\Gamma$, and $\omega$ is a real-valued function assuming both positive and negative values. For our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem. We also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces.

Elliptic problems involving an indefinite weight

We consider a selfadjoint elliptic eigenvalue problem, which is derived formally from a variational problem, of the form L u = λ ω ( x ) u Lu = \lambda \omega (x)u in Ω \Omega , B j u = 0 {B_j}u = 0 on Γ \Gamma , j = 1 , … , m j = 1, \ldots ,m , where L L is a linear elliptic operator of order 2 m 2m defined in a bounded open set Ω ⊂ R n ( n ≥ 2 ) \Omega \subset {{\mathbf {R}}^n}\quad (n \geq 2) with boundary Γ \Gamma , the B j {B_j} are linear differential operators defined on Γ \Gamma , and ω \omega is a real-valued function assuming both positive and negative values. For our problem we prove the completeness of the eigenvectors and associated vectors in two function spaces which arise naturally in such an indefinite problem. We also establish some results concerning the eigenvalues of the problem which complement the known results and investigate the structure of the principal subspaces.

Amendments and Corrections: Common Principal Component Subspaces in Two Groups

Amendments and Corrections: Common Principal Component Subspaces in Two Groups

Common Principal Component Subspaces in Two Groups

Common Principal Component Subspaces in Two Groups

Common principal component subspaces in two groups

SUMMARY One important practical application of principal component analysis is to reduce a large number of variables, say p, to a smaller number, m, by making use of the first m principal components. This technique can easily be extended to two or more groups if the subspaces spanned by the first m principal components are the same for all groups. In this paper we develop an approximate procedure for testing such a hypothesis of common subspaces when two groups are involved. The adequacy of the approximation is investigated by a simulation and the method is illustrated by a numerical example.

Principal functions and invariant subspaces of hyponormal operators

Principal functions and invariant subspaces of hyponormal operators