An Introduction to Wishart Matrix Moments

Bishop, Adrian N.; Moral, Pierre Del; Niclas, Angèle

doi:10.1561/2200000072

Cited by 13 publications

(4 citation statements)

References 85 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The last assertion is a consequence of the Ando-Hemmen inequality (1) and the estimate (8). This ends the proof of the Taylor expansion at rank n " 1.…”

Section: Introductionsupporting

confidence: 67%

“…The formulae (5) for higher terms in the Taylor series for the square root provide a polynomial-type perturbation approximation of the square root at any order. These non asymptotic expansions have been used in [7,8] to analyze the fluctuation as well as the bias of the square root function of Wishart matrices and sample covariance matrices associated with stochastic Riccati equations arising in Ensemble-Kalman-Bucy filter theory.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Taylor expansion of the square root matrix function

Moral

Niclas

2018

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

“…The last assertion is a consequence of the Ando-Hemmen inequality (1) and the estimate (8). This ends the proof of the Taylor expansion at rank n " 1.…”

Section: Introductionsupporting

confidence: 67%

Section: Introductionmentioning

confidence: 99%

A Taylor expansion of the square root matrix function

Moral

Niclas

2018

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

“…more detailed discussion on this multivariate central limit theorem can be found in [39]; see also the more recent study [12] as well as [15] for non-necessarily Gaussian variables. As verified later in corollary 2.9, the sample covariance satisfies the required moment condition (1.14).…”

Section: Ensemble Kalman-bucy Filtersmentioning

confidence: 99%

A perturbation analysis of stochastic matrix Riccati diffusions

Bishop¹,

Moral²,

Niclas³

2020

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

Matrix differential Riccati equations are central in filtering and optimal control theory. The purpose of this article is to develop a perturbation theory for a class of stochastic matrix Riccati diffusions. Diffusions of this type arise, for example, in the analysis of ensemble Kalman-Bucy filters since they describe the flow of certain sample covariance estimates. In this context, the random perturbations come from the fluctuations of a mean field particle interpretation of a class of nonlinear diffusions equipped with an interacting sample covariance matrix functional. The main purpose of this article is to derive non-asymptotic Taylor-type expansions of stochastic matrix Riccati flows with respect to some perturbation parameter. These expansions rely on an original combination of stochastic differential analysis and nonlinear semigroup techniques on matrix spaces. The results here quantify the fluctuation of the stochastic flow around the limiting deterministic Riccati equation, at any order. The convergence of the interacting sample covariance matrices to the deterministic Riccati flow is proven as the number of particles tends to infinity. Also presented are refined moment estimates and sharp bias and variance estimates. These expansions are also used to deduce a functional central limit theorem at the level of the diffusion process in matrix spaces.Résumé: Les équations de Riccati matricielles jouent un rôle important dans la théorie du filtrage et du contrôle optimal. Cet article présente une théorie des perturbations d'une classe d'équations de Riccati matricielles stochastiques. Ces modèles probabilistes sont d'un usage courant dans la théorie des filtres de Kalman d'Ensemble. Ils représentent dans ce contexte l'évolution des matrices de covariance empiriques associées à un ensemble de diffusions en interaction. Les perturbations aléatoires résultent des fluctuations stochastiques d'un système de particules de type champ moyen interagissant avec la mesure empirique du système. Nous présentons dans cet article une formule de Taylor non asymptotique pour des flots stochastiques de diffusion de Riccati matricelles par rapport à un paramètre de fluctuation. Ces développements sont fondés sur un nouveau calcul différentiel stochastique et une analyse fine de semigroupes non linéaires dans des espaces de matrices. Ces résultats permettent de quantifier avec précision les fluctuations des flots de matrices stochastiques autour des systèmes limites à tout ordre. Nous illustrons ces résultats avec une preuve de la convergence des matrices empiriques de filtres de Kalman d'Ensemble vers la solution d'équations de Riccati déterministes lorsque le nombre de particules tends vers l'infini. Nous présentons dans ce cadre des estimations fines des biais et des variances, ainsi qu'un theorème de la limite centrale fonctionnel au niveau du processus matriciel.

show abstract

“…The fluctuation analysis of this third class of EnKF model can also be developed easily by combining the Lipschitz-type estimates w.r.t. the initial state presented in [9,10], with conventional sample mean error estimates based on independent copies of the initial values, see for instance [13] for χ 2 -type estimates associated with sample covariance estimates.…”

Section: Mean Field Particle Interpretationsmentioning

confidence: 99%

On one-dimensional Riccati diffusions

Bishop¹,

Moral²,

Kamatani³

et al. 2019

Ann. Appl. Probab.

View full text Add to dashboard Cite

This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynman-Kac path integration, and exponential change of measures, to derive sharp exponential decays to equilibrium. We also provide uniform estimates with respect to the time horizon, quantifying with some precision the fluctuations of these diffusions around a limiting deterministic Riccati differential equation. These results provide a stronger and almost sure version of the conventional central limit theorem. We illustrate these results in the context of ensemble Kalman-Bucy filtering. To the best of our knowledge, the exponential stability and the fluctuation analysis developed in this work are the first results of this kind for this class of nonlinear diffusions.

show abstract

An Introduction to Wishart Matrix Moments

Abstract: These lecture notes provide a comprehensive, self-contained introduction to the analysis of Wishart matrix moments. This study may act as an introduction to some particular aspects of random matrix theory, or as a self-contained exposition of Wishart matrix moments.

Cited by 13 publications

References 85 publications

A Taylor expansion of the square root matrix function

A Taylor expansion of the square root matrix function

A perturbation analysis of stochastic matrix Riccati diffusions

On one-dimensional Riccati diffusions

Contact Info

Product

Resources

About