On a Class of Covariance Operators

1998 ◽  
Vol 5 (5) ◽  
pp. 415-424
Author(s):  
T. Chantladze ◽  
N. Kandelaki
Keyword(s):  
Complete Classification ◽  
Covariance Operator ◽  
Random Vectors ◽  
Covariance Operators

Abstract This paper is the continuation of [Vakhania and Kandelaki, Teoriya Veroyatnost. i Primenen 41: 31–52, 1996] in which complex symmetries of distributions and their covariance operators are investigated. Here we also study the most general quaternion symmetries of random vectors. Complete classification theorems on these symmetries are proved in terms of covariance operator spectra.

On Bernstein Type and Maximal Inequalities for Dependent Banach-Valued Random Vectors and Applications

2018 ◽  
Author(s):  
Noureddine Rhomari
Keyword(s):  
Recursive Estimation ◽  
Covariance Operator ◽  
Partial Sums ◽  
Strong Mixing ◽  
Random Vectors ◽  
Bernstein Type ◽  
Strong Law ◽  
Maximal Inequalities ◽  
Mixing Coefficients ◽  
Measure Of Dependence

This article discusses some results on Bernstein type and maximal inequalities for partial sums of dependent random vectors taking their values in separable Hilbert or Banach spaces of finite or infinite dimension. Two types of measure of dependence are considered: strong mixing coefficients (α-mixing) and absolutely regular mixing coefficients (β-mixing). These inequalities, which are similar to those in the dependent real case, are used to derive the strong law of large numbers (SLLN) and the bounded law of the iterated logarithm (LIL) for absolutely regular Hilbert- or Banach-valued processes under minimal mixing conditions. The article first introduces the relevant notation and definitions before presenting the maximal inequalities in the strong mixing case, followed by the absolutely regular mixing case. It concludes with some applications to the SLLN, the bounded LIL for Hilbertian or Banachian absolutely regular processes, the recursive estimation of probability density, and the covariance operator estimations.

Double-scale diffusive wave model dedicated to spatial river observation and associated covariance kernel for variational data assimilation 

2021 ◽  
Author(s):  
Thibault Malou ◽  
Jérome Monnier
Keyword(s):  
Data Assimilation ◽  
Control Variable ◽  
Variational Data Assimilation ◽  
Covariance Operator ◽  
Background Error ◽  
Green Kernel ◽  
Covariance Kernel ◽  
Covariance Operators ◽  
Physically Based ◽  
Double Scale

<p>The spatial altimetry provides an important amount of water surface height data from multi-missions satellites (especially Jason-3, Sentinel-3A/B and the forthcoming NASA-CNES SWOT mission). To exploit at best the potential of spatial altimetry, the present study proposes on the derivation of a model adapted to spatial observations scale; a diffusive-wave type model but adapted to a double scale [1].</p><p>Moreover, Green-like kernel can be employed to derived covariance operators, therefore they may provide an approximation of the covariance kernel of the background error in Variational Data Assimilation processes. Following the derivation of the aforementioned original flow model, we present the derivation of a Green kernel which provides an approximation of the covariance kernel of the background error for the bathymetry (i.e. the control variable) [2].</p><p>This approximation of the covariance kernel is used to infer the bathymetry in the classical Saint-Venant’s (Shallow-Water) equations with better accuracy and faster convergence than if not introducing an adequate covariance operator [3].</p><p>Moreover, this Green kernel helps to analyze the sensitivity of the double-scale diffusive waves (or even the Saint-Venant’s equations) with respect to the bathymetry.</p><p>Numerical results are analyzed on real like datasets (derived from measurements of the Rio Negro, Amazonia basin).</p><p>The double-scale diffusive wave provide more accurate results than the classical version. Next, in terms of inversions, the derived physically-based covariance operators enable to improve the inferences, compared to the usual exponential one.</p><p>[1] T. Malou, J. Monnier "Double-scale diffusive wave equations dedicated to spatial river observations". In prep.</p><p>[2] T. Malou, J. Monnier "Physically-based covariance kernel for variational data assimilation in spatial hydrology". In prep.</p><p>[3] K. Larnier, J. Monnier, P.-A. Garambois, J. Verley. "River discharge and bathymetry estimations from SWOT altimetry measurements". Inv. Pb. Sc. Eng (2020).</p>

Orthogonal Random Vectors and the Hurwitz-Radon-Eckmann Theorem

1994 ◽  
Vol 1 (1) ◽  
pp. 99-113
Author(s):  
N. Vakhania
Keyword(s):  
Hilbert Space ◽  
Nauk Sssr ◽  
Adjoint Operator ◽  
Covariance Operator ◽  
Random Vectors ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
And Topology ◽  
Dimensional Hilbert Space ◽  
Akad Nauk Sssr

Abstract In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operators Ui in with the property UiUj = –UjUi (i ≠ j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operators Ui must commute with a given arbitrary self-adjoint operator and H can be infinite-dimensional. Our second main result deals with the conditions for almost sure orthogonality of two random vectors taking values in a finite or infinite-dimensional Hilbert space H. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors in H with the same covariance operator and each pair having a linear support in H ⊕ H. The paper is based on the results obtained jointly with N.P.Kandelaki (see [Vakhania and Kandelaki, Dokl. Akad. Nauk SSSR 294: 528-531, 1987, Dokl. Akad. Nauk SSSR 296: 265-266, 1988, Trudy Inst. Vychisl. Mat. Akad. Nauk Gruz. SSR 28: 11-37, 1988]).

The Multigrid Beta Function Approach for Modeling of Background Error Covariance in the Real-Time Mesoscale Analysis (RTMA)

2022 ◽  
Keyword(s):  
Spatial Scales ◽  
Beta Function ◽  
Parallel Computer ◽  
Covariance Operator ◽  
General Covariance ◽  
Background Error ◽  
Recursive Filters ◽  
Error Covariance ◽  
Covariance Operators ◽  
Helmholtz Operators

Abstract We describe a method for the efficient generation of the covariance operators of a variational data assimilation scheme which is suited to implementation on a massively parallel computer. The elementary components of this scheme are what we call ‘beta filters’, since they are based on the same spatial profiles possessed by the symmetric beta distributions of probability theory. These approximately Gaussian (bell-shaped) polynomials blend smoothly to zero at the ends of finite intervals, which makes them better suited to parallelization than the present quasi-Gaussian ‘recursive filters’ used in operations at NCEP. These basic elements are further combined at a hierarchy of different spatial scales into an overall multigrid structure formulated to preserve the necessary self-adjoint attribute possessed by any valid covariance operator. This paper describes the underlying idea of the beta filter and discusses how generalized Helmholtz operators can be enlisted to weight the elementary contributions additively in such a way that the covariance operators may exhibit realistic negative sidelobes, which are not easily obtained through the recursive filter paradigm. The main focus of the paper is on the basic logistics of the multigrid structure by which more general covariance forms are synthesized from the basic quasi-Gaussian elements. We describe several ideas on how best to organize computation, which led us to a generalization of this structure which made it practical so that it can efficiently perform with any rectangular arrangement of processing elements. Some simple idealized examples of the applications of these ideas are given.

scholarly journals Bayesian posterior contraction rates for linear severely ill-posed inverse problems

2014 ◽  
Vol 22 (3) ◽  
Author(s):  
Sergios Agapiou ◽  
Andrew M. Stuart ◽  
Yuan-Xiang Zhang
Keyword(s):  
Inverse Problems ◽  
Singular Values ◽  
Covariance Operator ◽  
Posterior Consistency ◽  
Constant Multiplier ◽  
Covariance Operators ◽  
Noise Covariance ◽  
Noise Limit ◽  
Ill Posed ◽  
Observational Noise

Abstract.We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals On Randomized Trace Estimates for Indefinite Matrices with an Application to Determinants

2021 ◽  
Author(s):  
Alice Cortinovis ◽  
Daniel Kressner
Keyword(s):  
Large Scale ◽  
Gaussian Process Regression ◽  
Likelihood Estimation ◽  
Random Vectors ◽  
Symmetric Positive Definite ◽  
Matrix Logarithm ◽  
The Matrix ◽  
Trace Estimation ◽  
Scale Matrix ◽  
Existing Result

AbstractRandomized trace estimation is a popular and well-studied technique that approximates the trace of a large-scale matrix B by computing the average of $$x^T Bx$$ x T B x for many samples of a random vector X. Often, B is symmetric positive definite (SPD) but a number of applications give rise to indefinite B. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite B with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite B, reducing the number of required samples by a factor n or even more, where n is the size of B. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.

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