Wishart Distribution Research Articles

Bayesian hierarchical modeling on covariance valued data

Analysis of structural and functional connectivity (FC) of human brains is of pivotal importance for diagnosis of cognitive ability. The Human Connectome Project (HCP) provides an excellent source of neural data across different regions of interest (ROIs) of the living human brain. Individual specific data were available in the form of time varying covariance matrices representing the brain activity as the subjects perform a specific task. As a preliminary objective of studying the heterogeneity of brain connectomics across the population, we develop a probabilistic model for a sample of covariance matrices using a scaled Wishart distribution. We stress here that our data units are available in the form of covariance matrices, and we use the Wishart distribution to create our likelihood function rather than its more common usage as a prior on covariance matrices. Based on empirical explorations suggesting the data matrices to have a low effective rank, we further model the center of the Wishart distribution using an orthogonal factor model type decomposition. We encourage shrinkage toward a low rank structure through a novel shrinkage prior and discuss strategies to sample from the posterior distribution using a combination of Gibbs and slice sampling. The efficacy of the approach is explored in various simulation settings and exemplified on several case studies including our motivating HCP data. We extend our modeling framework to a dynamic setting to detect change points.

Dirichlet process mixture models using matrix‐generalized half‐t distribution

Dirichlet process mixture models (or mixture of Dirichlet process [MDP]) are Bayesian non‐parametric mixture models that can solve the problem of determining the number of components in mixture models by assuming infinitely many components. We propose a new approach for MDP by using a matrix‐generalized half‐ distribution as a non‐informative prior for the covariance in the base distribution. This new approach aims to improve the performance of MDP in cases where conjugacy constraints cause underperformance of current priors, such as Wishart distribution or inverse‐Wishart. The effectiveness of this approach is demonstrated through simulations and by comparing its performance on the Old Faithful geyser dataset with existing MDP specifications.

Tuning-Free Bayesian Estimation Algorithms for Faulty Sensor Signals in State-Space

Sensors provide insights into the industrial processes, while misleading sensor outputs may result in inappropriate decisions or even catastrophic accidents. In this article, the Bayesian estimation algorithms are developed to estimate unforeseen signals in sensor outputs without tuning. The optimal Bayesian estimation method is first derived by incorporating a Gaussian distribution specifying potential unmodeled dynamics into the measurement equation. Since its performance depends on tuning parameters, an iterative Bayesian estimation algorithm is developed using the variational inference technique. Specifically, an inverse Wishart distribution is introduced to describe the predicted covariance of abnormal signals. We then estimate it together with the other independent Gaussian distributions to conditionally approximate the joint posterior distribution, by which the effects of tuning parameters can be replaced adaptively. Testing the proposed algorithms through a simulated electromechanical brake model and a real experimental system shows that the proposed algorithm can satisfactorily estimate additive sensor faults online and services as a sensor monitor that simultaneously provides the locations and magnitudes of faulty signals without tuning.

An Adaptive and Scalable Multi-Object Tracker Based on the Non-Homogeneous Poisson Process

This paper proposes a new adaptive framework for tracking multiple objects in the presence of data association uncertainty and heavy clutter, either with or without knowledge of the measurement rates and/or target shapes. Built upon an online Gibbs sequential Markov chain Monte Carlo sampling scheme, the adaptive tracker is Bayesian optimal and robust, requiring no additional approximations or measurement partition steps. With a non-homogeneous Poisson process measurement model, our tracker can tackle the data association task with linear computational complexity. Meanwhile, we study generalised inverse Gaussian and inverse Wishart distributions for modelling Poisson rates and object shapes, respectively; these prior models ensure closed-form full conditionals in our online Gibbs sampling steps, under which object states and shapes can be jointly estimated with associations and Poisson rates in a parallel fashion. Furthermore, a fast Rao-Blackwellisation scheme for linear Gaussian dynamics is designed and demonstrated to significantly improve both tracking efficiency and accuracy. We validate the efficacy of our method on real and simulated data.

Variational Bayesian-Based Moving Horizon Estimation of Toolface for Rotary Steerable Drilling Tool Systems

This article is concerned with the estimation problem of Toolface for dynamic point-the-bit rotary steerable drilling tool systems. First, considering the unknown frequency and amplitude of vibration during the drilling process, the Toolface system is modeled as a time-varying stochastic system with unknown and time-varying noise covariance matrices. Under the assumption that the noises and their covariance matrices, respectively, obey the Gaussian distribution and the inverse Wishart distribution, the variational Bayesian-based moving horizon estimation algorithm is proposed. Then, the state and the noise covariance matrices are inferred by iteratively updating their approximate posterior probability distributions. Finally, simulations and experiments are provided to demonstrate the effectiveness and superiority of the developed estimation scheme.

Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices

This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei [32] and Liu et al. [15] are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.

Infinitely divisible matrix gamma distribution: Asymptotic behaviour and parameters estimation

In this research paper, we investigate an infinitely divisible p×p matrix gamma distribution AΓp(η,Σ), with parameters η>(p−1)/2 and Σ, concentrated on the cone of symmetric positive definite matrices. The parameter Σ is supposed to be a symmetric positive definite p×p matrix. We also display some of its fundamental properties. Additionally, we identify the link between this multivariate gamma distribution and the Wishart one, which leads us to prove that AΓp(η,Σ) distribution is asymptotically a stochastic linear combination of Wishart matrices. Moreover, we provide an explicit expression of the parameters estimators using the method of moments. Eventually, we exhibit a new simulation algorithm, grounded on the obtained results, in order to illustrate the performance of these estimators.

The Eigenvectors of Single-Spiked Complex Wishart Matrices: Finite and Asymptotic Analyses

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {W}\in \mathbb {C}^{n\times n}$ </tex-math></inline-formula> be a single-spiked Wishart matrix in the class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {W}\sim \mathcal {CW}_{n}(m,\mathrm {I}_{n}+ \theta \mathrm {v}\mathrm {v}^{\dagger}) $ </tex-math></inline-formula> with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq n$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {I}}_{n}$ </tex-math></inline-formula> is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> identity matrix, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathrm {v}\in \mathbb {C}^{n\times 1}$ </tex-math></inline-formula> is an arbitrary vector with unit Euclidean norm, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\theta \geq 0$ </tex-math></inline-formula> is a non-random parameter, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\cdot)^{\dagger} $ </tex-math></inline-formula> represents the conjugate-transpose operator. Let u1 and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathrm {u}}_{n}$ </tex-math></inline-formula> denote the eigenvectors corresponding to the smallest and the largest eigenvalues of W, respectively. This paper investigates the probability density function (p.d.f.) of the random quantity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{\ell }^{(n)}=\left |{\mathrm {v}^{\dagger} \mathrm {u}_\ell }\right |^{2}\in (0,1)$ </tex-math></inline-formula> for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell =1,n$ </tex-math></inline-formula> . In particular, we derive a finite dimensional closed-form p.d.f. for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{1}^{(n)}$ </tex-math></inline-formula> which is amenable to asymptotic analysis as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n$ </tex-math></inline-formula> diverges with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-n$ </tex-math></inline-formula> fixed. It turns out that, in this asymptotic regime, the scaled random variable <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$nZ_{1}^{(n)}$ </tex-math></inline-formula> converges in distribution to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi ^{2}_{2}/2(1+\theta)$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\chi _{2}^{2}$ </tex-math></inline-formula> denotes a chi-squared random variable with two degrees of freedom. This reveals that u1 can be used to infer information about the spike. On the other hand, the finite dimensional p.d.f. of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$Z_{n}^{(n)}$ </tex-math></inline-formula> is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n-2)$ </tex-math></inline-formula> . Although a simple solution to this double integral seems intractable, for special configurations of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n=2,3$ </tex-math></inline-formula> , and 4, we obtain closed-form expressions.

Properties of eigenvalues and eigenvectors of large-dimensional sample correlation matrices

This paper is to study the properties of eigenvalues and eigenvectors of high-dimensional sample correlation matrices. We first improve the result of Jiang (Sankhyā 66 (2004) 35–48), Xiao and Zhou (J. Theoret. Probab. 23 (2010) 1–20) and the Theorem 1 of El Karoui (Ann. Appl. Probab. 19 (2009) 2362–2405), both concerning the limiting spectral distribution and the extreme eigenvalues of sample correlation matrices, by allowing a more general fourth moment condition. Then, we establish a central limit theorem (CLT) for the linear statistics of the eigenvectors of large sample correlation matrices. We discover that the difference between the functional CLT of the sample covariance matrix and the sample correlation matrix is fundamentally influenced by the direction of a nonrandom projection vector. In the special case where the square root of the correlation matrix is identity, the difference will be determined by the sum of the fourth powers of the entries of the projection vector. These results also indicate that the eigenmatrix of sample correlation matrices is not asymptotically Haar if the underlying distribution is Gaussian. In other words, the normalization based on the sample variances affects the asymptotic properties of the eigenmatrix of the Wishart matrix. Furthermore, we establish a theorem concerning CLT for the linear statistics of the eigenvectors of large sample covariance matrices. This theorem improves the main results in Bai, Miao and Pan (Ann. Probab. 35 (2007) 1532–1572), which requires the assumption that the fourth moment of the underlying variable matches the one of Gaussian distribution, as well as Theorem 1.3 in Pan and Zhou (Ann. Appl. Probab. 18 (2008) 1232–1270), which relaxed the Gaussian like fourth moment requirement but assumes the maximum entries of the projection vector converge to 0 uniformly. We illustrate the usefulness of the theoretical results through an application in communications.

Fast Line Segment Detection and Large Scene Airport Detection for PolSAR

In this paper, we propose a fast Line Segment Detection algorithm for Polarimetric synthetic aperture radar (PolSAR) data (PLSD). We introduce the Constant False Alarm Rate (CFAR) edge detector to obtain the gradient map of the PolSAR image, which tests the equality of the covariance matrix using the test statistic in the complex Wishart distribution. A new filter configuration is applied here to save time. Then, the Statistical Region Merging (SRM) framework is utilized for the generation of line-support regions. As one of our main contributions, we propose a new Statistical Region Merging algorithm based on gradient Strength and Direction (SRMSD). It determines the merging predicate with consideration of both gradient strength and gradient direction. For the merging order, we set it by bucket sort based on the gradient strength. Furthermore, the pixels are restricted to belong to a unique region, making the algorithm linear in time cost. Finally, based on Markov chains and a contrario approach, the false alarm control of line segments is implemented. Moreover, a large scene airport detection method is designed based on the proposed line segment detection algorithm and scattering characteristics. The effectiveness and applicability of the two methods are demonstrated with PolSAR data provided by UAVSAR.

Random matrices theory elucidates the nonequilibrium critical phenomena

The earlier times of the evolution of a magnetic system contain more information than we can imagine. Capturing correlation matrices built from different time evolutions of a simple testbed spin system, as the spin-1/2 and spin-1 Ising models, we analyzed the density of eigenvalues for different temperatures of the so called Wishart matrices. We observe a transition in the shape of the distribution that presents a gap of eigenvalues for temperatures lower than the critical temperature, or in its roundness, with a continuous migration to the Marchenko–Pastur law in the paramagnetic phase. We consider the analysis a promising method to be applied in other spin systems, with or without defined Hamiltonian, to characterize phase transitions. Our approach differs from the alternatives in literature since it uses the concept of magnetization matrix, not the spatial matrix of single spins.

Shrinkage estimators of large covariance matrices with Toeplitz targets in array signal processing

The problem of estimating a large covariance matrix arises in various statistical applications. This paper develops new covariance matrix estimators based on shrinkage regularization. Individually, we consider two kinds of Toeplitz-structured target matrices as the data come from the complex Gaussian distribution. We derive the optimal tuning parameter under the mean squared error criterion in closed form by discovering the mathematical properties of the two target matrices. We get some vital moment properties of the complex Wishart distribution, then simplify the optimal tuning parameter. By unbiasedly estimating the unknown scalar quantities involved in the optimal tuning parameter, we propose two shrinkage estimators available in the large-dimensional setting. For verifying the performance of the proposed covariance matrix estimators, we provide some numerical simulations and applications to array signal processing compared to some existing estimators.

On Data Augmentation for Models Involving Reciprocal Gamma Functions

In this article, we introduce a new and efficient data augmentation approach to the posterior inference of the models with shape parameters when the reciprocal gamma function appears in full conditional densities. Our approach is to approximate full conditional densities of shape parameters by using Gauss’s multiplication formula and Stirling’s formula for the gamma function, where the approximation error can be made arbitrarily small. We use the techniques to construct efficient Gibbs and Metropolis–Hastings algorithms for a variety of models that involve the gamma distribution, Student’s t-distribution, the Dirichlet distribution, the negative binomial distribution, and the Wishart distribution. The proposed sampling method is numerically demonstrated through simulation studies. Supplementary materials for this article are available online.

Realized BEKK-CAW Models

Abstract Estimating time-varying conditional covariance matrices of financial returns play important role in portfolio analysis, risk management, and financial econometrics research. The availability of high-frequency financial data can provide an additional data source for dynamic covariance modeling. In this paper, we propose to use the information of asset return vector and realized covariance measures simultaneously to develop a new conditional covariance matrix model. We derive the stationary condition of the new model. We use the normal and Wishart distributions to construct the quasi-log-likelihood function. We also consider the variance targeting (VT) method, which plugs in the weighted average of the sample covariance matrix of returns and the sample mean of realized covariance measure for the unconditional covariance matrix, in order to maximize the quasi-log-likelihood function. We show the consistency and asymptotic normality of the quasi-maximum likelihood (QML) and VT estimators. We investigate the finite sample property of these estimators via Monte Carlo experiments. The empirical example for the bivariate data of the Nikkei 225 index and its futures indicates that the first-step VT estimation could have non-negligible effects on the standard errors of the second-step VT estimates.

Spectral properties of the generalized diluted Wishart ensemble

The celebrated Marčenko–Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as , where X and Y are diluted N × P rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution , with the probability density ϱ(x, y) controlling the correlation between the matrices entries of X and Y . Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph d, the rectangularity of the matrices α = N/P, and the degree of correlation of the matrix entries via the density ϱ(x, y). Finally, we compare our findings to numerical diagonalisation showing excellent agreement.

PolSAR Models with Multimodal Intensities

Polarimetric synthetic aperture radar (PolSAR) systems are an important remote sensing tool. Such systems can provide high spacial resolution images, but they are contaminated by an interference pattern called multidimensional speckle. This fact requires that PolSAR images receive specialised treatment; particularly, tailored models which are close to PolSAR physical formation are sought. In this paper, we propose two new matrix models which arise from applying the stochastic summation approach to PolSAR, called compound truncated Poisson complex Wishart (CTPCW) and compound geometric complex Wishart (CGCW) distributions. These models offer the unique ability to express multimodal data. Some of their mathematical properties are derived and discussed—characteristic function and Mellin-kind log-cumulants (MLCs). Moreover, maximum likelihood (ML) estimation procedures via expectation maximisation algorithm for CTPCW and CGCW parameters are furnished as well as MLC-based goodness-of-fit graphical tools. Monte Carlo experiment results indicate ML estimates perform at what is asymptotically expected (small bias and mean square error) even for small sample sizes. Finally, our proposals are employed to describe actual PolSAR images, presenting evidence that they can outperform other well-known distributions, such as WmC, Gm0, and Km.

Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}\in \mathbb {C}^{n\times m}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq n$ </tex-math></inline-formula> ) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single-spiked</i> covariance matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {I}_{n}$ </tex-math></inline-formula> is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> identity matrix, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {u}\in \mathbb {C}^{n\times 1}$ </tex-math></inline-formula> is an arbitrary vector with unit Euclidean norm, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta \geq 0$ </tex-math></inline-formula> is a non-random parameter, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\cdot)^{*}$ </tex-math></inline-formula> represents the conjugate-transpose. This paper investigates the distribution of the random quantity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})=\sum _{k=1}^{n} \lambda _{k}/\lambda _{1}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0\le \lambda _{1}\le \lambda _{2}\le \ldots \leq \lambda _{n} &lt; \infty $ </tex-math></inline-formula> are the ordered eigenvalues of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}\mathbf {X}^{*}$ </tex-math></inline-formula> (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">scaled condition number</i> or the Demmel condition number (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}(\mathbf {X})$ </tex-math></inline-formula> ) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{-2}(\mathbf {X})$ </tex-math></inline-formula> ). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n\to \infty $ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-n$ </tex-math></inline-formula> is fixed and when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> scales on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> scales on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n^{3}$ </tex-math></inline-formula> . In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n\to \infty $ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n/m\to c\in (0,1)$ </tex-math></inline-formula> , properly centered <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> fluctuates on the scale <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m^{\frac {1}{3}}$ </tex-math></inline-formula> .

Enriched standard conjugate priors and the right invariant prior for Wishart distributions

The prediction of the variance–covariance matrix of the multivariate normal distribution is important in the multivariate analysis. We investigated Bayesian predictive distributions for Wishart distributions under the Kullback–Leibler divergence. The conditional reducibility of the family of Wishart distributions enables us to decompose the risk of a Bayesian predictive distribution. We considered a recently introduced class of prior distributions, which is called the family of enriched standard conjugate prior distributions, and compared the Bayesian predictive distributions based on these prior distributions. Furthermore, we studied the performance of the Bayesian predictive distribution based on the reference prior distribution in the family and showed that there exists a prior distribution in the family that dominates the reference prior distribution. Our study provides new insight into the multivariate analysis when there exists an ordered inferential importance for the independent variables.

Adaptive maximum correntropy based robust CKF with variational Bayesian for covariance estimation

To address the interference of outliers on the estimation of state and measurement noise covariance matrix, an adaptive maximum correntropy cubature Kalman filter with variational Bayesian approximation over a sliding window is proposed. The multiple kernel size is adjusted for different noise within a reasonable range based on the squared Mahalanobis distance of innovation, which overcomes the excessive convergence problem in the adjustment process. The correntropy matrix is established using the adaptive multiple kernel size to achieve measurement-specific outliers processing. Then the measurement noise covariance matrix is updated as inverse Wishart distribution exploiting the posterior smoothing-based variational Bayesian approximations with correntropy matrix, suppressing the disturbance of measurement outliers to the modification of the measurement noise covariance matrix. Finally, the target tracking simulation and cooperative positioning experiment demonstrate that the proposed method can effectively achieve the robust state estimation with accurate modification of MNCM in the presence of outliers.

Improving cosmological covariance matrices with machine learning

Cosmological covariance matrices are fundamental for parameter inference, since they are responsible for propagating uncertainties from the data down to the model parameters. However, when data vectors are large, in order to estimate accurate and precise covariance matrices we need huge numbers of observations, or rather costly simulations - neither of which may be viable. In this work we propose a machine learning approach to alleviate this problem in the context of the covariance matrices used in the study of large-scale structure. With only a small amount of data (matrices built with samples of 50-200 halo power spectra) we are able to provide significantly improved covariance matrices, which are almost indistinguishable from the ones built from much larger samples (thousands of spectra). In order to perform this task we trained convolutional neural networks to denoise the covariance matrices, using in the training process a data set made up entirely of spectra extracted from simple, inexpensive halo simulations (mocks). We then show that the method not only removes the noise in the covariance matrices of the cheap simulation, but it is also able to successfully denoise the covariance matrices of halo power spectra from N-body simulations. We compare the denoised matrices with the noisy sample covariance matrices using several metrics, and in all of them the denoised matrices score significantly better, without any signs of spurious artifacts. With the help of the Wishart distribution we show that the end product of the denoiser can be compared with an effective sample augmentation in the input matrices. Finally, we show that, by using the denoised covariance matrices, the cosmological parameters can be recovered with nearly the same accuracy as when using covariance matrices built with a sample of 30,000 spectra in the case of the cheap simulations, and with 15,000 spectra in the case of the N-body simulations. Of particular interest is the bias in the Hubble parameter H 0, which was significantly reduced after applying the denoiser.