Wishart Distribution Research Articles

Equilibria of large random Lotka-Volterra systems with vanishing species: a mathematical approach.

Ecosystems with a large number of species are often modelled as Lotka-Volterra dynamical systems built around a large interaction matrix with random part. Under some known conditions, a global equilibrium exists and is unique. In this article, we rigorously study its statistical properties in the large dimensional regime. Such an equilibrium vector is known to be the solution of a so-called Linear Complementarity Problem. We describe its statistical properties by designing an Approximate Message Passing (AMP) algorithm, a technique that has recently aroused an intense research effort in the fields of statistical physics, machine learning, or communication theory. Interaction matrices based on the Gaussian Orthogonal Ensemble, or following a Wishart distribution are considered. Beyond these models, the AMP approach developed in this article has the potential to describe the statistical properties of equilibria associated to more involved interaction matrix models.

Bayesian detection for distributed targets in compound Gaussian sea clutter with lognormal texture

This article investigates the Bayesian detection problem for the distributed targets in the compound Gaussian (CG) sea clutter. The CG sea clutter is formulated as a product of lognormal texture and speckle component with an inverse Wishart distribution covariance matrix (CM). A generalized likelihood ratio test (GLRT) based Bayesian detector, which can operate without training data, is proposed by integrating the speckle CM and estimating the texture using the maximum a posteriori (MAP) criterion. Additionally, three other Bayesian detectors are designed for distributed targets by exploiting the two-step GLRT, the complex-valued Rao, and Wald tests. We first derive the test statistics assuming known texture and speckle CM. Then, by incorporating the MAP-estimated texture components and speckle CM into the test statistics, we present three Bayesian detectors for distributed targets. Finally, simulation experiments validate the detection performance of the proposed Bayesian detectors using both simulated and real sea clutter data.

A Robust Trajectory Multi-Bernoulli Filter for Superpositional Sensors

This paper proposes a trajectory multi-Bernoulli filter applied to the superpositional sensor model for multi-target tracking in the presence of unknown measurement noise. This filter can provide a Multi-Bernoulli approximation of the posterior density on a set of alive trajectories at the current time step. We also provide a Gaussian mixture (GM) implementation of this filter, employing a mixture of Gaussian and inverse Wishart distributions to represent the combined state of measurement noise and target information. Subsequently, the variational Bayesian (VB) method is employed to approximate the posterior distribution, ensuring its form remains consistent with the prior distribution. This method is capable of directly generating trajectory estimates and can jointly estimate both multi-object tracking and measurement noise covariance. The performance of this algorithm is verified through simulation. Finally, a computationally more efficient L-scan approximation is provided. The simulation results indicate that the filter can achieve robust tracking performance, adapting to unknown measurement noise.

The Conditional Autoregressive F-Riesz Model for Realized Covariance Matrices

Abstract We introduce a new model for the dynamics of fat-tailed (realized) covariance-matrix-valued time-series using the F-Riesz distribution. The model allows for heterogeneous tail behavior across the coordinates of the covariance matrix via two vector-valued degrees of freedom parameters, thus generalizing the familiar Wishart and matrix-F distributions. We show that the filter implied by the new model is invertible and that a two-step targeted maximum likelihood estimator is consistent. Applying the new F-Riesz model to U.S. stocks, both tail heterogeneity and tail fatness turn out to be empirically relevant: they produce significant in-sample and out-of-sample likelihood increases, ex-post portfolio standard deviations that are in the global minimum variance model confidence set, and economic differences that are either in favor of the new model or competitive with a range of benchmark models.

Relations between generalised Wishart matrices, the Muttalib–Borodin model and matrix spherical functions

Relations between generalised Wishart matrices, the Muttalib–Borodin model and matrix spherical functions

Exploring Multivariate Statistics: Unveiling the Power of Eigenvalues in Wishart Distribution Analysis

This research examines how degrees of freedom and covariance matrix configurations affect Wishart distributed matrix eigenvalue distributions. We use the energy distance metric to compare eigenvalue distributions in Identity, Diagonal, and Structured covariance matrices. Through extensive simulations, we show that degrees of freedom and covariance matrix architectures greatly affect eigenvalue dispersion and energy distance distributions. Statistical models are more reliable with smaller, more stable distributions from higher degrees of freedom. We analyze the eigenvalue distributions of NextEra Energy (NEE), Enphase Energy (ENPH), First Solar (FSLR), SunPower (SPWR), and Brookfield Renewable Partners (BEP) stocks using this analytical methodology. Daily returns and covariance matrices were calculated using daily closing prices from January 1, 2018, to January 1, 2023. Our findings support simulation studies indicating larger degrees of freedom lead to more stable energy distance distributions.The findings of this study are useful in finance, genetics, and environmental studies, where stability and variability of the covariance structure are important. The requirement for higher-dimensional settings and real-world datasets to validate the theoretical framework are our research’s constraints. This research expands Wishart distribution knowledge and provides a solid analytical foundation for data analysis and model fitting in numerous scientific fields.

The Exact Density of the Eigenvalues of the Wishart and Matrix-Variate Gamma and Beta Random Variables

The determination of the distributions of the eigenvalues associated with matrix-variate gamma and beta random variables of either type proves to be a challenging problem. Several of the approaches utilized so far yield unwieldy representations that, for instance, are expressed in terms of multiple integrals, functions of skew symmetric matrices, ratios of determinants, solutions of differential equations, zonal polynomials, and products of incomplete gamma or beta functions. In the present paper, representations of the density functions of the smallest, largest and jth largest eigenvalues of matrix-variate gamma and each type of beta random variables are explicitly provided as finite sums when certain parameters are integers and, as explicit series, in the general situations. In each instance, both the real and complex cases are considered. The derivations initially involve an orthonormal or unitary transformation whereby the wedge products of the differential elements of the eigenvalues can be worked out from those of the original matrix-variate random variables. Some of these results also address the distribution of the eigenvalues of a central Wishart matrix as well as eigenvalue problems arising in connection with the analysis of variance procedure and certain tests of hypotheses in multivariate analysis. Additionally, three numerical examples are provided for illustration purposes.

Efficient computational method using random matrices describing critical thermodynamics

Our study emphasizes the efficacy of employing matrices resembling Wishart matrices, derived from magnetization time series data within specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. Through the application of appropriate statistical methods, we not only identify second-order transitions but also distinguish weaker first-order transitions by carefully analyzing the density of eigenvalues and their fluctuations. Additionally, we investigate the method’s sensitivity to stronger first-order transition points. Notably, we establish a robust correlation between the system’s actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further supported by correlation histograms of the time series data, revealing insightful patterns. Building upon our core results, we provide a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.

Passive underwater tracking with unknown measurement noise statistics using variational Bayesian approximation

This paper considers a bearings-only tracking problem with unknown measurement noise statistics. It is assumed that the measurement noise follows a Gaussian probability density function where the mean and the covariance of the noise are unknown. Here, an adaptive nonlinear filtering technique is proposed, where the joint distribution of the measurement noise mean and its covariance are considered to follow a normal inverse Wishart (NIW) distribution. Using the variational Bayesian (VB) approximation, joint distribution of the target state, the measurement noise mean and covariance is factorized as the product of their individual probability density function (pdf). Minimizing the Kullback-Leibler divergence (KLD) between the factorized and true joint pdfs, probability distributions of the noise mean, covariance and the target states are evaluated. The estimation of states with the proposed VB based method is compared with the maximum a posteriori (MAP) and the maximum likelihood estimation (MLE) based adaptive filtering. Deterministic sigma points are used to realize the filtering algorithms. The proposed adaptive filter with VB approximation is found to be more accurate compared to their corresponding MAP-MLE based counterparts.

Seebeck Coefficient of Ionic Conductors from Bayesian Regression Analysis.

We propose a novel approach to evaluating the ionic Seebeck coefficient in electrolytes from relatively short equilibrium molecular dynamics simulations, based on the Green-Kubo theory of linear response and Bayesian regression analysis. By exploiting the probability distribution of the off-diagonal elements of a Wishart matrix, we develop a consistent and unbiased estimator for the Seebeck coefficient, whose statistical uncertainty can be arbitrarily reduced in the long-time limit. We assess the efficacy of our method by benchmarking it against extensive equilibrium molecular dynamics simulations conducted on molten CsF using empirical force fields. We then employ this procedure to calculate the Seebeck coefficient of molten NaCl, KCl, and LiCl using neural network force fields trained on ab initio data over a range of pressure-temperature conditions.

Matrix‐variate risk measures under Wishart and gamma distributions

AbstractMatrix‐variate distribution theory has been instrumental across various disciplines for the past seven decades. However, a comprehensive examination of financial literature reveals a notable gap concerning the application of matrix‐variate extensions to Value‐at‐Risk (VaR). However, from a mathematical perspective, the core requirement for VaR lies in determining meaningful percentiles within the context of finance, necessitating the consideration of matrix c.d.f. This paper introduces the concept of “matrix‐variate VaR” for both Wishart and Gamma distributions. To achieve this, we leverage the theory of hypergeometric functions of matrix argument and integrate over positive definite matrices. Our proposed approach adeptly characterizes a company's exposure by into a comprehensive risk measure. This facilitates a readily computable estimation of the total incurred risk. Notably, this approach enables efficient computation of risk measures under Wishart, exponential, Erlang, gamma, and chi‐square distributions. The resulting risk measures are expressed in closed analytic forms, enhancing their practical utility for day‐to‐day risk management.

On Wilks' joint moment formulas for embedded principal minors of Wishart random matrices

AbstractIn 1934, the American statistician Samuel S. Wilks derived remarkable formulas for the joint moments of embedded principal minors of sample covariance matrices in multivariate Gaussian populations, and he used them to compute the moments of sample statistics in various applications related to multivariate linear regression. These important but little‐known moment results were extended in 1963 by the Australian statistician A. Graham Constantine using Bartlett's decomposition. In this note, a new proof of Wilks' results is derived using the concept of iterated Schur complements, thereby bypassing Bartlett's decomposition. Furthermore, Wilks' open problem of evaluating joint moments of disjoint principal minors of Wishart random matrices is related to the Gaussian product inequality conjecture.

An Extension of the Non-central Wishart Distribution with Integer Shape Vector

An Extension of the Non-central Wishart Distribution with Integer Shape Vector

Brain MR image segmentation for tumor detection based on Riesz probability distributions

ABSTRACT This research introduces a new approach using the Riesz mixture model for medical image segmentation, specifically for diagnosing and treating brain tumors. We developed a novel technique for pixel classification based on the Riesz distribution, which is generated using an extended Bartlett decomposition. Our work is pioneering, as there are no existing studies addressing this issue in the literature. We aim to demonstrate the effectiveness of this distribution for brain image segmentation. We used the Expectation-Maximization algorithm to estimate the mixture parameters. To validate our segmentation algorithm, we conducted a comparative study with a recent method based on the Wishart distribution using Matlab software. Experiments with the Riesz mixture model showed that our method produces more intuitive results with a recognition rate of 94.52%. These results confirm the reliability of our method in detecting tumors using both synthetic and real brain images.

Loss-based prior for the degrees of freedom of the Wishart distribution

Motivated by the proliferation of extensive macroeconomic and health datasets necessitating accurate forecasts, a novel approach is introduced to address Vector Autoregressive (VAR) models. This approach employs the global-local shrinkage-Wishart prior. Unlike conventional VAR models, where degrees of freedom are predetermined to be equivalent to the size of the variable plus one or equal to zero, the proposed method integrates a hyperprior for the degrees of freedom to account for the uncertainty in the parameter values. Specifically, a loss-based prior is derived to leverage information regarding the data-inherent degrees of freedom. The efficacy of the proposed prior is demonstrated in a multivariate setting both for forecasting macroeconomic data, and Dengue infection data.

Bayesian detection for distributed target with limited training data

The issue of subspace-based distributed target detection with limited training data is addressed in this study. We use the Bayesian method to tackle the issue by assuming that the covariance matrix follows an inverse Wishart distribution. According to the generalized likelihood ratio test, Rao test, and Wald test, three Bayesian detectors are designed. Real data and simulations both attest to the usefulness of the proposed detectors.

Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations.

Underlying Topography Estimation over Forest Using Maximum a Posteriori Inversion with Spaceborne Polarimetric SAR Interferometry

This paper presents a method for extracting the digital elevation model (DEM) of forested areas from polarimetric interferometric synthetic aperture radar (PolInSAR) data. The method models the ground phase as a Von Mises distribution, with a mean of the topographic phase computed from an external DEM. By combining the prior distribution of the ground phase with the complex Wishart distribution of the observation covariance matrix, we derive the maximum a posterior (MAP) inversion method based on the RVoG model and analyze its Cramer–Rao Lower Bound (CRLB). Furthermore, considering the characteristics of the objective function, this paper introduces a Four-Step Optimization (FSO) method based on gradient optimization, which solves the inefficiency problem caused by exhaustive search in solving ground phase using the MAP method. The method is validated using spaceborne L-band repeat-pass SAOCOM data from a test forest area. The test results for FSO indicate that it is approximately 5.6 times faster than traditional methods without compromising accuracy. Simultaneously, the experimental results demonstrate that the method effectively solves the problem of elevation jumps in DEM inversion when modeling the ground phase with the Gaussian distribution. ICESAT-2 data are used to evaluate the accuracy of the inverted DEM, revealing that our method improves the root mean square error (RMSE) by about 23.6% compared to the traditional methods.

Parameter estimation and sensitivities of a Bayesian source localization method in an urban environment

Acoustic signals propagating in urban environments are influenced by rough-surface scattering, multipath reflections, and diffraction. The performance of conventional source localization algorithms often suffers when these effects are present. Bayesian approaches, however, are particularly well suited to incorporating physics-based statistical models for the signal propagation. Previously, we found that the complex Wishart distribution, which describes fully saturated scattered signals across a network of receivers, can be readily employed in a Bayesian framework. In this presentation, a new source localization algorithm based on the Wishart distribution signal model is described and tested on real acoustic data. The experimental data were collected within a mock urban environment as part of a NATO urban acoustics-seismics experiment in Walenstadt, Switzerland. Four acoustic arrays recorded signatures from gunshots and ground vehicles. The present study uses the measured signals to investigate the sensitivity and relative importance of the model parameters, including source and noise amplitudes, frequency band, prior signal sampling, and sensor-sensor correlation behavior. Results are presented in the form of source location probability maps and localization error metrics. Interpretations of the study, including strengths and weaknesses of the new method, are discussed.

On the expectations of equivariant matrix‐valued functions of Wishart and inverse Wishart matrices

AbstractMany matrix‐valued functions of an Wishart matrix , , say, are homogeneous of degree in , and are equivariant under the conjugate action of the orthogonal group , that is, , . It is easy to see that the expectation of such a function is itself homogeneous of degree in , the covariance matrix, and are also equivariant under the action of on . The space of such homogeneous, equivariant, matrix‐valued functions is spanned by elements of the type , where and, for each , varies over the partitions of , and denotes the power‐sum symmetric function indexed by . In the analogous case where is replaced by , these elements are replaced by . In this paper, we derive recurrence relations and analytical expressions for the expectations of such functions. Our results provide highly efficient methods for the computation of all such moments.