Wishart Distribution Research Articles

Maximum likelihood estimation of a TVP-VAR

This paper proposes the maximum likelihood estimation of a vector autoregression with drifting coefficients and multivariate stochastic volatility. The coefficients are assumed to follow heteroscedastic random walks and the volatility of the system is modeled as a Wishart process, increasing the flexibility in describing the behavior of stochastic covariances. Exploiting the conjugacy between Normal, Wishart and multivariate beta distributions, filtering formulas for tracking the latent states and expression for the likelihood function can be obtained in closed form.

Large deviations for the largest eigenvalue of Rademacher matrices

In this article, we consider random Wigner matrices, that is, symmetric matrices such that the subdiagonal entries of $X_{n}$ are independent, centered and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This is satisfied by the Rademacher law and the uniform law on $[-\sqrt{3},\sqrt{3}]$. We extend our result to complex entries Wigner matrices and Wishart matrices.

Interference Coordination for 3-D Beamforming-Based HetNet Exploiting Statistical Channel-State Information

In this paper, we investigate the inter-tier interference coordination for a heterogeneous network (HetNet), where both macro and small cells work in the frequency-division duplexing mode and share the same spectrum. A 2-D large-scale antenna array is deployed at a macro-base station (BS), while conventional multiple antenna array is deployed at each small cell. First, we derive an approximation of macro-users’ signal-to-interference-plus-noise ratio (SINR), under the assumption of only statistical channel-state information (CSI) at macro-BS, and a 3-D beamforming transmission scheme is applied for macro-users. A lower-bound of small-cell users’ expected SINR is also derived using the property of complex Wishart matrix. Based on these, simple metrics are derived to measure the inter-tier interference. Then, new interference coordination algorithms are proposed to achieve a good tradeoff for the performance and traffic between macro-cells and small cells. The proposed algorithms require only a few additional statistical CSI. Moreover, we derive tractable ergodic rate approximation of both the macro-cell and small-cell users which are shown to match well with the Monte Carlo results.

Weighted distributions of eigenvalues

In this article, the weighted version of a probability density function is considered as a mapping of the original distribution. Generally, the properties of the distribution of a random matrix and the distributions of its eigenvalues are closely related. Therefore, the weighted versions of the distributions of the eigenvalues of the Wishart distribution are introduced and their properties are discussed. We propose the concept of rotation invariance for the weighted distributions of the eigenvalues of the Wishart and non-central Wishart distributions. We also introduce here, the concept of a “mirror”, meaning, looking at the distribution of a random matrix through the distribution of its eigenvalues. Some graphical representations are given, to visualize the weighted distributions of the eigenvalues for specific cases.

Numerical and experimental assessment of random matrix theory to quantify uncertainty in aerospace structures

Abstract The increased complexities of modernized structures have catalyzed additional research over the past few decades to accurately quantify the variabilities within static designs, which can be categorized into parametric, non-parametric and experimental uncertainties. The majority of works have dealt with parametric uncertainty alone and utilize algorithms that require priori knowledge of the input uncertainty. This is problematic since there is a large number of uncertain variables that typically percolate physical engineering problems which are difficult to characterize using precise probability distributions. A unified method to encompass these stochastic properties in a computationally efficient manner is needed and one such approach is using random matrix theory (RMT), which has integrated and validated its importance through various fields, however largely excluding the aerospace sector. This study utilizes RMT and the Wishart distribution to characterize modal uncertainties that arise due to parametric, non-parametric and experimental variabilities. Numerical studies are undertaken with finite element models of a rectangular wing and an aircraft wing box to validate RMT and the Marchenko-Pastur (MP) density function. An experimental study is also conducted on six nominally designed aircraft T-tails, where RMT is used to provide a quantitative bound on the unknown uncertainties that exist within the physical system.

Finiteness of inverse moments of (m,n,β)-Laguerre matrices

Wishart matrices are one of the fundamental matrix models in multivariate statistics. The classical Wishart ensemble has been generalized to [Formula: see text]-Laguerre ensemble for the values of [Formula: see text]. Many properties such as eigenvalue densities, moments and inverse moments of [Formula: see text]-Laguerre matrices are important in various fields of mathematics and physics. The moments and inverse moments of Wishart matrices have been studied rigorously and the explicit formulas were given in [P. Graczyk, G. Letac and H. Massam, Ann. Statist. 31 (2003) 287–309; S. Matsumoto, General moments of the inverse real Wishart distribution and orthogonal Weingarten functions, J. Theoret. Probab. 25 (2012) 798–822]. We give a necessary and sufficient condition for the existence of finite inverse moments of the [Formula: see text]-Laguerre matrix. Moreover, we extend our result for inverse compound Wishart matrices for the values of [Formula: see text] and [Formula: see text].

New Insights Into the Statistical Properties of <inline-formula> <tex-math notation="LaTeX">$M$</tex-math> </inline-formula>-Estimators

This paper proposes an original approach to better understanding the behavior of robust scatter matrix Mestimators. Scatter matrices are of particular interest for many signal processing applications since the resulting performance strongly relies on the quality of the matrix estimation. In this context, M-estimators appear as very interesting candidates, mainly due to their flexibility to the statistical model and their robustness to outliers and/or missing data. However, the behavior of such estimators still remains unclear and not well understood since they are described by fixed-point equations that make their statistical analysis very difficult. To fill this gap, the main contribution of this work is to prove that these estimators distribution is more accurately described by a Wishart distribution than by the classical asymptotical Gaussian approximation. To that end, we propose a new “Gaussian-core” representation for complex elliptically symmetric distributions and we analyze the proximity between M-estimators and a Gaussian-based sample covariance matrix, unobservable in practice and playing only a theoretical role. To confirm our claims, we also provide results for a widely used function of M-estimators, the Mahalanobis distance. Finally, Monte Carlo simulations for various scenarios are presented to validate theoretical results.

A New Variational Bayesian Adaptive Extended Kalman Filter for Cooperative Navigation.

To solve the problem of unknown state noises and uncertain measurement noises inherent in underwater cooperative navigation, a new Variational Bayesian (VB)-based Adaptive Extended Kalman Filter (VBAEKF) for master–slave Autonomous Underwater Vehicles (AUV) is proposed in this paper. The Inverse Wishart (IW) distribution is used to model the predicted error covariance and measurement noise covariance matrix. The state, together with the predicted error covariance and measurement noise covariance matrix, can be adaptively estimated based on VB approximation. The performance of the proposed algorithm is demonstrated through a lake trial, which shows the advantage of the proposed algorithm.

A consistent estimator for spectral density matrix of a discrete time periodically correlated process

In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectral density matrix of a periodically correlated process. We derive its limiting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed periodogram and exhibit its asymptotic consistency using simulated data.

On the asymptotic distribution for the periodograms of almost periodically correlated (cyclostationary) processes

Almost periodically correlated (APC) processes have almost periodic mean and auto-covariance functions. These processes have spectral mass on lines parallel to the diagonal, Tj(x)=x±αj, j=1,2,…, in the two-dimensional spectral plane [0,2π)2, and contain stationary and periodically correlated processes. The aim of this article is to make almost periodically correlated processes practical in time series modeling. The main results are on the asymptotic unbiasedness of the periodogram, and the limiting distribution for the finite Fourier transform and the periodogram of APC processes. First, we put light on APC processes in more details. The periodogram is introduced and by using an auxiliary operator, we prove that the limiting distribution of the finite Fourier transform and the periodogram are multivariate complex normal and complex Wishart distributions, respectively. Finally, the accuracy of theoretical results is investigated through simulation study.

Region-based classification of PolSAR data using radial basis kernel functions with stochastic distances

ABSTRACTRegion-based classification of PolSAR data can be effectively performed by seeking for the assignment that minimizes a distance between prototypes and segments. Silva et al. [“Classification of segments in PolSAR imagery by minimum stochastic distances between wishart distributions.” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing 6 (3): 1263–1273] used stochastic distances between complex multivariate Wishart models which, differently from other measures, are computationally tractable. In this work we assess the robustness of such approach with respect to errors in the training stage, and propose an extension that alleviates such problems. We introduce robustness in the process by incorporating a combination of radial basis kernel functions and stochastic distances with Support Vector Machines (SVM). We consider several stochastic distances between Wishart: Bhatacharyya, Kullback-Leibler, Chi-Square, Rényi, and Hellinger. We perform two case studies with PolSAR images, both simulated and from actual sensors, and different classification scenarios to compare the performance of Minimum Distance and SVM classification frameworks. With this, we model the situation of imperfect training samples. We show that SVM with the proposed kernel functions achieves better performance with respect to Minimum Distance, at the expense of more computational resources and the need of parameter tuning. Code and data are provided for reproducibility.

Statistical Theory of Shape Under Elliptical Models via Polar Decompositions

A new model of statistical shape theory under elliptical models is proposed by using the polar decomposition. This work completes the group of SVD and QR shape densities obtained from the transpose of the square root of a non singular Wishart matrix. The associated non isotropic and non central polar shape distributions are set in the context of consistent computable series of zonal polynomials. Then the inference procedures with elliptical assumptions can be performed at the same computational cost of the published routines based on Gaussian models. As an example of the technique, a classical application in Biology is studied under three models, the usual Gaussian and two Kotz type models; then the best model is selected by a modified BIC∗ criterion, and a test for equality in polar shapes is performed. The published results for this landmark data under isotropic Gaussian models and procrustes theory are also discussed.

Standard Condition Number Distributions of Finite Wishart Matrices for Cognitive Radio Networks

The standard condition number (SCN) in random matrix theory (RMT) has widely been employed in cognitive radio networks (CRNs). The paper studies the SCN distributions of central and complex Wishart matrices with finite dimensions. The exact and closed-form expressions of SCN distributions are first proposed in sum of multiple polynomials. The proposed formulations provide an efficient way to determine probability density function and cumulative distribution function of the SCN in finite RMT. In particular, the specific SCN distributions of triple Wishart matrix are initially presented. The theoretical and precise sensing thresholds for cooperative spectrum sensing (CSS) systems in CRNs are calculated using the proposed SCN distributions. Numerical results indicate that the proposed theoretical SCN distributions match the empirical SCN distributions very well. Furthermore, the exact SCN-based CSS systems are able to obtain higher sensing performance than the asymptotic SCN-based schemes because precise sensing thresholds can be determined.

Flood Detection in PolSAR Images Based on Level Set Method Considering Prior Geoinformation

This letter presents a novel flood detection approach using full polarimetric synthetic aperture radar (PolSAR) images based on a level set method considering prior geoinformation. The prior geoinformation includes information derived from vector data and topography data. The main approach accomplishes flood detection by the improved level set method, an active contour segmentation model, based on the classical Wishart distribution. Vector data are used to generate the zero initial level set curves. To investigate the separability between water and nonwater low–backscattering objects in PolSAR images, topography information is incorporated into the level set function as a constraint. Moreover, we introduce a piecewise statistical method to refine the result with the Kullback–Leibler divergence of circular polarization coherence. In addition, we design a new quantitative evaluation index to assess flood detection results. For validation, three real PolSAR images of flooded area are tested. The experimental results confirm the effectiveness of the proposed method.

DESPECKLING POLSAR IMAGES BASED ON RELATIVE TOTAL VARIATION MODEL

Abstract. Relatively total variation (RTV) algorithm, which can effectively decompose structure information and texture in image, is employed in extracting main structures of the image. However, applying the RTV directly to polarimetric SAR (PolSAR) image filtering will not preserve polarimetric information. A new RTV approach based on the complex Wishart distribution is proposed considering the polarimetric properties of PolSAR. The proposed polarization RTV (PolRTV) algorithm can be used for PolSAR image filtering. The L-band Airborne SAR (AIRSAR) San Francisco data is used to demonstrate the effectiveness of the proposed algorithm in speckle suppression, structural information preservation, and polarimetric property preservation.

Estimating tail probabilities of the ratio of the largest eigenvalue to the trace of a Wishart matrix

This paper develops an efficient Monte Carlo method to estimate the tail probabilities of the ratio of the largest eigenvalue to the trace of the Wishart matrix, which plays an important role in multivariate data analysis. The estimator is constructed based on a change-of-measure technique and it is proved to be asymptotically efficient for both the real and complex Wishart matrices. Simulation studies further show the improved performance of the proposed method over existing approaches based on asymptotic approximations, especially when estimating probabilities of rare events.

Enhanced edge detection for polarimetric SAR images using a directional span-driven adaptive window

ABSTRACTAutomatic edge detection for polarimetric synthetic aperture radar (PolSAR) images plays a fundamental role in various PolSAR applications. The classic methods apply the fixed-shape windows to detect the edges, whereas their performance is limited in heterogeneous areas. This article presents an enhanced edge detection method for PolSAR data based on the directional span-driven adaptive (DSDA) window. The DSDA window has variable sizes and flexible shapes, and is constructed by adaptively selecting samples that follow the same statistical distribution. Therefore, it can overcome the limitation of classic fixed-shape windows. To obtain refined and reliable edge detection results in heterogeneous urban areas, we adopt the spherically invariant random vector (SIRV) product model since the complex Wishart distribution is often not met. In addition, a span ratio is combined with the SIRV distance to highlight the dissimilarity measure and to improve the robustness of the proposed method. The simulated PolSAR data and three real data sets from experimental synthetic aperture radar, electromagnetics institute synthetic aperture radar, and Radarsat-2 systems are used to validate the performance of the enhanced edge detector. Both quantitative evaluation and visual presentation of the results demonstrate the effectiveness of the proposed method and its superiority over the classic edge detectors.

Fast Converging Weighted Neumann Series Precoding for Massive MIMO Systems

Neumann series (NS) expansion-based precoder in massive multiple input multiple output systems suffers from slow convergence. To solve this problem, this letter proposes a weighted NS expansion precoder. The weights are designed to minimize the error between the exact inverse and the weighted NS inverse. The optimal weights are deduced analytically. Moreover, an approximation of these optimal weights is proposed, based on the properties of large Wishart matrices, which saves the re-computation of these weights. The weighted NS precoding provides near optimal performance at only four weighted expanded NS terms and has lower complexity than recently proposed approximate precoders.

Spectra of overlapping Wishart matrices and the Gaussian free field

Consider a doubly-infinite array of i.i.d. centered variables with moment conditions, from which one can extract a finite number of rectangular, overlapping submatrices, and form the corresponding Wishart matrices. We show that under basic smoothness assumptions, centered linear eigenstatistics of such matrices converge jointly to a Gaussian vector with an interesting covariance structure. This structure, which is similar to those appearing in [A. Borodin, Clt for spectra of submatrices of Wigner random matrices, Mosc. Math. J. 14(1) (2014) 29–38; A. Borodin and V. Gorin, General beta Jacobi corners process and the Gaussian free field, preprint (2013), arXiv:1305.3627; T. Johnson and S. Pal, Cycles and eigenvalues of sequentially growing random regular graphs, Ann. Probab. 42(4) (2014) 1396–1437], can be described in terms of the height function, and leads to a connection with the Gaussian Free Field on the upper half-plane. Finally, we generalize our results from univariate polynomials to a special class of planar functions.

Distributed Averaging of Exponential-Class Densities With Discrete-Time Event-Triggered Consensus

The paper addresses discrete-time event-driven consensus on exponential-class probability densities (including Gaussian, binomial, Poisson, Rayleigh, Wishart, Inverse Wishart, and many other distributions of interest) completely specified by a finite-dimensional vector of so-called natural parameters. First, it is proved how such exponential classes are closed under Kullback-Leibler fusion (average), and how the latter is equivalent to a weighted arithmetic average over the natural parameters. Then, a novel event-driven transmission strategy is proposed in order to trade off the data-communication rate and, hence, energy consumption, versus consensus speed and accuracy. A theoretical analysis of the convergence properties of the proposed algorithm is provided by exploiting the Fisher metric as a local approximation of the Kullback-Leibler divergence. Some numerical examples are presented in order to demonstrate the effectiveness of the proposed event-driven consensus. It is expected that the latter can be successfully exploited for energy- and/or bandwidth-efficient networked state estimation.