Wishart Distribution Research Articles

Asymptotic analysis for extreme eigenvalues of principal minors of random matrices

Consider a standard white Wishart matrix with parameters n and p. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the m×m principal minors, under the asymptotic regime that n, p, m go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing in signal processing and high-dimensional linear regression in statistics.

A Comparison of Three Bayesian Point Estimates for Cross-Classified Multiple Membership Growth Curve Modeling

본 모의실험 연구에서는 다양한 조건에서 교차분류 다중소속 다층성장모형의 모수 복원력을 살펴보고자 하였다. 이를 위해 사전 분포(역 Wishart 분포, 균일 분포), 측정 횟수(3, 5), 집단 수(30, 50, 100), 집단 크기(20, 40), 비순수 위계적 자료 비율(20%, 40%) 조건이 고려되었다. 세 개의 베이즈 추정치(사후 평균, 사후 중앙값 및 사후 최빈값)를 비교한 결과, 역 Wishart 분포를 적용하였을 때 절편 및 기울기의 학교 간 분산에 대한 사후 평균 추정치는 집단 수가 50개 이하일 때 과대 추정된 반면, 사후 최빈값 추정치는 상당히 과소 추정되었다. 사후 중앙값 추정치는 30개 이하의 그룹에서 편향이 나타났다. 균일 분포를 적용하였을 경우에는 절편 및 기울기에서 학교 간 분산의 사후 평균, 사후 중앙값 및 사후 최빈값 추정치가 상당히 과대 추정되는 것으로 나타났다. 따라서 역 Wishart 분포를 적용한 경우 일부 조건을 제외하고 교차분류 다중소속 다층성장모형 모수추정 시 마코프체인 몬테카를로(MCMC) 사후 중앙값을 사용하는 것이 권장되며, 균일 분포를 적용한 경우 교차분류 다중소속 다층성장모형 모수추정 시 MCMC 사후 평균 또는 중앙값보다 MCMC 최빈값을 활용하는 것이 권장된다.This simulation study evaluated parameter recovery in cross-classified multiple membership growth curve modeling under a variety of manipulated conditions, including prior distributions (inverse Wishart and uniform distributions), number of measurement occasions (3, 5), number of groups (30, 50, 100), average group size (20, 40), and non-pure hierarchical data rates (20%, 40%). Three Bayesian point estimates were compared: the posterior mean, the posterior median, and the posterior mode. With the inverse Wishart prior, the posterior mean estimates for the between-school variances in the intercept and slope were substantially overestimated with 50 groups or less, while the posterior mode estimates were substantially underestimated. The posterior median estimates were substantially biased with 30 groups or less. With the uniform prior, the posterior mean, posterior median, and posterior mode estimates of the between-school variances in the intercept and slope were substantially overestimated. In general, with the inverse Wishart prior, the Markov chain Monte Carlo (MCMC) posterior median is recommended for cross-classified multiple membership growth curve modeling, except under certain conditions. With the uniform prior, the MCMC mode is recommended for estimating cross-classified multiple membership growth curve modeling rather than the MCMC posterior mean or posterior median.

Lower bound on the capacity of distributed MIMO systems with spatially correlated Rician/Lognormal fading

This paper presents an analytic lower bound of ergodic capacity for distributed MIMO (D-MIMO) systems that experience not only Rician fading but also shadowing effects of Lognormal distributed. In particularly, we consider that the Rician fading channel is spatially correlated at both transmitter and receiver. In the communication environment corresponding to this setting, the angle spread at the transmitter and the angle spread at the receiver are both insufficient, and the non-fading components co-exists with the fading components. Such communication environment is very common. In the process of deriving the analytic lower bound, in order to avoid Hayakawa polynomials that cannot be analytically expressed, non-central quadratic forms are transformed to non-central Wishart matrices by use of inequality. The validity of the presented lower bound is verified by computer simulations. The simulation results show the influence of the number of radio ports at the transmitter, the number of antennas at the transmitter or receiver, correlation level (angle spreads) and Rician K-factor on the capacity of systems. In all cases, the presented bound remains tight across the entire signal-to-noise ratio (SNR) regime.

Moments of a Wishart Matrix

The paper discusses the moments of Wishart matrices, in both the central and noncentral cases. The first part of the paper shows that the expectation map has certain homogeneity and equivariance properties which impose considerable structure on the moments, hitherto unrecognised. The second part of the paper explains how the moments may be computed efficiently. The two parts of the paper are completely independent, but the computations produce precisely the algebraic structure predicted in the first part, as well as reproducing all previously known formulae. A number of examples are given for the more manageable cases.

Matrix-variate Lindley distributions and its applications

Restring on the fact that the definition of multivariate analogs of the real gamma distribution is replaced by the Wishart distribution on symmetric matrices, and based on the notion of mixture models which is a flexible and powerful tool for treating data taken from multiple subpopulations, we set forward a multivariate analog of the real Lindley distributions of the first and second kinds on the modern framework of symmetric cones which can be used to model waiting and survival times matrix data. Within this framework, we first construct a new probability distributions, named the matrix-variate Lindley distributions. Some fundamental properties of these new distributions are established. Their statistical properties including moments, the coefficient of variation, skewness and the kurtosis are discussed. We then create an iterative hybrid Expectation-Maximization Fisher-Scoring (EM-FS) algorithm to estimate the parameters of the new class of probability distributions. Through simulation as well as comparative studies with respect to the Wishart distribution, the effectiveness and reliability of the proposed distributions are proved. Finally, the usefulness and the applicability of the new models are elaborated and illustrated by means of two real data sets from biological sciences and medical image segmentation which is one of the most important and popular tasks in medical image analysis.

Average Achievable Rate and Average BLER Analyses for MIMO Short-Packet Communication Systems

Multiple-input-multiple-output (MIMO) systems significantly increase the network throughput and coverage probability for short-packet communications. In this paper, a tight lower bound on the average achievable rate of MIMO short-packet communication systems is theoretically derived, considering flat uncorrelated Rayleigh fading MIMO channels. The derivations are based on the eigenvalue analysis of a Wishart matrix and Jensen's inequality. Given the actual transmission rate, a tight approximation on the average block error rate (BLER) of MIMO short-packet communication systems is also provided, which is based on the moment generating function and inverse Laplace transform. Numerical results demonstrate that both our theoretically derived analytical average achievable rate lower bound and approximate average BLER are almost the same as the simulation results.

Gaussian fluctuation for Gaussian Wishart matrices of overall correlation

In this note, we study the Gaussian fluctuations for the Wishart matrices d−1Xn,dXn,dT, where Xn,d is a n×d random matrix whose entries are jointly Gaussian and correlated with row and column covariance functions given by r and s respectively such that r(0)=s(0)=1. Under the assumptions s∈ℓ4/3(Z) and ‖r‖ℓ1(Z)<6/2, we establish the n3/d convergence rate for the Wasserstein distance between a normalization of d−1Xn,dXn,dT and the corresponding Gaussian ensemble. This rate is the same as the optimal one computed in Bubeck et al. (2016), Bubeck and Ganguly (2018) and Jiang and Li (2015) for the total variation distance, in the particular case where the Gaussian entries of Xn,d are independent. Similarly, we obtain the n2p−1/d convergence rate for the Wasserstein distance in the setting of symmetric random tensor of order p with overall correlation. Our analysis is based on the Malliavin–Stein approach.

Asymptotic Behavior of Large Gaussian Correlated Wishart Matrices

We consider high-dimensional Wishart matrices \(d^{-1}{\mathcal {X}}_{n,d}{\mathcal {X}}_{n,d}^T\), associated with a rectangular random matrix \({\mathcal {X}}_{n,d}\) of size \(n\times d\) whose entries are jointly Gaussian and correlated. Even if we will consider the case of overall correlation among the entries of \({\mathcal {X}}_{n,d}\), our main focus is on the case where the rows of \({\mathcal {X}}_{n,d}\) are independent copies of an n-dimensional stationary centered Gaussian vector of correlation function s. When s belongs to \(\ell ^{4/3}({\mathbb {Z}})\), we show that a proper normalization of \(d^{-1}{\mathcal {X}}_{n,d}{\mathcal {X}}_{n,d}^T\) is close in Wasserstein distance to the corresponding Gaussian ensemble as long as d is much larger than \(n^3\), thus recovering the main finding of Bubeck et al. (Random Struct Algorithms 49(3):503–532, 2016) and Jiang and Li (J Theor Probab 28(3):804–847, 2015) and extending it to a larger class of matrices. We also investigate the case where s is the correlation function associated with the fractional Brownian noise of parameter H. This example is very rich, as it gives rise to a great variety of phenomena with very different natures, depending on how H is located with respect to 1/2, 5/8 and 3/4. Notably, when \(H>3/4\), our study highlights a new probabilistic object, which we have decided to call the Rosenblatt–Wishart matrix. Our approach crucially relies on the fact that the entries of the Wishart matrices we are dealing with are double Wiener–Itô integrals, allowing us to make use of multivariate bounds arising from the Malliavin–Stein method and related ideas. To conclude the paper, we analyze the situation where the row-independence assumption is relaxed and we also look at the setting of random p-tensors (\(p\ge 3\)), a natural extension of Wishart matrices.

Outage Performance Analysis of Widely Linear Receivers in Uplink Multi-User MIMO Systems

This paper considers the application of widely linear (WL) receivers in an uplink multi-user system using real-valued modulation schemes, where the cellular base station (BS) with multiple antennas provides connectivity for randomly deployed single-antenna users. The targeted use case is massive machine type communication (mMTC) with grant-free access in the uplink, where the network is required to host a large number of low data rate devices transmitting in an uncoordinated fashion. Four types of WL receivers are investigated, namely the WL zero-forcing (ZF) and the WL minimum mean-squared error (MMSE) receivers, along with their enhanced versions employing successive interference cancellation (SIC) with channel-dependent ordering, i.e., the WL-ZF-SIC and WL-MMSE-SIC receivers. The outage performances of these receivers are analytically characterized in the high signal-to-noise ratio (SNR) regime and compared to those of conventional linear (CL) receivers using complex-valued modulation schemes. For the non-SIC receivers, we show that, when compared to the CL counterparts, the WL receivers yield a higher diversity gain when decoding the same number of users and have the same diversity gain but a decreased coding gain when the number of users is nearly doubled. The outage performance analysis of WL-SIC receivers is facilitated by the marginal distribution of ordered eigenvalues of a real-valued Wishart matrix. It is shown that the SIC operation with channel-dependent ordering brings no additional diversity gain to the WL receivers but instead increases the coding gain. Moreover, the coding gain of WL-SIC receivers grows as the number of users increases and even exceeds that of CL-SIC receivers under suitable conditions. For the mMTC scenario with grant-free transmission, it is demonstrated that the WL receivers outperform their CL counterparts in terms of offering a lower outage (and packet drop) probability and a higher system throughput for a given packet drop probability.

Bayesian modeling and forecasting of vector autoregressive moving average processes

The article proposes a Bayesian methodology to implement complete and cohesive analysis of vector time series. Assuming the series are generated by vector autoregressive moving average process, the identification, diagnostic checking, estimation and forecasting phases of time series analysis are done by referring to the appropriate posterior or predictive distributions. The identification phase is based on approximating the posterior distribution of the coefficients of the largest possible model by a matric-variate generalization of the t-distribution (matrix t-distribution). Then the insignificant coefficients are eliminated by a sequence of F or Chi-square tests using a procedure similar to the backward elimination technique used in regression analysis. The diagnostic checking phase is done using overfitting tests, the estimation phase is done using the matrix t and Wishart distributions, and the forecasting phase is performed using multivariate t-distribution. Using the proposed posterior distributions, the article proposes the machinery necessary to implement the four phases of multivariate time series analysis. The proposed Bayesian methodology has been illustrated and checked by a simulated data used by distinguished researchers. The Initial examination of the numerical results supports the adequacy of using the proposed methodology.

Hurst exponent estimation of fractional surfaces for mammogram images analysis

The work presents a methodology to precise simulation and parameter estimation of multidimensional fractional Brownian motion (fBm). The simulation approach uses circulant embedding algorithm and solution of Poisson equation, while generalizing it to multiple dimensions. For estimation, a method using Wishart distribution and maximum likelihood is presented and verified on simulated data. Unlike approximate methods for generating multidimensional fBm and its Hurst exponent estimation, this approach shows unbiased results for all processes with short memory and majority of cases with long memory. The methodology is applied to mammography screening images to find significant differences between benign and cancerous breast lumps.

A variational Bayesian based robust cubature Kalman filter under dynamic model mismatch and outliers interference

This paper proposes a variational Bayesian-based cubature Kalman filter to solve the state estimation problem of nonlinear discrete-time systems under dynamic model mismatch and outliers interference, which occurs in the maneuvering target tracking system. In the proposed filter, the measurement noise is divided into the normal measurement noise and the outlier noise, and two inverse Wishart distributions with different degree of freedom parameters are used to model the unknown normal measurement noise covariance and the unknown outlier noise covariance, respectively. To overcome outliers interference, the judgment factor which follows the Bernoulli-beta distribution is designed to identify the type of measurement noise. Meanwhile, the input vector which follows the Gaussian distribution is introduced to realize the dynamic model revision. The state, the judgment factor, the normal measurement noise covariance, the outlier noise covariance, and the input vector are jointly estimated under the variational Bayesian framework. The numerical simulation is illustrated to verify the performance of the proposed filter and the corresponding results show that the proposed filter has good state estimation performance and robustness.

Limit theorem associated with Wishart matrices with application to hypothesis testing for common principal components

This paper describes the derivation of a new property of the Wishart distribution when the degrees of freedom and the sizes of scale matrices grow simultaneously. In particular, the asymptotic normality of the trace of the product of four independent Wishart matrices is demonstrated for a high-dimensional asymptotic regime. As an application of the result, a statistical test procedure for the common principal components hypothesis is proposed. For this problem, the proposed test statistic is asymptotically normal under the null hypothesis and diverges to positive infinity in probability under the alternative hypothesis.

The volume-of-tube method for Gaussian random fields with inhomogeneous variance

The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.

An Improved Invariant Kalman Filter for Lie Groups Attitude Dynamics with Heavy-Tailed Process Noise

Attitude estimation is a basic task for most spacecraft missions in aerospace engineering and many Kalman type attitude estimators have been applied to the guidance and navigation of spacecraft systems. By building the attitude dynamics on matrix Lie groups, the invariant Kalman filter (IKF) was developed according to the invariance properties of symmetry groups. However, the Gaussian noise assumption of Kalman theory may be violated when a spacecraft maneuvers severely and the process noise might be heavy-tailed, which is prone to degrade IKF’s performance for attitude estimation. To address the attitude estimation problem with heavy-tailed process noise, this paper proposes a hierarchical Gaussian state-space model for invariant Kalman filtering: The probability density function of state prediction is defined based on student’s t distribution, while the conjugate prior distributions of the scale matrix and degrees of freedom (dofs) parameter are respectively formulated as the inverse Wishart and Gamma distribution. For the constructed hierarchical Gaussian attitude estimation state-space model, the Lie groups rotation matrix of spacecraft attitude is inferred together with the scale matrix and dof parameter using the variational Bayesian iteration. Numerical simulation results illustrate that the proposed approach can significantly improve the filtering robustness of invariant Kalman filter for Lie groups spacecraft attitude estimation problems with heavy-tailed process uncertainty.

Discrimination of multi-crop scenarios with polarimetric SAR data using Wishart mixture model

Discrimination of crop varieties spanned over heterogeneous agriculture land is a vital application of polarimetric SAR images for agriculture monitoring and assessment. The covariance matrix of polarimetric SAR images is observed to follow a complex Wishart distribution for major classification tasks. It is true for homogeneous regions, but for heterogeneous regions, the covariance matrix follows a mixture of multiple Wishart distributions. We aim to improve the classification accuracy when the terrain under observation is heterogeneous. For this purpose, Wishart mixture model is employed along with expectation-maximization (EM) algorithm for parameter estimation. Elbow method helps us to devise the number of mixtures. The convergence of the EM algorithm depends on the choice of initial points. So, to improve the robustness of the model, different initialization approaches, such as random, K-means, and global K-means, are embedded in the EM algorithm. Further, the degrees of freedom is one of the crucial parameters of Wishart distribution. Therefore, the impact of different degrees of freedom is analyzed on classification accuracy. The method that is equipped with initialization technique along with optimum degrees of freedom is assessed using three full polarimetric SAR data sets of agriculture lands. The first two are benchmark data sets of Flevoland, Netherlands, region acquired by AIRSAR sensor, and third is our study area of Mysore, India, acquired by RADARSAT-2 sensor.

The Laplace transform of the integrated Volterra Wishart process

AbstractWe establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and resolvent. Furthermore, we link the characteristic exponents to a system of non‐standard infinite dimensional matrix Riccati equations. This leads to a second representation of the Laplace transform for a special case of convolution kernel. In practice, we show that both representations can be approximated by either closed form solutions of conventional Wishart distributions or finite dimensional matrix Riccati equations stemming from conventional linear‐quadratic models. This allows fast pricing in a variety of highly flexible models, ranging from bond pricing in quadratic short rate models with rich autocorrelation structures, long range dependence and possible default risk, to pricing basket options with covariance risk in multivariate rough volatility models.

Random Matrix Theory and Its Applications

This article reviews the important ideas behind random matrix theory (RMT), which has become a major tool in a variety of disciplines, including mathematical physics, number theory, combinatorics and multivariate statistical analysis. Much of the theory involves ensembles of random matrices that are governed by some probability distribution. Examples include Gaussian ensembles and Wishart–Laguerre ensembles. Interest has centered on studying the spectrum of random matrices, especially the extreme eigenvalues, suitably normalized, for a single Wishart matrix and for two Wishart matrices, for finite and infinite sample sizes in the real and complex cases. The Tracy–Widom Laws for the probability distribution of a normalized largest eigenvalue of a random matrix have become very prominent in RMT. Limiting probability distributions of eigenvalues of a certain random matrix lead to Wigner’s Semicircle Law and Marc˘enko–Pastur’s Quarter-Circle Law. Several applications of these results in RMT are described in this article.

Regression conditions that characterize free-Poisson and free-Kummer distributions

We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free-Kummer and free-Poisson (Marchenko–Pastur) non-commutative random variables.

A Brief Derivation of Necessary and Sufficient Conditions for a Family of Matrix Quadratic Forms to Have Mutually Independent Non-Central Wishart Distributions

A Brief Derivation of Necessary and Sufficient Conditions for a Family of Matrix Quadratic Forms to Have Mutually Independent Non-Central Wishart Distributions