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Many matrix‐valued functions of an m×m Wishart matrix W, Fk(W), say, are homogeneous of degree k in W, and are equivariant under the conjugate action of the orthogonal group 풪(m), that is, Fk(HWHT)=HFk(W)HT, H∈풪(m). It is easy to see that the expectation of such a function is itself homogeneous of degree k in ∑, the covariance matrix, and are also equivariant under the action of 풪(m) on ∑. The space of such homogeneous, equivariant, matrix‐valued functions is spanned by elements of the type Wrpλ(W), where r∈{0,...,k} and, for each r, λ varies over the partitions of k−r, and pλ(W) denotes the power‐sum symmetric function indexed by λ. In the analogous case where W is replaced by W−1, these elements are replaced by W−rpλ(W−1). 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. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

Random matrix theory, particularly using matrices akin to the Wishart ensemble, has proven successful in elucidating the thermodynamic characteristics of critical behavior in spin systems across varying interaction ranges. This paper explores the applicability of such methods in investigating critical phenomena and the crossover to tricritical points within the Blume–Capel model. Through an analysis of eigenvalue mean, dispersion, and extrema statistics, we demonstrate the efficacy of these spectral techniques in characterizing critical points in both two and three dimensions. Crucially, we propose a significant modification to this spectral approach, which emerges as a versatile tool for studying critical phenomena. Unlike traditional methods that eschew diagonalization, our method excels in handling short timescales and small system sizes, widening the scope of inquiry into critical behavior. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper aims to improve and extend the improved spatial Moran's I theory by analyzing multi-observation samples. By constructing an expanded spatial weight matrix, a vector definition of the improved spatial Moran's I is given. In order to improve the judgment basis of the improved spatial Moran's I, the range of the improved spatial Moran's I is derived using the non-negativity of variance. Since the improved Moran's I is only applicable to the analysis of a single variable with unknown distribution, a Moran's I matrix suitable for analyzing the spatial autocorrelation of multiple variables is proposed. The distribution of the elements of the Moran's I matrix is studied by Monte Carlo simulation. The simulation results show that only the elements on the non-main diagonal follow a normal distribution when the sample size is small. Any element follows a normal distribution when the sample size is large. Then it is proved that the Moran's I matrix follows a Wishart distribution when the spatial weight matrix is a positive definite matrix. Finally, several comprehensive evaluation indicators suitable for the theory of multivariate spatial autocorrelation are proposed based on the algebraic meaning of the Moran's I matrix. Spatial autocorrelation analysis is carried out in combination with multi-dimensional air pollution data. [ABSTRACT FROM AUTHOR]

Some might say that the eigenproblem is one of the examples people discovered by looking at the sky and wondering. Even though it was formulated to explain the movement of the planets, today it has become the ansatz of solving many linear and nonlinear problems. Formulation in the terms of the eigenproblem is one of the key tools to solve complex problems, especially in the area of molecular geometry. However, the basic concept is difficult without proper preparation. A review paper covering basic concepts and algorithms is very useful. This review covers the basics of the topic. Definitions are provided for defective, Hermitian, Hessenberg, modal, singular, spectral, symmetric, skew-symmetric, skew-Hermitian, triangular, and Wishart matrices. Then, concepts of characteristic polynomial, eigendecomposition, eigenpair, eigenproblem, eigenspace, eigenvalue, and eigenvector are subsequently introduced. Faddeev–LeVerrier, von Mises, Gauss–Jordan, Pohlhausen, Lanczos–Arnoldi, Rayleigh–Ritz, Jacobi–Davidson, and Gauss–Seidel fundamental algorithms are given, while others (Francis–Kublanovskaya, Gram–Schmidt, Householder, Givens, Broyden–Fletcher–Goldfarb–Shanno, Davidon–Fletcher–Powell, and Saad–Schultz) are merely discussed. The eigenproblem has thus found its use in many topics. The applications discussed include solving Bessel's, Helmholtz's, Laplace's, Legendre's, Poisson's, and Schrödinger's equations. The algorithm extracting the first principal component is also provided. [ABSTRACT FROM AUTHOR]

We study a discrete-time Markov process on triangular arrays of matrices of size d ≥ 1 , driven by inverse Wishart random matrices. The components of the right edge evolve as multiplicative random walks on positive definite matrices with one-sided interactions and can be viewed as a d-dimensional generalisation of log-gamma polymer partition functions. We establish intertwining relations to prove that, for suitable initial configurations of the triangular process, the bottom edge has an autonomous Markovian evolution with an explicit transition kernel. We then show that, for a special singular initial configuration, the fixed-time law of the bottom edge is a matrix Whittaker measure, which we define. To achieve this, we perform a Laplace approximation that requires solving a constrained minimisation problem for certain energy functions of matrix arguments on directed graphs. [ABSTRACT FROM AUTHOR]

It is necessary to adequately extract scattering characteristics from polarimetric synthetic aperture radar (PolSAR) data for land-cover classification. Current interpretation technologies, such as target decompositions, polarimetric signatures, and polarimetric coherence patterns, have been widely used to explore polarization information. However, because limited scattering models are used to characterize all ground targets, these methods are occasionally limited in the interpretation of PolSAR data. This may reduce the accuracy of the land-cover classification. In this study, we attempted to extract the scattering characteristics through another novel method. We understand the PolSAR image interpretation as the projection of the scattering model on the measured matrix. Current methods project only a limited series of scattering models onto a measured matrix. To complement current methods of PolSAR image interpretation, we propose a polarimetric projection-based scattering characteristics extraction (PPSCE) method. It projects scattering models consisting of four changing independent polarization parameters onto the measured matrix. These scattering models, associated with the geometric and physical properties of the ground targets, correspond to the entire polarization space. The PPSCE method, meanwhile, extracts polarization information sufficiently. L-band UAVSAR, C-band Radarsat-2, and P-band AIRSAR images of classical experimental areas are used to validate the PPSCE method. The results show that the proposed PPSCE method can present polarization information that is difficult to investigate by the existing methods. In addition, the scattering features derived using the PPSCE method are superior for land-cover classification. Compared to the classification results based on the coherency matrix and Wishart classifier, the proposed method improves the overall classification accuracy by 23.56% for UAVSAR data. Compared to target decompositions and polarization signatures, the overall classification accuracy of the proposed method is increased by 29.32% (RF) and 22.63% (SVM) for UAVSAR data, 7.96% (RF) and 4.36% (RF) for Radarsat-2 and AIRSAR datasets, respectively. Furthermore, the proposed method has significant potential for soil moisture inversion. https://rgdoi.net/10.13140/RG.2.2.14091.26409. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The complex Wishart distribution has ample applications in science and engineering. In this paper, we give explicit expressions for E (tr (W h)) g (tr (W j)) i and E (tr (W − h)) g (tr (W − j)) i , respectively, for particular values of g, h, i, j, g + h + i + j ≤ 5 , where W follows a complex Wishart distribution. For specific values of g, h, i, j, we first write (tr (W h)) g (tr (W j)) i and (tr (W − h)) g (tr (W − j)) i in terms of zonal polynomials and then by using results on integration evaluate resulting expressions. Several expected values of matrix-valued functions of a complex Wishart matrix have also been derived. [ABSTRACT FROM AUTHOR]

Wishart matrices play an important role in normal multivariate statistical analysis. In this work, we present an approach that has been already used for normal vectors and is now applied to noncentral Wishart matrices. We show that, under general conditions, the vec of the Wishart matrix and a large class of its statistics have asymptotic normal distributions when the norm of the noncentrality parameter diverges ∞. These statistics are called smooth and are given by functions whose component functions have continuous second-order partial derivatives in a neighbourhood of a 'pivot' point. Moreover, we derive the application domain of the asymptotic normal distributions for the vec of the Wishart matrix and its smooth statistics. Thus we have an attraction to the normal model, for the increasing predominance of noncentrality and not for increasing sample sizes. A simulation study shows that the threshold for the use of asymptotic normal distributions is quite acceptable. [ABSTRACT FROM AUTHOR]

Let $\mathrm {W}\in \mathbb {C}^{n\times n}$ be a single-spiked Wishart matrix in the class $\mathrm {W}\sim \mathcal {CW}_{n}(m,\mathrm {I}_{n}+ \theta \mathrm {v}\mathrm {v}^{\dagger}) $ with $m\geq n$ , where ${\mathrm {I}}_{n}$ is the $n\times n$ identity matrix, $\mathrm {v}\in \mathbb {C}^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $\theta \geq 0$ is a non-random parameter, and $(\cdot)^{\dagger} $ represents the conjugate-transpose operator. Let u1 and ${\mathrm {u}}_{n}$ 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 $Z_{\ell }^{(n)}=\left |{\mathrm {v}^{\dagger} \mathrm {u}_\ell }\right |^{2}\in (0,1)$ for $\ell =1,n$. In particular, we derive a finite dimensional closed-form p.d.f. for $Z_{1}^{(n)}$ which is amenable to asymptotic analysis as $m,n$ diverges with $m-n$ fixed. It turns out that, in this asymptotic regime, the scaled random variable $nZ_{1}^{(n)}$ converges in distribution to $\chi ^{2}_{2}/2(1+\theta)$ , where $\chi _{2}^{2}$ 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 $Z_{n}^{(n)}$ is expressed as a double integral in which the integrand contains a determinant of a square matrix of dimension $(n-2)$. Although a simple solution to this double integral seems intractable, for special configurations of $n=2,3$ , and 4, we obtain closed-form expressions. [ABSTRACT FROM AUTHOR]

Clustering time series into similar groups can improve models by combining information across like time series. While there is a well developed body of literature for clustering of time series, these approaches tend to generate clusters independently of model training, which can lead to poor model fit. We propose a novel distributed approach that simultaneously clusters and fits autoregression models for groups of similar individuals. We apply a Wishart mixture model so as to cluster individuals while modelling the corresponding autocovariance matrices at the same time. The fitted Wishart scale matrices map to cluster‐level autoregressive coefficients through the Yule–Walker equations, fitting robust parsimonious autoregressive mixture models. This approach is able to discern differences in underlying autocorrelation variation of time series in settings with large heterogeneous datasets. We prove consistency of our cluster membership estimator, asymptotic distributions of coefficients and compare our approach against competing methods through simulation as well as by fitting a COVID‐19 forecast model. [ABSTRACT FROM AUTHOR]

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

We analyze statistical features of the ‘optimization landscape’ in a random version of one of the simplest constrained optimization problems of the least-square type: finding the best approximation for the solution of a system of M linear equations in N unknowns: ( a k , x ) = b k , k = 1, …, M on the N -sphere x 2 = N. We treat both the N -component vectors a k and parameters b k as independent mean zero real Gaussian random variables. First, we derive the exact expressions for the mean number of stationary points of the least-square loss function in the overcomplete case M > N in the framework of the Kac-Rice approach combined with the random matrix theory for Wishart ensemble. Then we perform its asymptotic analysis as N â†' ∞ at a fixed α = M / N > 1 in various regimes. In particular, this analysis allows to extract the large deviation function for the density of the smallest Lagrange multiplier λ min associated with the problem, and in this way to find its most probable value. This can be further used to predict the asymptotic mean minimal value E min of the loss function as N â†' ∞. Finally, we develop an alternative approach based on the replica trick to conjecture the form of the large deviation function for the density of E min at N ≫ 1 and any fixed ratio α = M / N > 0. As a by-product, we find the compatibility threshold α c < 1 which is the value of α beyond which a large random linear system on the N -sphere becomes typically incompatible. [ABSTRACT FROM AUTHOR]

In noninteracting limit, the density of states (dos) of a many body system can be expressed as a convolution of the single body dos of its subunits. We use the formulation to derive, in the edge of the spectrum, a differential equation for the ensemble averaged many body dos that is relatively easier to solve. Our analysis, based on the systems in which the subunits can be modelled by a Gaussian or Wishart random matrix ensemble, indicates that a rescaling of energy by the number of subunits leaves the many body dos in a mathematically invariant form. [ABSTRACT FROM AUTHOR]

We study the a central limit theorem for sums of independent tensor powers, |$\frac{1}{\sqrt{d}}\sum \limits _{i=1}^d X_i^{\otimes p}$|⁠. We focus on the high-dimensional regime where |$X_i \in{\mathbb{R}}^n$| and |$n$| may scale with |$d$|⁠. Our main result is a proposed threshold for convergence. Specifically, we show that, under some regularity assumption, if |$n^{2p-1}\ll d$|⁠ , then the normalized sum converges to a Gaussian. The results apply, among others, to symmetric uniform log-concave measures and to product measures. This generalizes several results found in the literature. Our main technique is a novel application of optimal transport to Stein's method, which accounts for the low-dimensional structure, which is inherent in |$X_i^{\otimes p}$|⁠. [ABSTRACT FROM AUTHOR]

A robust adaptive probability hypothesis density (PHD) filter is proposed to address the degradation of PHD performance due to an unknown process noise and measurement noise covariance matrix. Therefore, the inverse Wishart distribution is introduced to model the prior distribution of process noise and measurement noise. Meanwhile, the multi‐target posterior intensity is approximated as a mixture of the inverse Wishart distribution and Gaussian distribution. The closed solution of the robust PHD filter is derived by the variational Bayes approach. Simulation results show that the proposed algorithm outperforms the Gaussian‐mixed PHD filter and the variational Bayesian PHD filter in terms of target number estimation accuracy and optimal sub‐pattern assignment distance. [ABSTRACT FROM AUTHOR]

We report on the emergence of scaling laws in the temporal evolution of the daily closing values of the S&P 500 index prices and its modeling based on the Lévy flights in two dimensions (2D). The efficacy of our proposed model is verified and validated by using the extreme value statistics in the random matrix theory. We find that the random evolution of each pair of stocks in a 2D price space is a scale-invariant complex trajectory whose tortuosity is governed by a 2 / 3 geometric law between the gyration radius R g (t) and the total length ℓ (t) of the path, i.e., R g (t) ∼ ℓ (t) 2 / 3 . We construct a Wishart matrix containing all stocks up to a specific variable period and look at its spectral properties for over 30 years. In contrast to the standard random matrix theory, we find that the distribution of eigenvalues has a power-law tail with a decreasing exponent over time—a quantitative indicator of the temporal correlations. We find that the time evolution of the distance of 2D Lévy flights with index α = 3 / 2 from origin generates the same empirical spectral properties. The statistics of the largest eigenvalues of the model and the observations are in perfect agreement. [ABSTRACT FROM AUTHOR]

We present a simple mean-field approach for calculating spectral densities for random matrix ensembles in the thermodynamic limit. Our approach is based on constructing a linear system of equations and calculating how the solutions to these equation change in response to a small perturbation using the zero-temperature cavity method. We illustrate the power of the method by providing simple analytic derivations of the Wigner Semi-circle Law for symmetric matrices, the Marchenko–Pastur Law for Wishart matrices, the spectral density for a product Wishart matrix composed of two square matrices, and the Circle and elliptic laws for real random matrices. [ABSTRACT FROM AUTHOR]

We consider the problem of detecting a signal of interest corrupted by Gaussian noise with unknown mean and covariance matrix when the training samples available have a different mean. We evaluate the robustness of well-known adaptive detectors designed under the assumption of a same mean. More precisely, statistical representations of the generalized likelihood ratio test, the adaptive matched filter and the adaptive coherence estimator are derived for both an additive model with an arbitrary mismatch between the means, and a replacement model which is widely used in hyperspectral imaging. The new representations are given in terms of simple $F$ distributions and are shown to depend in a simple way on the norm of the whitened mean difference and its angle with the whitened signal of interest signature, or on the replacement factor. These new representations allow to identify the key parameters that impact most the probability of false alarm and probability of detection. Numerical simulations illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We consider estimation of the inverse scatter matrix Σ − 1 for a scale mixture of Wishart matrices under various Efron–Morris type losses, tr [ { Σ ˆ − 1 − Σ − 1 } 2 S k ] for k = 0 , 1 , 2... , where S is the sample covariance matrix. We improve on the standard estimators a S + , where S + denotes the Moore–Penrose inverse of S and a is a positive constant, through an unbiased estimator of the risk difference between the new estimators and a S + . Thus we demonstrate that improvements over the standard estimators under a Wishart distribution can be extended under mixing. We give a unified treatment of the two cases where S is invertible ( S + = S − 1 ) and where S is singular. • The standard estimators of an inverse scatter matrix are not satisfactory. • These estimators can be improved under a class of Efron–Morris type losses. • Improvements are available for scale mixtures of Wishart distributions. [ABSTRACT FROM AUTHOR]

This paper analyzes the detection of a $M$ –dimensional useful signal modeled as the output of a $M \times K$ MIMO filter driven by a $K$ –dimensional white Gaussian noise, and corrupted by a $M$ –dimensional Gaussian noise with mutually uncorrelated components. The study is focused on frequency domain test statistics based on the eigenvalues of an estimate of the spectral coherence matrix (SCM), obtained as a renormalization of the frequency-smoothed periodogram of the observed signal. If $N$ denotes the sample size and $B$ the smoothing span, it is proved that in the high-dimensional regime where $M,B,N$ converge to infinity while $K$ remains fixed, the SCM behaves as a certain correlated Wishart matrix. Exploiting well-known results on the behaviour of the eigenvalues of such matrices, it is deduced that the standard tests based on linear spectral statistics of the SCM fail to detect the presence of the useful signal in the high-dimensional regime. A new test based on the SCM, which is proved to be consistent, is also proposed, and its statistical performance is evaluated through numerical simulations. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Traces of Wishart matrices appear in many applications, for example in finance, discriminant analysis, Mahalanobis distances and angles, loss functions and many more. These applications typically involve mixtures of traces of Wishart and inverse Wishart matrices that are concerned in this paper. Of particular interest are the sampling moments and their limiting joint distribution. The covariance matrix of the marginal positive and negative spectral moments is derived in closed form (covariance matrix of Y = [ p − 1 Tr { W − 1 } , p − 1 Tr { W } , p − 1 Tr { W 2 } ] ′ , where W ∼ W p (Σ = I , n) ). The results are obtained through convenient recursive formulas for E [ ∏ i = 0 k Tr { W − m i } ] and E [ Tr { W − m k } ∏ i = 0 k − 1 Tr { W m i } ]. Moreover, we derive an explicit central limit theorem for the scaled vector Y, when p / n → d < 1 , p , n → ∞ , and present a simulation study on the convergence to normality and on a skewness measure. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The paper derives the Jacobians of the matrix decompositions such as SVD, spectral decomposition, QR decomposition, L ′ D L decomposition and so on, for singular random matrix. These results are established for the random matrix whose elements can be linearly dependent mutually. Furthermore, the density function is restated for the singular Wishart random matrix, and the density function of its M−P inverse is also derived. [ABSTRACT FROM AUTHOR]

We consider a n x d random matrix Xn,d whoses entries can be expressed as Skorohod integrals. By using the techniques of the Malliavin calculus, we study the fluctuations under the Wasserstein distance, as n, d → ∞, of the renormalized Wishart matrix ... where In is the n x n identity matrix. [ABSTRACT FROM AUTHOR]

In this work, we consider the weighted difference of two independent complex Wishart matrices and derive the joint probability density function of the corresponding eigenvalues in a finite-dimension scenario using two distinct approaches. The first derivation involves the use of unitary group integral, while the second one relies on applying the derivative principle. The latter relates the joint probability density of eigenvalues of a matrix drawn from a unitarily invariant ensemble to the joint probability density of its diagonal elements. Exact closed form expressions for an arbitrary order correlation function are also obtained and spectral densities are contrasted with Monte Carlo simulation results. Analytical results for moments as well as probabilities quantifying positivity aspects of the spectrum are also derived. Additionally, we provide a large-dimension asymptotic result for the spectral density using the Stieltjes transform approach for algebraic random matrices. Finally, we point out the relationship of these results with the corresponding results for difference of two random density matrices and obtain some explicit and closed form expressions for the spectral density and absolute mean. [ABSTRACT FROM AUTHOR]

For non-orthogonal multiple access (NOMA) based cellular communication networks, cache enabled interference cancelation not only reduces the complexity of successive interference cancelation (SIC), but also improves the received signal-to-interference-and-noise ratios. In this paper, a cache and SIC enabled downlink multiple-input-multiple-output NOMA cellular network with randomly distributed users is studied. To theoretically derive its outage performance, the Wishart matrices are introduced to express the achievable rates of users. From the joint probability density function of the unordered eigenvalues of Wishart matrices, we obtain the moment generating function of the achievable rates. The exact outage probabilities are obtained by employing the inverse Laplace transform. Numerical results demonstrate that our theoretically derived outage probabilities match the simulation results. [ABSTRACT FROM AUTHOR]

The study of the statistical distribution of the eigenvalues of Wishart matrices finds application in many fields of physics and engineering. Here, we consider a special case of finite dimensions correlated complex central Wishart matrices, characterized by the fact that the covariance matrix has all eigenvalues equal, except for one which is the largest. Starting from the knowledge of the joint probability distribution function (p.d.f.) of this kind of Wishart matrices, we focus on the evaluation of a tractable form for the distribution of each individual eigenvalue. In particular, we derive an expression for the p.d.f. of the lthlargest eigenvalue as a sum of terms of the type xβ/e-xδ, which allows us to write a large class of statistical averages involving functions of eigenvalues in closed form. [ABSTRACT FROM AUTHOR]

This study analyzed the channel capacity of multiple-input multiple-output (MIMO) reception by large-scale arrays at both ends in spatially correlated fading environments. The environments considered were Rayleigh fading for representing the non-line-of-sight condition and Nakagami-Rice fading for the line-of-sight condition. The transmission characteristics of massive MIMO were evaluated using the asymptotic eigenvalue distribution of a Wishart matrix composed of the channel matrix. The transmission scheme adopted here was eigen-mode multi-stream transmission that selected L from N streams of N-by-M (M ≥ N) massive MIMO. From the analysis, we determined that the effect of spatial correlation for the case of L/N being less than approximately 20% enhances the channel capacity, unlike spatial diversity operation. In this way, we clarified the effect of spatial correlation on channel capacity of massive MIMO quantitatively through asymptotic eigenvalue analysis. [ABSTRACT FROM AUTHOR]

We propose a new dynamic principal component CAW model (DPC-CAW) for time-series of high-dimensional realized covariance matrices of asset returns (up to 100 assets). The model performs a spectral decomposition of the scale matrix of a central Wishart distribution and assumes independent dynamics for the principal components' variances and the eigenvector processes. A three-step estimation procedure makes the model applicable to high-dimensional covariance matrices. We analyze the finite sample properties of the estimation approach and provide an empirical application to realized covariance matrices for 100 assets. The DPC-CAW model has particularly good forecasting properties and outperforms its competitors for realized covariance matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study the local law for eigenvalues of large random regular bipartite graphs with degree growing moderately fast. We prove that the empirical spectral distribution of the adjacency matrix converges in probability to a scaled down copy of the Marchenko–Pastur distribution on intervals of short length. [ABSTRACT FROM AUTHOR]

This article addresses the problem of estimating a covariance matrix in a multivariate skew-normal distribution relative to two different losses. The estimation problem can be reduced to that of a scale matrix of a noncentral Wishart distribution. The noncentrality parameter matrix, which is a nuisance parameter, brings about non optimality of the best triangular invariant estimators which are minimax under normality. Some improving techniques under normality are proven to remain robust under the multivariate skew-normal distribution. [ABSTRACT FROM AUTHOR]

Graphical models are powerful tools to describe high-dimensional data; they provide a compact graphical representation of the interactions between different variables and such representation enables efficient inference. In particular for Gaussian graphical models, such representation is encoded by the zero pattern of the precision matrix (i.e., inverse covariance). Existing approaches to learning Gaussian graphical models often leverage the framework of penalized likelihood, and therefore suffer from the issue of regularization selection. In this paper, we address the structure learning problem of Gaussian graphical models from a variational Bayesian perspective. Specifically, sparse promoting priors are imposed on the off-diagonal elements of the precision matrix. We then approximate the posterior distribution of the precision matrix by a Wishart distribution using the framework of variational Bayes, and derive efficient natural gradient based algorithms to learn the model. We consider both monoscale and multiscale graphical models. Numerical results show that the proposed method can learn sparse graphs that can reliably describe the data in an automated fashion. [ABSTRACT FROM AUTHOR]

We present a combinatorial approach to the infinitesimal distribution of the Gaussian orthogonal ensemble (goe). In particular, we show how the infinitesimal moments are described by non‐crossing pairings, but not those of type B. We demonstrate the asymptotic infinitesimal freeness of independent complex Wishart matrices and compute their infinitesimal cumulants. Using our combinatorial picture, we compute the infinitesimal cumulants of the goe and demonstrate the lack of asymptotic infinitesimal freeness of independent Gaussian orthogonal ensembles. [ABSTRACT FROM AUTHOR]

A novel unsupervised image classification algorithm which based on the sparse representation theory for polarimetric synthetic aperture radar (PolSAR) image is introduced in this paper. The algorithm conjunctively uses sparse representation-based classification (SRC) theory, dictionary updating method, and label smoothness constraint to update class labels. The unsupervised H/ /A Wishart classification method is introduced to provide the preliminary classification result, from which the initial dictionary and class labels can be extracted. An energy function is defined, and it contains two terms. The first term is based on the sparse representation theory. It reflects the cost of assigning different class labels to a pixel. The second term is label smoothness constraint. It constrains that class labels of neighbouring pixels in flat regions should be the same. By alternately minimizing the energy function, two unknown variables, dictionary and class labels are updated. Optimized class labels are the outputs to compose the final classification result. Extensive experimental results for three PolSAR datasets are analysed to verify the validity of the proposed method. Comparison with other unsupervised/supervised classification methods indicates its superiority. [ABSTRACT FROM AUTHOR]

This paper derives and studies Bayesian Cramér-Rao lower bounds for the mean squared error of covariance matrices that are structured as weighted sums of symmetric positive definite matrices associated with a circularly-symmetric Gaussian statistical model. This model naturally appears in a number of important applications, including multivariate multifractal analysis and vector-valued additive Gaussian processes. As an intermediary result, we derive a novel expression for the expectation of compositions of Wishart random matrices. We provide extensive numerical simulation results for analyzing the derived bounds and their properties, and illustrate their use for the multifractal analysis of bivariate time series. [ABSTRACT FROM AUTHOR]

In this study, the authors consider the secrecy communication of one source–destination pair in a wireless multi-hop relay network, where each node is equipped with multiple antennas. They use the method of interference alignment for secrecy transmission in the presence of an eavesdropper. Firstly, they propose a decode-and-forward (DF) relaying protocol and give a new result on the joint probability density function of the first, second, …, kth (k = 1, 2,…) largest eigenvalues of the complex Wishart matrices HHH, where H is an matrix whose elements are independent identically distributed complex Gaussian variables with zero mean and unit variance. They then utilise this result to characterise the legitimate outage probability and the diversity order of the proposed DF protocol. The case where the relay works in the amplify-and-forward (AF) mode is considered at last, and the diversity order of the proposed AF protocol is given. [ABSTRACT FROM AUTHOR]

Let G be the graph corresponding to the graphical model of nearest neighbor interaction in a Gaussian character. We study Natural Exponential Families (NEF) of Wishart distributions on convex cones QG and PG, where PG is the cone of tridiagonal positive definite real symmetric matrices, and QG is the dual cone of PG. The Wishart NEF that we construct include Wishart distributions considered earlier for models based on decomposable(chordal) graphs. Our approach is, however, different and allows us to study the basic objects of Wishart NEF on the cones QG and PG. We determine Riesz measures generating Wishart exponential families on QG and PG, and we give the quadratic construction of these Riesz measures and exponential families. The mean, inverse-mean, covariance and variance functions, as well as moments of higher order, are studied and their explicit formulas are given. [ABSTRACT FROM AUTHOR]

Approximate symbol error rate (SER), outage probability, and rate expressions are derived for receive diversity system employing optimum combining when both the desired and the interfering signals are subjected to Rician fading, for the cases of equal power uncorrelated interferers, unequal power interferers, and interferer correlation. The derived expressions are applicable for an arbitrary number of receive antennas and interferers and for any quadrature amplitude modulation constellation. Furthermore, we derive a simple closed-form expression for SER in the interference-limited regime, for the special case of Rayleigh faded interferers. A close match is observed between the SER, outage probability, and rate results obtained through the derived analytical expressions and the ones obtained from Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

One of the key issues in the acquisition of sparse data by means of compressed sensing is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper, we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy–Widom distribution. [ABSTRACT FROM AUTHOR]

For colocated multiple-input multiple-output (MIMO) radar, we investigate the target detection problem in Gaussian clutter with unknown covariance matrix with known inverse complex Wishart distribution as its prior probability density function. We propose three detectors in the Bayesian framework according to the criteria of the generalized likelihood ratio test, Rao test, and Wald test. The two main advantages of the proposed Bayesian detectors are as follows: first, no training data are required; and second, a priori knowledge about the clutter is incorporated in the decision rules to achieve detection performance gains. Numerical simulations show that the proposed Bayesian detectors outperform their non-Bayesian counterparts, especially when the sample number of the transmitted waveform is small. In addition, the proposed Bayesian Wald test is the most robust against the mismatch in the receive steering vector, and the proposed Bayesian Rao test exhibits the strongest rejection capability of mismatched signals. [ABSTRACT FROM AUTHOR]

This paper deals with the detection problem of a moving target for the distributed airborne multi-input multi-output radar embedded in the compound-Gaussian and non-homogeneous clutters, without assuming that training data are available. A novel detector combing the Bayesian approach and the generalized likelihood ratio test is proposed, where we model the covariance of clutter as an inverse Wishart distribution with an unknown average clutter covariance matrix (ACCM). More precisely, we regard the unknown ACCM as a structured matrix with the Hadamard product form involving two independent parts, composing of the covariance matrix taper (CMT) and the Doppler spectrum component. Further, the proposed detector exploits a nonlinear processing in an iteration fashion to reconstruct sparse signals with the aim of estimating the unknown spectrum of clutters. As to the CMT component, we resort to generalized CMT model to improve estimation accuracy. Numerical simulations are provided to assess the capability of the proposed detector in different complicated scenarios. [ABSTRACT FROM AUTHOR]

Abstract Spectrum sensing is an initial task for the successful operation of cognitive radio networks (CRN). During cooperative spectrum sensing, malicious secondary user (SU) may report false sensing data which would degrade the final aggregated sensing outcome. In this paper, we propose a distributed cooperative spectrum sensing (CSS) method based on reinforcement learning (RL) to remove data fusion between users with different reputations in CRN. This method regards each SU as an agent, which is selected from the adjacent nodes of CRN participating in CSS. The reputation value is used as reward to ensure that the agent tends to merge with high reputation nodes. The conformance fusion is adopted to promote consensus of the whole network, while it’s also compared with the decision threshold to complete CSS. Simulation results show that the proposed method can identify malicious users effectively. As a result, the whole CRN based on RL is more intelligent and stable. [ABSTRACT FROM AUTHOR]

Wishart matrices are one of the fundamental matrix models in multivariate statistics. The classical Wishart ensemble has been generalized to (m , n , β) -Laguerre ensemble for the values of β > 0. Many properties such as eigenvalue densities, moments and inverse moments of (m , n , β) -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 (m , n , β) -Laguerre matrix. Moreover, we extend our result for inverse compound Wishart matrices for the values of β = 1 and 2. [ABSTRACT FROM AUTHOR]

Use of the Gaussian inverse Wishart probability hypothesis density (GIW-PHD) filter has demonstrated promise as an approach used to track an unknown number of ellipsoidal extended targets. However, based on the assumption that the shape of a target is not ellipsoidal, the performance of the GIW-PHD filter will degrade. This study presents an improved GIW-PHD filter, called a non-ellipsoidal model GIW-PHD (NEM-GIW-PHD) filter, to solve the problem of tracking multiple non-ellipsoidal extended targets. First, an additional target shape estimating approach using B-spline curve is presented for use in the GIW-PHD filter to provide non-ellipsoidal shape parameters. Second, these shape parameters can be used to improve the likelihood function of the GIW-PHD filter. Thus, the identifiability of closely spaced targets with different shapes can be improved. Third, we present an improved shape selection partitioning approach to partition the measurements sets generated by the non-ellipsoidal targets. The simulation results show that the NEM-GIW-PHD filter can achieve higher accuracy than that with the standard GIW-PHD filter in tracking non-ellipsoidal extended targets. In addition, it can provide target shape estimations using the B-spline curve. [ABSTRACT FROM AUTHOR]

This paper proposes an original approach to better understanding the behavior of robust scatter matrix $M$ -estimators. 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. [ABSTRACT FROM AUTHOR]