Wishart Random Matrices Research Articles

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

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

On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices

On the Gaussian product inequality conjecture for disjoint principal minors of Wishart random matrices

Bounds for the estimation of matrix-valued parameters of a Gaussian random process

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.

Matrix Whittaker processes

We study a discrete-time Markov process on triangular arrays of matrices of size dge 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.

Joint DOA tracking and spectral signature estimation of wideband sources using a multi-spectral TBD-CPHD filter

In this paper, a novel method for direction of arrival (DOA) tracking of multiple unknown time-varying number of wideband emitters is presented using a random finite set (RFS) framework. This method combines multiple narrowband covariance-based measurements to form a wideband cardinalized probability hypothesis density (CPHD) filter using the parallel combination (PC) Lemma generally used in multisensor fusion. By doing this, each narrowband frequency bin acts as a separate virtual spectral sensor. The covariance-based measurements, represented by complex Wishart random matrices, also allow the method to have a track-before-detect (TBD) property, which has shown to be advantageous in very low signal-to-noise ratios (SNR). Furthermore, the proposed method can simultaneously monitor the spectrum of the environment and extract the spectral characteristics of the active sources, i.e. can work as some sort of cognitive radio (CR). The suggested algorithm is implemented using an auxiliary particle filter. The simulation results and the comparison with the known state-of-the-art approaches confirm the superiority of the proposed method in negative SNR scenarios and low number of snapshots.

SporTran: A code to estimate transport coefficients from the cepstral analysis of (multivariate) current time series

SporTran is a Python utility designed to estimate generic transport coefficients in extended systems, based on the Green-Kubo theory of linear response and the recently introduced cepstral analysis of the current time series generated by molecular dynamics simulations. SporTran can be applied to univariate as well as multivariate time series. Cepstral analysis requires minimum discretion from the user, in that it weakly depends on two parameters, one of which is automatically estimated by a statistical model-selection criterion that univocally determines the resulting accuracy. In order to facilitate the optimal refinement of these parameters, SporTran features an easy-to-use graphical user interface. A command-line interface and a Python API, easy to embed in complex data-analysis workflows, are also provided. Program summaryProgram Title:SporTranCPC Library link to program files:https://doi.org/10.17632/hm48f8kgj9.1Licensing provisions: GPLv3Programming language: PythonNature of problem: Given an M-variate time series, Jj(t), j=0,…M−1, typically describing a number of currents resulting from a molecular-dynamics simulation, SporTran estimates the transport coefficient κ=1/(Λ−1)00, where Λij=∫0∞〈Jj(t)Jk(0)〉dt is the matrix of the Onsager linear-response coefficients, and 〈⋅〉 indicates an equilibrium average over initial conditions.Solution method:i)It is first observed that the Onsager transport coefficients are the zero-frequency values of the cross power spectra of the currents under scrutiny: Λij=12Sij(ω=0), where Sij(ω)=∫−∞∞〈Ji(t)Jj(0)〉eiωtdt.ii)We next define the (cross) periodogram as the product of pairs of Fourier transforms of the current time series: Skij=ϵNJ˜kiJ˜kj⁎, where ϵ is the time step of the time series, N the number of their terms, and J˜kj=∑n=0N−1Jnje2πiknN their discrete Fourier transforms, and Jnj=Jj(ϵn).iii)As the current time series are realisations of a Gaussian process, in the long-time limit and for k≠k′ the Skij are uncorrelated complex Wishart random matrices (a matrix generalisation of the χ2 distribution) whose expectation, according to the Wiener-Khintchine theorem, is the cross power spectrum we are after. It follows that (Sk−1)00 is proportional to a set of uncorrelated χ2 deviates;iv)A consistent estimator for log⁡(κ)=−log⁡((Λ−1)00) is finally obtained by applying a low-pass filter to the process log⁡((Sk−1)00). The theoretical background of the methodology implemented in SporTran is thoroughly presented in Refs. [1-3].

The partial transpose and asymptotic free independence for Wishart random matrices, II

Using new combinatorial techniques, we significantly improve the previous results on asymptotic distributions and asymptotic free independence relations of partial transposes of Wishart random matrices. In particular, we give a necessary and sufficient condition for the asymptotic free independence of partial transposes of Wishart matrices with difference block sizes.

Extreme-value statistics of stochastic transport processes

We derive exact expressions for the finite-time statistics of extrema (maximum and minimum) of the spatial displacement and the fluctuating entropy flow of biased random walks. Our approach captures key features of extreme events in molecular motor motion along linear filaments. For one-dimensional biased random walks, we derive exact results which tighten bounds for entropy production extrema obtained with martingale theory and reveal a symmetry between the distribution of the maxima and minima of entropy production. Furthermore, we show that the relaxation spectrum of the full generating function, and hence of any moment, of the finite-time extrema distributions can be written in terms of the Marčenko–Pastur distribution of random-matrix theory. Using this result, we obtain efficient estimates for the extreme-value statistics of stochastic transport processes from the eigenvalue distributions of suitable Wishart and Laguerre random matrices. We confirm our results with numerical simulations of stochastic models of molecular motors.

Volume of the set of LOCC-convertible quantum states

The class of quantum operations known as local operations and classical communication (LOCC) induces a partial ordering on quantum states. We present the results of systematic numerical computations related to the volume (with respect to the unitarily invariant measure) of the set of LOCC-convertible bipartite pure states, where the ordering is characterised by an algebraic relation known as majorization. The numerical results, which exploit a tridiagonal model of random matrices, provide quantitative evidence that the proportion of LOCC-convertible pairs vanishes in the limit of large dimensions, and therefore support a previous conjecture by Nielsen. In particular, we show that the problem is equivalent to the persistence of a non-Markovian stochastic process and the proportion of LOCC-convertible pairs decays algebraically with a nontrivial persistence exponent. We extend this analysis by investigating the distribution of the maximal success probability of LOCC-conversions. We show a dichotomy in behaviour between balanced and unbalanced bipartitions. In the latter case the asymptotics is somehow surprising: in the limit of large dimensions, for the overwhelming majority of pairs of states a perfect LOCC-conversion is not possible; nevertheless, for most states there exist local strategies that succeed in achieving the conversion with a probability arbitrarily close to one. We present strong evidence of a universal scaling limit for the maximal probability of successful LOCC-conversions and we suggest a connection with the typical fluctuations of the smallest eigenvalue of Wishart random matrices.

Harmonic means of Wishart random matrices

We use free probability to compute the limiting spectral properties of the harmonic mean of [Formula: see text] i.i.d. Wishart random matrices [Formula: see text] whose limiting aspect ratio is [Formula: see text] when [Formula: see text]. We demonstrate an interesting phenomenon where the harmonic mean [Formula: see text] of the [Formula: see text] Wishart matrices is closer in operator norm to [Formula: see text] than the arithmetic mean [Formula: see text] for small [Formula: see text], after which the arithmetic mean is closer. We also prove some results for the general case where the expectation of the Wishart matrices are not the identity matrix.

Competition of noise and collectivity in global cryptocurrency trading: Route to a self-contained market.

Cross correlations in fluctuations of the daily exchange rates within the basket of the 100 highest-capitalization cryptocurrencies over the period October 1, 2015-March 31, 2019 are studied. The corresponding dynamics predominantly involve one leading eigenvalue of the correlation matrix, while the others largely coincide with those of Wishart random matrices. However, the magnitude of the principal eigenvalue, and thus the degree of collectivity, strongly depends on which cryptocurrency is used as a base. It is largest when the base is the most peripheral cryptocurrency; when more significant ones are taken into consideration, its magnitude systematically decreases, nevertheless preserving a sizable gap with respect to the random bulk, which in turn indicates that the organization of correlations becomes more heterogeneous. This finding provides a criterion for recognizing which currencies or cryptocurrencies play a dominant role in the global cryptomarket. The present study shows that over the period under consideration, the Bitcoin (BTC) predominates, hallmarking exchange rate dynamics at least as influential as the U.S. dollar (USD). Even more, the BTC started dominating around the year 2017, while other cryptocurrencies, such as the Ethereum and even Ripple, assumed similar trends. At the same time, the USD, an original value determinant for the cryptocurrency market, became increasingly disconnected, and its related characteristics eventually started approaching those of a fictitious currency. These results are strong indicators of incipient independence of the global cryptocurrency market, delineating a self-contained trade resembling the Forex.

On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

An Alternative Matrix Skew-Normal Random Matrix and Some Properties

We propose an alternative skew-normal random matrix, which is an extension of the multivariate skew-normal vector parameterized in Vernic (A Stiint Univ Ovidius Constanta. 13, 83–96 2005, Insur. Math. Econ.38, 413–426 2006). We define the density function and then derive and apply the corresponding moment generating function to determine the mean matrix, covariance matrix, and third and fourth moments of the new skew-normal random matrix. Additionally, we derive eight marginal and two conditional density functions and provide necessary and sufficient conditions such that two pairs of sub-matrices are independent. Finally, we derive the moment generating function for a skew-normal random matrix-based quadratic form and show its relationship to the moment generating function of the noncentral Wishart and central Wishart random matrices.

Freeness and The Partial Transposes of Wishart Random Matrices

AbstractWe show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.

Nonnegative definite and Re-nonnegative definite solutions to a system of matrix equations with statistical applications

Necessary and sufficient conditions are given for the existence of a nonnegative definite solution, a Re-nonnegative definite solution, a positive definite solution and a Re-positive definite solution to the system of matrix equationsAXA*=CandBXB*=D,respectively. The expressions for these special solutions are given when the consistent conditions are satisfied. Based on the new results, the characterization of the covariance matrix such that a pair of multivariate quadratic forms are distributed as independent noncentral Wishart random matrices is derived. Many results existing in the literature are extended.

Large-deviation theory for diluted Wishart random matrices

Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology, and economy. In this work, we develop a theory for the eigenvalue fluctuations of diluted Wishart random matrices based on the replica approach of disordered systems. We derive an analytical expression for the cumulant generating function of the number of eigenvalues I_{N}(x) smaller than x∈R^{+}, from which all cumulants of I_{N}(x) and the rate function Ψ_{x}(k) controlling its large-deviation probability Prob[I_{N}(x)=kN]≍e^{-NΨ_{x}(k)} follow. Explicit results for the mean value and the variance of I_{N}(x), its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing very good agreement. The present work establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.104101] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.

Comments on “Cutset Bounds on the Capacity of MIMO Relay Channels”

We comment on the paper “Cutset Bounds on the Capacity of MIMO Relay Channels” by Jeong et al. and point out that, unlike what appears from a remark and some other contents by these authors, the matrix distribution for the sum of two complex random Wishart matrices has already been derived by Kumar for the general case of arbitrary covariance matrices and not only for the special case when one of them is assumed proportional to the identity matrix. The latter assumption has been made only for deriving the corresponding eigenvalue distribution. Furthermore, we draw attention to the result that when all covariance matrices are chosen proportional to the identity matrix, then it is possible to obtain exact and closed form expressions for the sum of an arbitrary number of Wishart matrices and not only for two as considered by Jeong et al.

A Matsumoto–Yor characterization for Kummer and Wishart random matrices

In the paper we resolve positively the conjecture on a characterization of matrix Kummer and Wishart laws through independence property, which was posed in Koudou (2012) [12]. Apart from the probabilistic result, we determine the general solution of the functional equation associated to the characterization problem under weak assumptions.

Tightness of Jensen’s Bounds and Applications to MIMO Communications

Due to the difficulty in manipulating the distribution of Wishart random matrices, the performance analysis of multiple-input-multiple-output (MIMO) channels has mainly focused on deriving capacity bounds via Jensen's inequality. However, to the best of our knowledge, the tightness of Jensen's bounds has not yet been rigorously quantified in the general MIMO context. This paper proposes a new methodology for measuring the tightness of Jensen's bounds via the sandwich theorem. In particular, we first compare the tightness of two different pairs of upper/lower bounds for a general class of MIMO channels based on the unordered eigenvalue of the instantaneous correlation matrix and for arbitrary numbers of antennas. The tightness of Jensen's bounds in different channel scenarios is investigated including multiuser MIMO with maximal ratio combining. Our analysis is facilitated by deriving some new results for finite-dimensional Wishart matrices, i.e., for the arbitrary moments of the unordered eigenvalue of central and non-central Wishart matrices. Our results provide very interesting insights into the implications of the system parameters, such as the number of antennas, and signal-to-noise ratio, on the tightness of Jensen's bounds, and showcase the suitability and limitations of Jensen's bounds.

Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy–Widom distribution

We derive efficient recursive formulas giving the exact distribution of the largest eigenvalue for finite dimensional real Wishart matrices and for the Gaussian Orthogonal Ensemble (GOE). In comparing the exact distribution with the limiting distribution of large random matrices, we also found that the Tracy-Widom law can be approximated by a properly scaled and shifted Gamma distribution, with great accuracy for the values of common interest in statistical applications.