Wishart Distribution Research Articles

Wishart planted ensemble: A tunably rugged pairwise Ising model with a first-order phase transition.

We propose the Wishart planted ensemble, a class of zero-field Ising models with tunable algorithmic hardness and specifiable (or planted) ground state. The problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign-inverted variant of the Hopfield model. The Hamiltonian contains only 2-spin interactions, with the coupler matrix following a type of Wishart distribution. The class exhibits a classical first-order phase transition in temperature. For some parameter settings the model has a locally stable paramagnetic state, a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape. We analytically probe the ensemble thermodynamic properties by deriving the Thouless-Anderson-Palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis; extensive Monte Carlo simulations confirm our predictions of the first-order transition temperature. The class exhibits a wide variation in algorithmic hardness as a generation parameter is varied, with a pronounced easy-hard-easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes. By deriving the ensemble-averaged energy distribution and taking into account finite-precision representation, we propose an analytical expression for the location of the hardness peak and show that at fixed precision, the number of constraints in the integer program must increase with system size to yield truly hard problems. The Wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms.

Channel Capacity of Massive MIMO With Selected Multi-Stream Transmission in Spatially Correlated Fading Environments

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.

Moments and Identities Involving Inverted Wishart Distribution

Moments of multivariate Wishart Distribution are known up to fourth order. But in many contexts, moments of some functions of Wishart distribution and Inverted Wishart Distribution have been found useful in risk theoretic estimation of covariance matrix and its characteristics. In this paper we review moments of some important functions of Wishart and Inverted Distributions.

Multilook Polarimetric SAR Change Detection Using Stochastic Distances Between Matrix-Variate Gd 0 Distributions

In this article, we propose an efficient heterogeneous change detection algorithm based on stochastic distance measure between two $\mathcal {G}_{d}^{0}$ distributions. Due to its flexibility and simplicity, the matrix-variate $\mathcal {G}_{d}^{0}$ distribution has been successfully used to model the multilook polarimetric synthetic aperture radar (PolSAR) data and has been tested for classification, segmentation, and image analysis. Concretely, closed-form expressions for the Kullback–Leibler, Renyi of order $\beta $ , Bhattacharyya, and Hellinger distances are provided to compute the stochastic distance between $\mathcal {G}_{d}^{0}$ distributions. In this context, we resort to the expectation–maximization (EM) to estimate accurately with low complexity for the parameters of the probability distribution of the two multilook polarimetric covariance matrices to be compared. Finally, the performance of the method is compared firstly to the performance of other known distributions, such as the scaled complex Wishart distribution, and secondly to other known statistical tests using simulated and real multilook PolSAR data.

On the mean and dispersion of the Moore-Penrose generalized inverse of a Wishart matrix

The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. The article complements the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart matrix are close to the degrees of freedom of the distribution.

Polarimetric Stationarity Omnibus Test (PSOT) for Selecting Persistent Scatterer Candidates with Quad-Polarimetric SAR Datasets.

In the traditional single polarimetric persistent scatterers interferometric (PSI) technology, the amplitude dispersion index (ADI) is usually used to select persistent scatterer candidates (PSC). Obviously, based on single polarimetric information, it is difficult to use the statistical characteristics for comprehensively describing the temporal stability of scatterers, which leads to a decrease in persistent scatterer (PS) density. Considering that the temporal polarimetric stationarity of PS, the paper is based on complex Wishart distribution and proposes the polarimetric stationarity omnibus test (PSOT) for identifying PSC. The nonstationary pixels can be removed by the preset significance threshold, which reduces the subsequent processing error and the calculation cost. Then, the exhaustive search polarimetric optimization (ESPO) method is selected for improving the phase quality of PSCs while suppressing the sidelobe of the strong scatterer effectively. For validating the effectiveness of the proposed method, we select a time-series quad-polarimetric ALOS PALSAR-1 images in an urban area as experimental data and mainly perform five group experiments for detailed analysis, including the PSOT+ESPO, ADI+ESPO, ADI+HH, ADI+HV, and ADI+VV. The results show that the proposed PSOT+ESPO method has a better performance on both PSC selection and interferometric phase optimization aspects than that of other methods. Specifically, compared to the last four methods, both the PSCs and PSs identified by the proposed PSOT+ESPO are more concentrated in the high-coherence region. The PSs with the standard deviation (STD) less than 5mm in the PSOT+ESPO method account for 94% of all PSs, which is greater than that of the ADI+ESPO, ADI+HH, ADI+HV, and ADI+VV methods, respectively.

A DCC-Type Approach for Realized Covariance Modelling With Score-Driven Dynamics

We propose a class of score-driven realized covariance models where volatilities and correlations are separately estimated. We can thus combine univariate realized volatility models with a recently introduced class of score-driven realized covariance models based on Wishart and matrix-F distributions. The proposed models are computationally simple to estimate in high dimensions and allow complete flexibility in the choice of the univariate specification. Through a Monte-Carlo study, we show that the two-step maximum likelihood procedure provides accurate parameter estimates in small samples. Empirically, we find that the proposed models outperform joint estimations, with forecasting gains that become more significant as dimension increases.

Coastline detection in polarimetric SAR images using Markov random field segmentation based on mixture Wishart distribution

Coastline detection in polarimetric SAR images using Markov random field segmentation based on mixture Wishart distribution

On the product of a singular Wishart matrix and a singular Gaussian vector in high dimension

In this paper we consider the product of a singular Wishart random matrix and a singular normal random vector. A very useful stochastic representation is derived for this product, in using which the characteristic function of the product and its asymptotic distribution under the double asymptotic regime are established. The application of obtained stochastic representation speeds up the simulation studies where the product of a singular Wishart random matrix and a singular normal random vector is present. We further document a good performance of the derived asymptotic distribution within a numerical illustration. Finally, several important properties of the singular Wishart distribution are provided.

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.

The Jacobians of matrix transformation about singular random matrices and its applications

In this paper, we give the Jacobians of matrix transformations and when the constant matrices A and are singular rectangular matrices, not limited to square matrices, and the random variables in the random matrices X, Y can be linearly dependent, W and are singular random matrices. Furthermore, some important properties about singular multivariate Normal distribution and singular Wishart distribution are proved by using these Jacobians. Finally, the density function of and the distribution of generalized Hotelling’s T 2 statistic are obtained.

Dynamic principal component CAW models for high-dimensional realized covariance matrices

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.

Improved generalized labeled multi-Bernoulli filter for non-ellipsoidal extended targets or group targets tracking based on random sub-matrices

For non-ellipsoidal extended targets and group targets tracking, using an ellipsoid to approximate the target extension may not be accurate enough because of lacking shape and orientation information. In consideration of this, we model a non-ellipsoidal extended target or target group as a combination of multiple ellipsoidal sub-objects, each represented by a random matrix. Besides, in practical tracking scenes, the knowledge of detection probability is of critical importance, which, however, is prior unknown due to the fluctuation of target parameters. Thus based on the aforementioned random sub-matrices model, an improved generalized labeled multi-Bernoulli (GLMB) filter is presented for the tracking of non-ellipsoidal extended targets or group targets in the case of unknown detection probability. In addition, a cheaper Gibbs sampling based joint prediction and updating implementation of the proposed GLMB filter is given, within which the extended targets or group targets are modeled using Beta gamma Gaussian inverse Wishart (BGGIW) distributions, achieving the estimation to detection probability, measurement rate, and extension state of each sub-object and the kinematic state with the positions of all sub-objects and unified kinematics for each extended target or target group. The simulation results demonstrate the effectiveness of the proposed filter.

On the Letac-Massam conjecture and existence of high dimensional Bayes estimators for graphical models

The Wishart distribution defined on the open cone of positive-definite matrices plays a central role in multivariate analysis and multivariate distribution theory. Its domain of parameters is often referred to as the Gindikin set. In recent years, varieties of useful extensions of the Wishart distribution have been proposed in the literature for the purposes of studying Markov random fields and graphical models. In particular, generalizations of the Wishart distribution, referred to as Type I and Type II (graphical) Wishart distributions introduced by Letac and Massam in Annals of Statistics (2007) play important roles in both frequentist and Bayesian inference for Gaussian graphical models. These distributions have been especially useful in high-dimensional settings due to the flexibility offered by their multiple-shape parameters. Concerning Type I and Type II Wishart distributions, a conjecture of Letac and Massam concerns the domain of multiple-shape parameters of these distributions. The conjecture also has implications for the existence of Bayes estimators corresponding to these high dimensional priors. The conjecture, which was first posed in the Annals of Statistics, has now been an open problem for about 10 years. In this paper, we give a necessary condition for the Letac and Massam conjecture to hold. More precisely, we prove that if the Letac and Massam conjecture holds on a decomposable graph, then no two separators of the graph can be nested within each other. For this, we analyze Type I and Type II Wishart distributions on appropriate Markov equivalent perfect DAG models and succeed in deriving the aforementioned necessary condition. This condition in particular identifies a class of counterexamples to the conjecture.

Robust Poisson Multi-Bernoulli Mixture Filter With Inaccurate Process and Measurement Noise Covariances

This paper proposes a robust Poisson multi-Bernoulli mixture (PMBM) filter with inaccurate process and measurement noise covariances. A derivation of the robust PMBM filter is provided for jointly estimating the kinematic state, the predicted state covariance, and the measurement noise covariance. By modeling the augmented state as a Gaussian inverse Wishart inverse Wishart (GIWIW) distribution, a computationally feasible implementation of the robust PMBM (GIWIW-PMBM) filter is given for linear Gaussian systems. To guarantee the conjugacy of the GIWIW distribution, the variational Bayesian (VB) approach is employed to approximate the posterior density. Finally, simulation results show that the GIWIW-PMBM filter has the best overall performance compared to existing state-of-the-art filters regarding computational cost and filtering performance.

Block-modified Wishart matrices: the easy case

Associated to any complex Wishart matrix $W$ of parameters $(dn,dm)$ and any linear map $\varphi:M_n(\mathbb C)\to M_n(\mathbb C)$ is the block-modified matrix $\tilde{W}=(id\otimes\varphi)W$. Following some previous work with Nechita, we study here the asymptotic $*$-distribution of $\tilde{W}$, in the $d\to\infty$ limit, in the case where the modification map $\varphi$ is easy, or more generally super-easy, in the quantum algebra/representation theory sense. Under suitable assumptions on $\varphi$ we obtain in this way a compound free Poisson law.

A Closed-Form Prediction Update for Extended Target Tracking Using Random Matrices

This paper proposes a new class of state transition models that afford closed-form predictions for the tracking of extended targets. A key innovation is to employ a non-central inverse Wishart distribution to model the state transition density of the target extent. Importantly, this results in a simplified prediction update that is computationally efficient and improves target tracking performance when compared to state-of-the-art alternatives on standard simulation scenarios.

A Generalized Labelled Multi-Bernoulli Filter for Extended Targets With Unknown Clutter Rate and Detection Profile

A prior knowledge of the background parameters such as clutter rate and detection profile is of critical importance in the tracking algorithms under the theory of random finite sets for extended objects which would lead to restrictions in the application. To accommodate this problem, a multiple extended target tracking algorithm based on the generalized labelled multi-Bernoulli (GLMB) filter under the circumstance of unknown clutter rate and detection profile is proposed in this article. After introducing a clutter generator, this new algorithm establishes augmented state space model for targets and clutter and propagates them in parallel by applying multi-class GLMB theory. We then employ Beta to describe detection probability. Target extension is modelled as an ellipse by using gamma Gaussian inverse Wishart distribution. Simulation results indicate that the proposed algorithm has better performance in estimating trajectories and extended shapes compared with the conventional filter having prior knowledge.

Robust Gaussian Approximate Fixed-Interval Smoother With Outlier Detection

We consider that existing nonlinear state estimators cannot always yield a satisfactory performance when the distribution of the measurement noise deviates from the assumed Gaussian distribution. This letter investigates the non-linear robust fixed-interval smoothing problem where the measurement vectors are partially disturbed by outliers. The proposed smoother employs a binary indicator variable modelled with a beta-Bernoulli distribution to determine whether the measurement vector is an outlier. Moreover, to address the inaccurate process noise covariance matrix, an inverse Wishart distribution was chosen as the conjugate prior of the process noise covariance matrix. Then, the variational Bayesian inference method was applied to estimate the state trajectory, inaccurate process noise covariance matrix, and a set of binary indicator variables. Traditional target tracking simulation results indicate the effectiveness and stability of the derived smoother, and the proposed smoother has demonstrated a marked improvement in estimation accuracy and robustness.

Explaining variability in tourist preferences: A Bayesian model well suited to small samples

Abstract Discrete choice experiments are becoming more popular in the tourism and travel literature. While Bayesian methods to analyze discrete choice experiment data have been used in other disciplines, they have not been used in the tourism literature. In this article, we develop a Bayesian Mixed Logit Model in which we use a little known prior distribution developed by Lewandowski, Kurowicka, and Joe (LKJ) and half Cauchy distributions as an alternative to the more traditionally used inverse Wishart distribution as a prior scheme for the covariance matrix of random parameters in mixed logit estimation. Using multiple simulated data sets, we show that use of the LKJ prior scheme improves the estimation of coefficients, especially for small data sets. Finally, we test the model with an actual small discrete choice data set examining tourist preferences for reducing glacier recession, and discuss the implications of the model for research and policy.