Wishart Model Research Articles

Forecasting high-dimensional realized volatility matrices using a factor model

Modelling and forecasting covariance matrices of asset returns play a crucial role in many financial fields, such as portfolio allocation and asset pricing. The availability of high-frequency intraday data enables the modelling of the realized covariance matrix directly. However, most models in the literature suffer from the curse of dimensionality, i.e. the number of parameters needed increases at the rate of the square of the number of assets. To solve the problem, we propose a factor model with a diagonal Conditional Autoregressive Wishart model for the factor realized covariance matrices. Consequently, the positive definiteness of the estimated covariance matrix is ensured with the proposed model. Asymptotic theory is derived for the estimated parameters. In the extensive empirical analysis, we find that the number of parameters can be reduced significantly; to only about one-tenth of the benchmark model. Furthermore, the proposed model maintains a comparable performance with a benchmark vector autoregressive model for different forecast horizons.

Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations

We review recent progress in modeling credit risk for correlated assets. We employ a new interpretation of the Wishart model for random correlation matrices to model non-stationary effects. We then use the Merton model in which default events and losses are derived from the asset values at maturity. To estimate the time development of the asset values, the stock prices are used, the correlations of which have a strong impact on the loss distribution, particularly on its tails. These correlations are non-stationary, which also influences the tails. We account for the asset fluctuations by averaging over an ensemble of random matrices that models the truly existing set of measured correlation matrices. As a most welcome side effect, this approach drastically reduces the parameter dependence of the loss distribution, allowing us to obtain very explicit results, which show quantitatively that the heavy tails prevail over diversification benefits even for small correlations. We calibrate our random matrix model with market data and show how it is capable of grasping different market situations. Furthermore, we present numerical simulations for concurrent portfolio risks, i.e., for the joint probability densities of losses for two portfolios. For the convenience of the reader, we give an introduction to the Wishart random matrix model.

Asymptotic coincidence of the statistics for degenerate and non-degenerate correlated real Wishart ensembles

The correlated Wishart model provides the standard benchmark when analyzing time series of any kind. Unfortunately, the real case, which is the most relevant one in applications, poses serious challenges for analytical calculations. Often these challenges are due to square root singularities which cannot be handled using common random matrix techniques. We present a new way to tackle this issue. Using supersymmetry, we carry out an anlaytical study which we support by numerical simulations. For large but finite matrix dimensions, we show that statistical properties of the fully correlated real Wishart model generically approach those of a correlated real Wishart model with doubled matrix dimensions and doubly degenerate empirical eigenvalues. This holds for the local and global spectral statistics. With Monte Carlo simulations we show that this is even approximately true for small matrix dimensions. We explicitly investigate the k-point correlation function as well as the distribution of the largest eigenvalue for which we find a surprisingly compact formula in the doubly degenerate case. Moreover we show that on the local scale the k-point correlation function exhibits the sine and the Airy kernel in the bulk and at the soft edges, respectively. We also address the positions and the fluctuations of the possible outliers in the data.

The Generalized Conditional Autoregressive Wishart Model for Multivariate Realized Volatility

It is well known that in finance variances and covariances of asset returns move together over time. Recently, much interest has been aroused by an approach involving the use of the realized covariance (RCOV) matrix constructed from high-frequency returns as the ex-post realization of the covariance matrix of low-frequency returns. For the analysis of dynamics of RCOV matrices, we propose the generalized conditional autoregressive Wishart (GCAW) model. Both the noncentrality matrix and scale matrix of the Wishart distribution are driven by the lagged values of RCOV matrices, and represent two different sources of dynamics, respectively. The GCAW is a generalization of the existing models, and accounts for symmetry and positive definiteness of RCOV matrices without imposing any parametric restriction. Some important properties such as conditional moments, unconditional moments, and stationarity are discussed. Empirical examples including sequences of daily RCOV matrices from the New York Stock Exchange illustrate that our model outperforms the existing models in terms of model fitting and forecasting.

Fully PolSAR image classification using machine learning techniques and reaction-diffusion systems

In this paper, we study the problem of supervised Fully PolSAR (polarimetric synthetic aperture radar) image classification. We estimate a complex Wishart model distribution for each class using training data, and we use such models to design a new classification procedure based on a diffusion-reaction equation. The method relies on simultaneously filtering and classifying pixels within the image. The diffusion term smooths the patches within the image, and the reaction term tends to move the pixel values towards the closest (in the sense of stochastic distances) representative class. We present a detailed study of the method accuracy using both simulated and true data, and we provide optimum parameters for its use. We show that the proposed method outperforms the results obtained using maximum likelihood and usual stochastic distance classification methods.

On the Ergodic Capacity of Distributed MIMO Antenna Systems

We consider distributed multiple-input---multiple-output (MIMO) antenna systems, along with their certain generalizations. We show that distributed MIMO configuration can be mapped to a semicorrelated (one side correlated) Wishart model. For a given set of large-scale fading parameters, associated with the path loss and shadow fading, we derive exact and closed-form results for the marginal density of eigenvalues of $$\mathbf{H}^\dag \mathbf{H}$$H?H (or $$\mathbf{H} \mathbf{H} ^\dag$$HH?), where $$\mathbf{H}$$H is the channel matrix. We also obtain exact and closed-form expressions for the ergodic channel capacity with the aid of Meijer G-function. The ergodic capacity of semicorrelated Rayleigh fading channel follows as a special case. All analytical results are validated by comparison with Monte-Carlo simulations.

Modeling and forecasting realized covariance matrices with accounting for leverage

ABSTRACTThe existing dynamic models for realized covariance matrices do not account for an asymmetry with respect to price directions. We modify the recently proposed conditional autoregressive Wishart (CAW) model to allow for the leverage effect. In the conditional threshold autoregressive Wishart (CTAW) model and its variations the parameters governing each asset's volatility and covolatility dynamics are subject to switches that depend on signs of previous asset returns or previous market returns. We evaluate the predictive ability of the CTAW model and its restricted and extended specifications from both statistical and economic points of view. We find strong evidence that many CTAW specifications have a better in-sample fit and tend to have a better out-of-sample predictive ability than the original CAW model and its modifications.

Eigenvalue density of the doubly correlated Wishart model: exact results

Data sets collected at different times and different observing points can possess correlations at different times and at different positions. The doubly correlated Wishart model takes both into account. We calculate the eigenvalue density of the Wishart correlation matrices using supersymmetry. In the complex case we obtain a new closed form expression which we compare to previous results in the literature. In the more relevant and much more complicated real case we derive an expression for the density in terms of a fourfold integral. Finally, we calculate the density in the limit of large correlation matrices.

Gamma Gaussian Inverse Wishart Probability Hypothesis Density for Extended Target Tracking Using X-Band Marine Radar Data

X-band marine radar systems represent a flexible and low-cost tool for the tracking of multiple targets in a given region of interest. Although suffering several sources of interference, e.g., the sea clutter, these systems can provide high-resolution measurements, both in space and time. Such features offer the opportunity to get accurate information not only about the target position/motion but also about the targets size. Accordingly, in this paper, we exploit emergent extended target tracking (ETT) methodologies in which the target state, typically position/velocity/acceleration, is augmented with the target length and width. In this paper, we propose an ETT procedure based on the popular probability hypothesis density filter, and in particular, we describe the extended target state through the gamma Gaussian inverse Wishart model. The comparative simplicity of the used models allows us to meet the real-time processing constraint required for the practical surveillance purposes. Real-world data from an experimental and operational campaign, collected during the recovery operations of the Costa Concordia wreckage in October 2013, are used to assess the performance of the proposed target tracking methodology. The full signal processing chain is implemented, and considerations of the experimental results are provided. Important nonideal effects, common to every marine radar, are observed and discussed in relation to the assumptions made for the tracking procedure.

Limiting statistics of the largest and smallest eigenvalues in the correlated Wishart model

The correlated Wishart model provides a standard tool for the analysis of correlations in a rich variety of systems. Although much is known for complex correlation matrices, the empirically much more important real case still poses substantial challenges. We put forward a new approach, which maps arbitrary statistical quantities, depending on invariants only, to invariant Hermitian matrix models. For completeness we also include the quaternion case and deal with all three cases in a unified way. As an important application, we study the statistics of the largest eigenvalue and its limiting distributions in the correlated Wishart model, because they help to estimate the behavior of large complex systems. We show that even for fully correlated Wishart ensembles, the Tracy-Widom distribution can be the limiting distribution of the largest as well as the smallest eigenvalue, provided that a certain scaling of the empirical eigenvalues holds.

Spectral domain of large nonsymmetric correlated Wishart matrices.

We study complex eigenvalues of the Wishart model for nonsymmetric correlation matrices. The model is defined for two statistically equivalent but different Gaussian real matrices, as C=AB(t)/T, where B(t) is the transpose of B and both matrices A and B are of dimensions N×T. If A and B are uncorrelated, or equivalently if C vanishes on average, it is known that at large matrix dimension the domain of the eigenvalues of C is a circle centered-at-origin and the eigenvalue density depends only on the radial distances. We consider actual correlation in A and B and derive a result for the contour describing the domain of the bulk of the eigenvalues of C in the limit of large N and T where the ratio N/T is finite. In particular, we show that the eigenvalue domain is sensitive to the correlations. For example, when C is diagonal on average with the same element c≠0, the contour is no longer a circle centered at origin but a shifted ellipse. In this case we explicitly derive a result for the spectral density which again depends only on the radial distances. For more general cases, we show that the contour depends on the symmetric and antisymmetric parts of the correlation matrix resulting from the ensemble-averaged C. If the correlation matrix is normal, then the contour depends only on its spectrum. We also provide numerics to justify our analytics.

The matrix ridge approximation: algorithms and applications

We are concerned with an approximation problem for a symmetric positive semidefinite matrix due to motivation from a class of nonlinear machine learning methods. We discuss an approximation approach that we call matrix ridge approximation. In particular, we define the matrix ridge approximation as an incomplete matrix factorization plus a ridge term. Moreover, we present probabilistic interpretations using a normal latent variable model and a Wishart model for this approximation approach. The idea behind the latent variable model in turn leads us to an efficient EM iterative method for handling the matrix ridge approximation problem. Finally, we illustrate the applications of the approximation approach in multivariate data analysis. Empirical studies in spectral clustering and Gaussian process regression show that the matrix ridge approximation with the EM iteration is potentially useful.

Spectral density of a Wishart model for nonsymmetric correlation matrices

The Wishart model for real symmetric correlation matrices is defined as W=AA^{t}, where matrix A is usually a rectangular Gaussian random matrix and A^{t} is the transpose of A. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices A and B as AB^{t}. The corresponding Wishart model, thus, is defined as C=AB^{t}BA^{t}. We study the spectral density of C for the case when A and B are not statistically independent. The ensemble average of such nonsymmetric matrices, therefore, does not simply vanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describes spectral density of C at large matrix dimension. We complement our analytic results with numerics.

Recovering networks from distance data

A fully probabilistic approach to reconstructing Gaussian graphical models from distance data is presented. The main idea is to extend the usual central Wishart model in traditional methods to using a likelihood depending only on pairwise distances, thus being independent of geometric assumptions about the underlying Euclidean space. This extension has two advantages: the model becomes invariant against potential bias terms in the measurements, and can be used in situations which on input use a kernel- or distance matrix, without requiring direct access to the underlying vectors. The latter aspect opens up a huge new application field for Gaussian graphical models, as network reconstruction is now possible from any Mercer kernel, be it on graphs, strings, probabilities or more complex objects. We combine this likelihood with a suitable prior to enable Bayesian network inference. We present an efficient MCMC sampler for this model and discuss the estimation of module networks. Experiments depict the high quality and usefulness of the inferred networks.

Speckle reduction in polarimetric SAR imagery with stochastic distances and nonlocal means

This paper presents a technique for reducing speckle in Polarimetric Synthetic Aperture Radar (PolSAR) imagery using nonlocal means and a statistical test based on stochastic divergences. The main objective is to select homogeneous pixels in the filtering area through statistical tests between distributions. This proposal uses the complex Wishart model to describe PolSAR data, but the technique can be extended to other models. The weights of the location-variant linear filter are function of the p-values of tests which verify the hypothesis that two samples come from the same distribution and, therefore, can be used to compute a local mean. The test stems from the family of (h–ϕ) divergences which originated in Information Theory. This novel technique was compared with the Boxcar, Refined Lee and IDAN filters. Image quality assessment methods on simulated and real data are employed to validate the performance of this approach. We show that the proposed filter also enhances the polarimetric entropy and preserves the scattering information of the targets.

Classification of Segments in PolSAR Imagery by Minimum Stochastic Distances Between Wishart Distributions

A new classifier for Polarimetric SAR (PolSAR) images is proposed and assessed in this paper. Its input consists of segments, and each one is assigned the class which minimizes a stochastic distance. Assuming the complex Wishart model, several stochastic distances are obtained from the h-phi family of divergences, and they are employed to derive hypothesis test statistics that are also used in the classification process. This article also presents, as a novelty, analytic expressions for the test statistics based on the following stochastic distances between complex Wishart models: Kullback-Leibler, Bhattacharyya, Hellinger, R\'enyi, and Chi-Square; also, the test statistic based on the Bhattacharyya distance between multivariate Gaussian distributions is presented. The classifier performance is evaluated using simulated and real PolSAR data. The simulated data are based on the complex Wishart model, aiming at the analysis of the proposal well controlled data. The real data refer to the complex L-band image, acquired during the 1994 SIR-C mission. The results of the proposed classifier are compared with those obtained by a Wishart per-pixel/contextual classifier, and we show the better performance of the region-based classification. The influence of the statistical modeling is assessed by comparing the results using the Bhattacharyya distance between multivariate Gaussian distributions for amplitude data. The results with simulated data indicate that the proposed classification method has a very good performance when the data follow the Wishart model. The proposed classifier also performs better than the per-pixel/contextual classifier and the Bhattacharyya Gaussian distance using SIR-C PolSAR data.

Multitemporal polarimetric RADARSAT-2 SAR data for urban land cover mapping through a dictionary-based and a rule-based model selection in a contextual SEM algorithm

This paper presents a dictionary- and rule-based model selection approach in an adaptive contextual semi-supervised algorithm for improving urban land cover classification using high-resolution multitemporal RADARSAT-2 polarimetric SAR (PolSAR) data. Six-date PolSAR data were acquired from June to September, 2008, over the Greater Toronto Area. Contextual information and the capabilities of different PolSAR distribution models were explored by the spatially variant Finite Mixture Model (FMM) with an adaptive Markov Random Field (MRF) in a Stochastic Expectation–Maximization (SEM) algorithm. This algorithm can obtain homogenous results while preserving shape details in the complex urban environment with high accuracy. Commonly used PolSAR distribution models such as Wishart, G0p, Kp, and KummerU were compared through the proposed approaches for urban land cover mapping. According to a Goodness-of-Fit test based on Mellin transformation, an accurate PolSAR distribution model could be selected with the dictionary-based classification. However, the results showed that improvement from the dictionary-based approach was limited. Therefore, further improvements were expected by exploring expert knowledge. The initial results showed that G0p and KummerU performed better for distinguishing between low density built-up areas and forest. G0p, Kp, and KummerU are better for the low scattering classes. The Wishart model has superior capacity in separating high density built-up areas and the adjacent roads. Based on such knowledge, a set of rules was developed to integrate the advantages of alternative models. Significant improvement on the overall classification accuracy could be observed by this rule-based approach. The biggest improvement was achieved using the HD–Road rule on the G0p model with the best overall classification accuracy at 89.99% (kappa: 0.87). This represented 4.1% (kappa: 0.045) improvement over that of G0p without model selection.

Simple matrix models for random Bergman metrics

Recently, the authors have proposed a new approach to the theory of random metrics,making an explicit link between probability measures on the space of metrics ona Kähler manifold and random matrix models. We consider simple examples ofsuch models and study the one- and two-point functions of the metric. Thesegeometric correlation functions correspond to new interesting types of matrix modelcorrelators. We provide in particular a detailed study of the Wishart model, where wedetermine the correlation functions explicitly. We find that the random measure inthis model turns out to be concentrated on the background metric in the largeN limit.

An Adaptive Contextual SEM Algorithm for Urban Land Cover Mapping Using Multitemporal High-Resolution Polarimetric SAR Data

This paper presents a semi-supervised Stochastic Expectation-Maximization (SEM) algorithm for detailed urban land cover mapping using multitemporal high-resolution polarimetric SAR (PolSAR) data. By applying an adaptive Markov Random Field (MRF) with the spatially variant Finite Mixture Model (SVFMM), spatial-temporal contextual information could be effectively explored to improve the mapping accuracy with homogenous results and preserved shape details. Further, a learning control scheme was proposed to ensure a robust semi-supervised mapping process thus avoiding the undesired class merges. Four-date RADARSAT-2 polarimetric SAR data over the Greater Toronto Area were used to evaluate the proposed method. Common PolSAR distribution models such as Wishart, G0p, Kp and KummerU were compared through this contextual SEM algorithm for detailed urban land cover mapping. Comparisons with Support Vector Machine (SVM) were also conducted to assess the potential of our parametric approach. The results show that the Kp, G0p and KummerU models could generate better urban land cover mapping results than the Wishart model and SVM.

The conditional autoregressive Wishart model for multivariate stock market volatility

We propose a Conditional Autoregressive Wishart (CAW) model for the analysis of realized covariance matrices of asset returns. Our model assumes an autoregressive moving average structure for the scale matrix of the Wishart distribution. It accounts for positive definiteness of covariance matrices without imposing parametric restrictions, and can be estimated by Maximum Likelihood. We also propose extensions of the CAW model obtained by including a Mixed Data Sampling (MIDAS) component and Heterogeneous Autoregressive (HAR) dynamics for long-run fluctuations. The CAW models are applied to realized variances and covariances for five New York Stock Exchange stocks.