Multivariate Student-t Distribution Research Articles

Consistency factor for the MCD estimator at the Student-t distribution

It is well known that trimmed estimators of multivariate scatter, such as the Minimum Covariance Determinant (MCD) estimator, are inconsistent unless an appropriate factor is applied to them in order to take the effect of trimming into account. This factor is widely recommended and applied when uncontaminated data are assumed to come from a multivariate normal model. We address the problem of computing a consistency factor for the MCD estimator in a heavy-tail scenario, when uncontaminated data come from a multivariate Student-t distribution. We derive a remarkably simple computational formula for the appropriate factor and show that it reduces to an even simpler analytic expression in the bivariate case. Exploiting our formula, we then develop a robust Monte Carlo procedure for estimating the usually unknown number of degrees of freedom of the assumed and possibly contaminated multivariate Student-t model, which is a necessary ingredient for obtaining the required consistency factor. Finally, we provide substantial simulation evidence about the proposed procedure and apply it to data from image processing and financial markets.

Prediction Distribution for Simultaneous Auto-Regressive Model with Multivariate Student-T Error under the Bayesian Approach

Prediction distribution is a basis for predictive inferences applied in many real world situations. The Bayesian approach under uniform prior is employed in this paper to derive the prediction distribution for Simultaneous Auto-regressive model with multivariate Student-t error distribution. Conditional on a set of realized responses, a single and a set of future responses have a univariate and multivariate Student-t distributions respectively, whose degrees of freedom depend on the size of the realized sample and the dimension of the auto-regression parameters' vector but do not depend on the degrees of freedom of the error distribution. Results are identical to those obtained under normal error distribution by a range of statistical approaches such as the normal distribution, autocorrelations and classical methods. This indicates not only the inference robustness with respect to departures from normal error to multivariate Student-t error distributions, but also indicates that the Bayesian approach with uniform prior is competitive with other statistical methods in the derivation of prediction distribution.

Multivariate Volatility Modeling of Nigerian Bank Share Prices

This study aims at finding the optimal Multivariate Generalized Autoregressive Conditional Heteroscedasticity (MGARCH) model among Diag-BEKK, Scalar-BEKK and CCC that captures the dynamics of returns in Nigerian bank share prices, using the data of daily share prices of two highly capitalized banks in Nigeria listed on the platform of Nigerian Stock Exchange (NSE) which span from 2nd January, 2009 to 28th December, 2019. Multivariate Normal and Multivariate student-t log-likelihood functions were simplified using the BHHH and Marquardt algorithm and the optimal solution was obtained using the information criteria. The BHHH and Marquardt algorithm was implemented in the GARCH 7 software of Laurent. Finally, the findings of this study showed that the Diag-Bekk (1,1) with Multivariate student-t distribution was the overall best model out of the six model combinations. Multivariate volatility modeling is therefore, recommended for bank share prices in Nigeria, and even in other banks from emerging economies.

Score-Driven Modeling of Spatio-Temporal Data

A simultaneous autoregressive score-driven model with autoregressive disturbances is developed for spatio-temporal data that may exhibit heavy tails. The model specification rests on a signal plus noise decomposition of a spatially filtered process, where the signal can be approximated by a nonlinear function of the past variables and a set of explanatory variables, while the noise follows a multivariate Student-t distribution. The key feature of the model is that the dynamics of the space-time varying signal are driven by the score of the conditional likelihood function. When the distribution is heavy-tailed, the score provides a robust update of the space-time varying location. Consistency and asymptotic normality of maximum likelihood estimators are derived along with the stochastic properties of the model. The motivating application of the proposed model comes from brain scans recorded through functional magnetic resonance imaging when subjects are at rest and not expected to react to any controlled stimulus. We identify spontaneous activations in brain regions as extreme values of a possibly heavy-tailed distribution, by accounting for spatial and temporal dependence.

Expressions for joint moments of elliptical distributions

Inspired by Stein’s lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive E[X12f(X)] for any measurable function f satisfying some regularity conditions. Then, by making use of this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of E[X12f(X)] for multivariate Student-t, logistic and Laplace distributions.

On moments of folded and truncated multivariate Student-t distributions based on recurrence relations

The use of the first two moments of the truncated multivariate Student-t distribution has attracted increasing attention from a wide range of applications. This paper develops recurrence relations for integrals that involve the density of multivariate Student-t distributions. The proposed techniques allow for fast computation of arbitrary-order product moments of folded and truncated multivariate Student-t distributions and offer explicit expressions of low-order moments of folded and truncated multivariate Student-t distributions. A real data example containing positive censored responses is applied to illustrate the effectiveness and importance of the proposed methods. An R MomTrunc package is developed and publicly available on the CRAN repository.

An Information-Theoretic Approach for Multivariate Skew-t Distributions and Applications

Shannon and Rényi entropies are two important measures of uncertainty for data analysis. These entropies have been studied for multivariate Student-t and skew-normal distributions. In this paper, we extend the Rényi entropy to multivariate skew-t and finite mixture of multivariate skew-t (FMST) distributions. This class of flexible distributions allows handling asymmetry and tail weight behavior simultaneously. We find upper and lower bounds of Rényi entropy for these families. Numerical simulations illustrate the results for several scenarios: symmetry/asymmetry and light/heavy-tails. Finally, we present applications of our findings to a swordfish length-weight dataset to illustrate the behavior of entropies of the FMST distribution. Comparisons with the counterparts—the finite mixture of multivariate skew-normal and normal distributions—are also presented.

Robust Generalized Labeled Multi-Bernoulli Filter for Multitarget Tracking With Unknown Non- Stationary Heavy-Tailed Measurement Noise

A robust generalized labeled multi-Bernoulli (GLMB) filter is presented to perform multitarget tracking (MTT) with unknown non-stationary heavy-tailed measurement noise (HTMN). The HTMN is modeled as a multivariate Student's t-distribution with unknown and time-varying mean. The proposed filter relaxes the restrictive assumption that the mean of HTMN is zero, and can effectively deal with MTT under the condition that the mean of HTMN is unknown and time-varying. The variational Bayesian (VB) approximation is applied in the GLMB filtering framework with the augmented state. The marginal likelihood function is obtained via minimizing the Kullback-Leibler divergence by the variational lower bound. The simulation results demonstrate that the proposed filter can effectively track multiple targets in both linear and nonlinear scenarios when the mean of HTMN is unknown and time-varying.

Towards a Deeper Understanding of Yield Curve Movements

From a modeling perspective, daily movements in risk-free yield curves deserve further exploration. In this paper, I study daily yield curve changes for three major liquid sovereign yield curves, namely US, UK and Germany. I focus on two main themes: (1) characterizing the statistical distribution of the multivariate vector of daily yield curve changes, and (2) exploring efficiency of deep learning based dimensionality reduction approaches for the daily yield curve changes. Specifically, I am able to reject the null hypothesis that daily yield curve changes are normally distributed. I also show that a multivariate student-t distribution with 2 degrees of freedom or a gaussian mixture model are more accurate at capturing the leptokurtic distribution of daily yield curve changes. In addition, on the dimensionality reduction front, I show that a deep neural network auto-encoder outperforms a PCA based linear decomposition approach.

A Probabilistic Distance Clustering Algorithm Using Gaussian and Student-t Multivariate Density Distributions

A new dissimilarity measure for cluster analysis is presented and used in the context of probabilistic distance (PD) clustering. The basic assumption of PD-clustering is that for each unit, the product between the probability of the unit belonging to a cluster and the distance between the unit and the cluster is constant. This constant is a measure of the classifiability of the point, and the sum of the constant over units is called joint distance function (JDF). The parameters that minimize the JDF maximize the classifiability of the units. The new dissimilarity measure is based on the use of symmetric density functions and allows the method to find clusters characterized by different variances and correlation among variables. The multivariate Gaussian and the multivariate Student-t distributions have been used, outperforming classical PD clustering, and its variation PD clustering adjusted for cluster size, on simulated and real datasets.

Bayesian estimation of a multivariate TAR model when the noise process follows a Student-t distribution

In this paper, we introduce a Bayesian methodology for the estimation of non-structural parameters (autoregressive matrices, covariance matrices and degrees of freedom) of a multivariate TAR model (MTAR) when noise process follows a multivariate Student-t distribution. For this, the use of non-informative prior distributions is proposed to obtain the full conditional distributions. MCMC methods are used to obtain samples of such distributions. The performance of the estimation is evaluated by means simulations. Finally, the model is applied to the returns data from the Bovespa, Colcap and Standard and Poor indexes.

Adaptive importance sampling for estimating multi-component chirp signal parameters in colored noise

The problem of estimating the center frequency and the chirp rate of multi-component chirp signals from data corrupted with colored noise often occurs in many signal processing applications. In the Bayesian framework, an algorithm based on adaptive importance sampling (AIS) and modified discrete chirp Fourier transform (MDCFT) was introduced in the present paper in order to calculate the center frequency and the chirp rate of chirp signals from observations corrupted by additive autoregressive (AR) random noise. First, importance functions (IFs) were constructed to approximate the marginal posterior distributions of interested parameters with observed data, and AIS was then employed to evaluate the posterior distributions. MDCFT was adopted to detect the spectral peaks of chirp signals, and the obtained results were applied to construct IFs with multivariate student-t distribution. It was found that the quality of the samples generated from multivariate student-t distribution was gradually improved. Finally, computer simulations validated the results obtained from the proposed algorithm. In comparison with the Markov chain-based method, the proposed method had better SNR adaptability and excellent convergence rate and required fewer samples. Furthermore, the proposed algorithm can be considered as a further expansion of previous works related to AIS method for estimating multi-component chirp signal parameters in white noise.

On hysteretic vector autoregressive model with applications

ABSTRACTThis paper proposes a new hysteretic vector autoregressive (HVAR) model in which the regime switching may be delayed when the hysteresis variable lies in a hysteresis zone. We integrate an adapted multivariate Student-t distribution from amending the scale mixtures of normal distributions. This HVAR model allows for a higher degree of flexibility in the degrees of freedom for each time series. We use the proposed model to test for a causal relationship between any two target time series. Using posterior odds ratios, we overcome the limitations of the classical approach to multiple testing. Both simulated and real examples herein help illustrate the suggested methods. We apply the proposed HVAR model to investigate the causal relationship between the quarterly growth rates of gross domestic product of United Kingdom and United States. Moreover, we check the pairwise lagged dependence of daily PM2.5 levels in three districts of Taipei.

Student-t mixture labeled multi-Bernoulli filter for multi-target tracking with heavy-tailed noise

Abstract A new labeled multi-Bernoulli (LMB) filter is proposed for multi-target tracking with joint heavy-tailed noises of the state and measurement. In contrast to the Gaussian assumption, the proposed method models both the process and measurement noises as multivariate Student-t distributions to handle the heavy-tailed noises. A closed form recursion of the LMB filter to propagate the parameters of Student-t mixture components is derived based on the multi-target Student-t models. Some approximations are applied to make the filter available in practice. The gating technique is also developed for the proposed method. Furthermore, a strategy of managing the number of Student-t components is introduced here to ensure the efficiency of the proposed method. Simulations with joint heavy-tailed noises of the state and measurement are performed to assess the proposed filter, and results demonstrate effectiveness of the new LMB filter.

Brain MR image segmentation based on an improved active contour model.

It is often a difficult task to accurately segment brain magnetic resonance (MR) images with intensity in-homogeneity and noise. This paper introduces a novel level set method for simultaneous brain MR image segmentation and intensity inhomogeneity correction. To reduce the effect of noise, novel anisotropic spatial information, which can preserve more details of edges and corners, is proposed by incorporating the inner relationships among the neighbor pixels. Then the proposed energy function uses the multivariate Student's t-distribution to fit the distribution of the intensities of each tissue. Furthermore, the proposed model utilizes Hidden Markov random fields to model the spatial correlation between neigh-boring pixels/voxels. The means of the multivariate Student's t-distribution can be adaptively estimated by multiplying a bias field to reduce the effect of intensity inhomogeneity. In the end, we reconstructed the energy function to be convex and calculated it by using the Split Bregman method, which allows our framework for random initialization, thereby allowing fully automated applications. Our method can obtain the final result in less than 1 second for 2D image with size 256 × 256 and less than 300 seconds for 3D image with size 256 × 256 × 171. The proposed method was compared to other state-of-the-art segmentation methods using both synthetic and clinical brain MR images and increased the accuracies of the results more than 3%.

Bayesian analysis of working capital management on corporate profitability: evidence from India

PurposeDecisions pertaining to working capital management have pivotal role for firms’ short-term financial decisions. The purpose of this paper is to examine impact of working capital on profitability for Indian corporate entities.Design/methodology/approachBoth classical panel analysis and Bayesian techniques have been employed that provides opportunity not only to perform comparative analysis but also allows flexibility in prior distribution assumptions.FindingsIt is found that longer cash conversion period has detrimental influence on profitability. Financial soundness indicators are playing significant role in determining firm profitability. Larger firms seem to be more profitable and significant as per Bayesian approach. Bayesian approach has led to considerable gain in estimation fit.Practical implicationsObserving the highly skewed distribution of dependent variable, Multivariate Student t-distribution has been considered along with normal distribution to model stochastic term. Accordingly, Bayesian methodology is applied.Originality/valueAnalysis of working capital for firms has been performed in Indian context. Application of Bayesian methodology is performed on balanced panel spanning from 2003 to 2012. As per author’s knowledge, this is the first study which applies Bayesian approach employing panel data for the analysis of working capital management for Indian firms.

Link sign prediction by Variational Bayesian Probabilistic Matrix Factorization with Student-t Prior

In signed social networks, link sign prediction refers to using the observed link signs to infer the signs of the remaining links, which is important for mining and analyzing the evolution of social networks. The widely used matrix factorization-based approach – Bayesian Probabilistic Matrix Factorization (BMF), assumes that the noise between the real and predicted entry is Gaussian noise, and the prior of latent features is multivariate Gaussian distribution. However, Gaussian noise model is sensitive to outliers and is not robust. Gaussian prior model neglects the differences between latent features, that is, it does not distinguish between important and non-important features. Thus, Gaussian assumption based models perform poorly on real-world (sparse) datasets. To address these issues, a novel Variational Bayesian Probabilistic Matrix Factorization with Student-t prior model (TBMF) is proposed in this paper. A univariate Student-t distribution is used to fit the prediction noise, and a multivariate Student-t distribution is adopted for the prior of latent features. Due to the high kurtosis of Student-t distribution, TBMF can select informative latent features automatically, characterize long-tail cases and obtain reasonable representations on many real-world datasets. Experimental results show that TBMF improves the prediction performance significantly compared with the state-of-the-art algorithms, especially when the observed links are few.

Volatility Spillovers and Heavy Tails: A Large T-Vector Autoregressive Approach

Volatility is a key measure of risk in financial analysis. The high volatility of one financial asset today could affect the volatility of another asset tomorrow. These lagged effects among volatilities - which we call volatility spillovers - are studied using the Vector AutoRegressive (VAR) model. We account for the possible fat-tailed distribution of the VAR model errors using a VAR model with errors following a multivariate Student t-distribution with unknown degrees of freedom. Moreover, we study volatility spillovers among a large number of assets. To this end, we use penalized estimation of the VAR model with t-distributed errors. We study volatility spillovers among energy, biofuel and agricultural commodities and reveal bidirectional volatility spillovers between energy and biofuel, and between energy and agricultural commodities.

Improved hypothesis testing in a general multivariate elliptical model

ABSTRACTThis paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.

An improved anisotropic hierarchical fuzzy c-means method based on multivariate student t-distribution for brain MRI segmentation

Accurate segmentation for magnetic resonance (MR) images is an essential step in quantitative brain image analysis. However, due to the existence of intensity inhomogeneity (also named as bias field) and noise in the MR images, many segmentation methods are suffer from limited robustness and hard to find accurate results. This paper proposes an improved anisotropic multivariate student t-distribution based hierarchical fuzzy c-means method (IAMTHFCM). Firstly, improved anisotropic spatial information, defined in the neighborhoods of each pixel, is proposed to overcome the effect of the noise and preserve more detail information, especially for points with less repetitive patterns, such as corner and end points. Secondly, the improved anisotropic spatially information is utilized into a negative multivariate student t-distribution based log-posterior as the dissimilarity function to improve the robustness and accuracy. Thirdly, we use the hierarchical strategy to construct a more flexible objective function by considering the improved dissimilarity function itself as a sub-FCM, to make the method more robust and accurate to outliers and weak edges. Finally, the intensity inhomogeneities is modeled as a linear combination of a set of orthogonal basis functions, and parameterized by the coefficients of the orthogonal basis functions. Then the objective function is integrated with the bias field estimation and makes the proposed method can estimate the bias field meanwhile segmenting images. The segmentation and the bias field estimation can obtain benefit from each other. Our statistical results on both synthetic and clinical images show that the proposed method can overcome the difficulties caused by noise and bias fields and obtain more accurate results.