Structured priors for multivariate time series

A class of prior distributions for multivariate autoregressive models is presented. This class of priors is built taking into account the latent component structure that characterizes a collection of autoregressive processes. In particular, the state-space representation of a vector autoregressive process leads to the decomposition of each time series in the multivariate process into simple underlying components. These components may have a common structure across the series. A key feature of the proposed priors is that they allow the modeling of such common structure. This approach also takes into account the uncertainty in the number of latent processes, consequently handling model order uncertainty in the multivariate autoregressive framework. Posterior inference is achieved via standard Markov chain Monte Carlo (MCMC) methods. Issues related to inference and exploration of the posterior distribution are discussed. We illustrate the methodology analyzing two data sets: a synthetic data set with quasi-periodic latent structure, and seasonally adjusted US monthly housing data consisting of housing starts and housing sales over the period 1965 to 1974.     

Multivariate geometric anisotropic Cox processes

This paper introduces a new modeling and inference framework for multivariate and anisotropic point processes. Building on recent innovations in multivariate spatial statistics, we propose a new family of multivariate anisotropic random fields, and from them a family of anisotropic point processes. We give conditions that make the proposed models valid. We also propose a Palm likelihood-based inference method for this type of point process, circumventing issues of likelihood tractability. Finally we illustrate the utility of the proposed modeling framework by analyzing spatial ecological observations of plants and trees in the Barro Colorado Island data.     

A class of weighted multivariate elliptical models useful for robust analysis of nonnormal and bimodal data

A class of weighted elliptical models useful for analyzing nonnormal and bimodal multivariate data is introduced. It is obtained from the marginal distribution of a centrally truncated multivariate elliptical distribution. As a special case, a finite mixture of weighted multinormal distribution is examined in detail, establishing connections with the multinormal and the finite mixture of multinormal. The special class of distributions is studied from several aspects such as weighting of probability density functions, association with centrally truncated distributions, and a finite scale mixture scheme. The relationships among these aspects are given, and various properties of the class are also discussed. For the inference of the class, an MCMC procedure and its numerical example are provided.     

A new family of multivariate skew slash distribution

In this article, the new family of multivariate skew slash distribution is defined. According to the definition, a stochastic representation of the multivariate skew slash distribution is derived. The first four moments and measures of skewness and kurtosis of a random vector with the multivariate skew slash distribution are obtained. The distribution of quadratic forms for the multivariate skew slash distribution and the non central skew slash χ² distribution are studied. Maximum likelihood inference and real data illustration are discussed. In the end, the potential extension of multivariate skew slash distribution is discussed.     

A MATLAB package for multivariate normality test

A MATLAB package testing for multivariate normality (TMVN) is implemented as an interactive and graphical tool to examine multivariate normality (MVN). Monte Carlo simulation studies have failed to find a uniformly most powerful MVN test, which requires a rather extensive statistical inference procedure. TMVN contains several competitive MVN tests and provides a flexible and extensive testing environment for univariate or multivariate data analyses. Simulated results provide information of which test may possess more power for the selected non-MVN alternatives. Fisher's Iris data are used to show how TMVN can be used in practice.     

Time-varying clustering of multivariate longitudinal observations

Abstract

We propose a statistical method for clustering multivariate longitudinal data into homogeneous groups. This method relies on a time-varying extension of the classical K-means algorithm, where a multivariate vector autoregressive model is additionally assumed for modeling the evolution of clusters' centroids over time. Model inference is based on a least-squares method and on a coordinate descent algorithm. To illustrate our work, we consider a longitudinal dataset on human development. Three variables are modeled, namely life expectancy, education and gross domestic product.     

Modeling biological data: Several vignettes

The author develops the theme of biology as a rich source of interesting problems where the data available are often noisy and multivariate. He presents various situations in which the data raise model building and inference issues. He emphasizes that appropriate modeling requires careful thought and usually provides methodological challenges. The data sets considered include bird count data, tracking data from leatherback turtles, and amino acid sequence data from primates and cetaceans.     

Bayesian Dynamic Factor Models and Portfolio Allocation

We discuss the development of dynamic factor models for multivariate financial time series, and the incorporation of stochastic volatility components for latent factor processes. Bayesian inference and computation is developed and explored in a study of the dynamic factor structure of daily spot exchange rates for a selection of international currencies. The models are direct generalizations of univariate stochastic volatility models and represent specific varieties of models recently discussed in the growing multivariate stochastic volatility literature. We discuss model fitting based on retrospective data and sequential analysis for forward filtering and short-term forecasting. Analyses are compared with results from the much simpler method of dynamic variance-matrix discounting that, for over a decade, has been a standard approach in applied financial econometrics. We study these models in analysis, forecasting, and sequential portfolio allocation for a selected set of international exchange-rate-return time series. Our goals are to understand a range of modeling questions arising in using these factor models and to explore empirical performance in portfolio construction relative to discount approaches. We report on our experiences and conclude with comments about the practical utility of structured factor models and on future potential model extensions.     

Bayesian factor models for multivariate categorical data obtained from questionnaires

Factor analysis is a flexible technique for assessment of multivariate dependence and codependence. Besides being an exploratory tool used to reduce the dimensionality of multivariate data, it allows estimation of common factors that often have an interesting theoretical interpretation in real problems. However, standard factor analysis is only applicable when the variables are scaled, which is often inappropriate, for example, in data obtained from questionnaires in the field of psychology, where the variables are often categorical. In this framework, we propose a factor model for the analysis of multivariate ordered and non-ordered polychotomous data. The inference procedure is done under the Bayesian approach via Markov chain Monte Carlo methods. Two Monte Carlo simulation studies are presented to investigate the performance of this approach in terms of estimation bias, precision and assessment of the number of factors. We also illustrate the proposed method to analyze participants'' responses to the Motivational State Questionnaire dataset, developed to study emotions in laboratory and field settings.     

Multivariate Poisson regression with covariance structure

In recent years the applications of multivariate Poisson models have increased, mainly because of the gradual increase in computer performance. The multivariate Poisson model used in practice is based on a common covariance term for all the pairs of variables. This is rather restrictive and does not allow for modelling the covariance structure of the data in a flexible way. In this paper we propose inference for a multivariate Poisson model with larger structure, i.e. different covariance for each pair of variables. Maximum likelihood estimation, as well as Bayesian estimation methods are proposed. Both are based on a data augmentation scheme that reflects the multivariate reduction derivation of the joint probability function. In order to enlarge the applicability of the model we allow for covariates in the specification of both the mean and the covariance parameters. Extension to models with complete structure with many multi-way covariance terms is discussed. The method is demonstrated by analyzing a real life data set.     

Linear or Nonlinear? Automatic Structure Discovery for Partially Linear Models

Partially linear models provide a useful class of tools for modeling complex data by naturally incorporating a combination of linear and nonlinear effects within one framework. One key question in partially linear models is the choice of model structure, that is, how to decide which covariates are linear and which are nonlinear. This is a fundamental, yet largely unsolved problem for partially linear models. In practice, one often assumes that the model structure is given or known and then makes estimation and inference based on that structure. Alternatively, there are two methods in common use for tackling the problem: hypotheses testing and visual screening based on the marginal fits. Both methods are quite useful in practice but have their drawbacks. First, it is difficult to construct a powerful procedure for testing multiple hypotheses of linear against nonlinear fits. Second, the screening procedure based on the scatterplots of individual covariate fits may provide an educated guess on the regression function form, but the procedure is ad hoc and lacks theoretical justifications. In this article, we propose a new approach to structure selection for partially linear models, called the LAND (Linear And Nonlinear Discoverer). The procedure is developed in an elegant mathematical framework and possesses desired theoretical and computational properties. Under certain regularity conditions, we show that the LAND estimator is able to identify the underlying true model structure correctly and at the same time estimate the multivariate regression function consistently. The convergence rate of the new estimator is established as well. We further propose an iterative algorithm to implement the procedure and illustrate its performance by simulated and real examples. Supplementary materials for this article are available online.     

Empirical best linear unbiased and empirical Bayes prediction in multivariate small area estimation

Small area estimation plays a prominent role in survey sampling due to a growing demand for reliable small area estimates from both public and private sectors. Popularity of model-based inference is increasing in survey sampling, particularly, in small area estimation. The estimates of the small area parameters can profitably ‘borrow strength’ from data on related multiple characteristics and/or auxiliary variables from other neighboring areas through appropriate models. Fay (1987, Small Area Statistics, Wiley, New York, pp. 91–102) proposed multivariate regression for small area estimation of multiple characteristics. The success of this modeling rests essentially on the strength of correlation of these dependent variables. To estimate small area mean vectors of multiple characteristics, multivariate modeling has been proposed in the literature via a multivariate variance components model. We use this approach to empirical best linear unbiased and empirical Bayes prediction of small area mean vectors. We use data from Battese et al. (1988, J. Amer. Statist. Assoc. 83, 28 –36) to conduct a simulation which shows that the multivariate approach may achieve substantial improvement over the usual univariate approach.     

Modelling human immunodeficiency virus ribonucleic acid levels with finite mixtures for censored longitudinal data

The measurement of human immunodeficiency virus ribonucleic acid levels over time leads to censored longitudinal data. Suitable models for dynamic modelling of these levels need to take this data characteristic into account. If groups of patients with different developments of the levels over time are suspected the model class of finite mixtures of mixed effects models with censored data is required. We describe the model specification and derive the estimation with a suitable expectation-maximization algorithm. We propose a convenient implementation using closed form formulae for the expected mean and variance of the truncated multivariate distribution. Only efficient evaluation of the cumulative multivariate normal distribution function is required. Model selection as well as methods for inference are discussed. The application is demonstrated on the clinical trial ACTG 315 data.     

大数据背景下网络调查样本的建模推断问题研究——以广义Boosted模型的倾向得分推断为例

随着大数据和网络的不断发展，网络调查越来越广泛，大部分网络调查样本属于非概率样本，难以采用传统的抽样推断理论进行推断，如何解决网络调查样本的推断问题是大数据背景下网络调查发展的迫切需求。本文首次从建模的角度提出了解决该问题的基本思路：一是入样概率的建模推断，可以考虑构建基于机器学习与变量选择的倾向得分模型来估计入样概率推断总体；二是目标变量的建模推断，可以考虑直接对目标变量建立参数、非参数或半参数超总体模型进行估计；三是入样概率与目标变量的双重建模推断，可以考虑进行倾向得分模型与超总体模型的加权估计与混合推断。最后，以基于广义Boosted模型的入样概率建模推断为例演示了具体解决方法。     

A MULTIVARIATE NORMAL MODEL FOR PEDIGREE AND LONGITUDINAL DATA AND THE SOFTWARE 'FISHER'

In 1918 R.A. Fisher published an interpretation of covariation between relatives in terms of Mendelian inheritance, which has allowed inference on genetic and environmental components of variation from plant, animal and human pedigree data. Fisher had introduced maximum likelihood six years earlier. His 1918 paper abo contained the basics of linear regression and decomposition of variance. These concepts have now been united to allow flexible modelling of the mean and covariance structure of non-independent data on continuous traits, using maximum likelihood under a multivariate normal assumption. FISHER is a software package, designed for pedigree analysis and easily adapted for repeated measures and longitudinal data analysis. A range of applications illustrate FISHER as a useful statistical tool. Issues related to assumptions, tests-of-fit, and robustness of inference are discussed.     

A flexible class of models for data arising from a ‘thorough QT/QTc study’

The standard methods for analyzing data arising from a ‘thorough QT/QTc study’ are based on multivariate normal models with common variance structure for both drug and placebo. Such modeling assumptions may be violated and when the sample sizes are small, the statistical inference can be sensitive to such stringent assumptions. This article proposes a flexible class of parametric models to address the above‐mentioned limitations of the currently used models. A Bayesian methodology is used for data analysis and models are compared using the deviance information criteria. Superior performance of the proposed models over the current models is illustrated through a real dataset obtained from a GlaxoSmithKline (GSK) conducted ‘thorough QT/QTc study’. Copyright © 2010 John Wiley & Sons, Ltd.     

Bayesian conditional inference for Rasch models

This paper is concerned with Bayesian inference in psychometric modeling. It treats conditional likelihood functions obtained from discrete conditional probability distributions which are generalizations of the hypergeometric distribution. The influence of nuisance parameters is eliminated by conditioning on observed values of their sufficient statistics, and Bayesian considerations are only referred to parameters of interest. Since such a combination of techniques to deal with both types of parameters is less common in psychometrics, a wider scope in future research may be gained. The focus is on the evaluation of the empirical appropriateness of assumptions of the Rasch model, thereby pointing to an alternative to the frequentists’ approach which is dominating in this context. A number of examples are discussed. Some are very straightforward to apply. Others are computationally intensive and may be unpractical. The suggested procedure is illustrated using real data from a study on vocational education.     

Random Effect Threshold Models for Dose–Response Relationships with Repeated Measurements

A random threshold model for dose--response designs with repeated measurements is introduced. Inference procedures for this model are discussed. The framework of generalized linear models is used to propose a quasi-likelihood inference procedure. The feasibility of this approach is illustrated with the analysis of food science data concerning a forced choice sensory experiment.     

Bayesian composite quantile regression for linear mixed-effects models

Longitudinal data are commonly modeled with the normal mixed-effects models. Most modeling methods are based on traditional mean regression, which results in non robust estimation when suffering extreme values or outliers. Median regression is also not a best choice to estimation especially for non normal errors. Compared to conventional modeling methods, composite quantile regression can provide robust estimation results even for non normal errors. In this paper, based on a so-called pseudo composite asymmetric Laplace distribution (PCALD), we develop a Bayesian treatment to composite quantile regression for mixed-effects models. Furthermore, with the location-scale mixture representation of the PCALD, we establish a Bayesian hierarchical model and achieve the posterior inference of all unknown parameters and latent variables using Markov Chain Monte Carlo (MCMC) method. Finally, this newly developed procedure is illustrated by some Monte Carlo simulations and a case analysis of HIV/AIDS clinical data set.     

Semiparametric transformation models for multivariate panel count data with dependent observation process

This article discusses regression analysis of multivariate panel count data in which the observation process may contain relevant information about or be related to the underlying recurrent event processes of interest. Such data occur if a recurrent event study involves several related types of recurrent events and the observation scheme or process may be subject-specific. For the problem, a class of semiparametric transformation models is presented, which provides a great flexibility for modelling the effects of covariates on the recurrent event processes. For estimation of regression parameters, an estimating equation-based inference procedure is developed and the asymptotic properties of the resulting estimates are established. Also the proposed approach is evaluated by simulation studies and applied to the data arising from a skin cancer chemoprevention trial.