Complex Dependence Structures Research Articles

Network Quantile Autoregression

It is a challenging task to understand the complex dependency structures in an ultra-high dimensional network, especially when one concentrates on the tail dependency. To tackle this problem, we consider a network quantile autoregres- sion model (NQAR) to characterize the dynamic quantile behavior in a complex system. In particular, we relate responses to its connected nodes and node spe- ci c characteristics in a quantile autoregression process. A minimum contrast estimation approach for the NQAR model is introduced, and the asymptotic properties are studied. Finally, we demonstrate the usage of our model by in- vestigating the nancial contagions in the Chinese stock market accounting for shared ownership of companies.

Fixed choice design and augmented fixed choice design for network data with missing observations.

The statistical analysis of social networks is increasingly used to understand social processes and patterns. The association between social relationships and individual behaviors is of particular interest to sociologists, psychologists, and public health researchers. Several recent network studies make use of the fixed choice design (FCD), which induces missing edges in the network data. Because of the complex dependence structure inherent in networks, missing data can pose very difficult problems for valid statistical inference. In this article, we introduce novel methods for accounting for the FCD censoring and introduce a new survey design, which we call the augmented fixed choice design (AFCD). The AFCD adds considerable information to analyses without unduly burdening the survey respondent, resulting in improvements over the FCD, and other existing estimators. We demonstrate this new method through simulation studies and an analysis of alcohol use in a network of undergraduate students living in a residence hall.

Flexible Bayesian Dynamic Modeling of Correlation and Covariance Matrices.

Modeling correlation (and covariance) matrices can be challenging due to the positive-definiteness constraint and potential high-dimensionality. Our approach is to decompose the covariance matrix into the correlation and variance matrices and propose a novel Bayesian framework based on modeling the correlations as products of unit vectors. By specifying a wide range of distributions on a sphere (e.g. the squared-Dirichlet distribution), the proposed approach induces flexible prior distributions for covariance matrices (that go beyond the commonly used inverse-Wishart prior). For modeling real-life spatio-temporal processes with complex dependence structures, we extend our method to dynamic cases and introduce unit-vector Gaussian process priors in order to capture the evolution of correlation among components of a multivariate time series. To handle the intractability of the resulting posterior, we introduce the adaptive Δ-Spherical Hamiltonian Monte Carlo. We demonstrate the validity and flexibility of our proposed framework in a simulation study of periodic processes and an analysis of rat's local field potential activity in a complex sequence memory task.

Clustering and classification problems in genetics through U-statistics

ABSTRACTGenetic data are frequently categorical and have complex dependence structures that are not always well understood. For this reason, clustering and classification based on genetic data, while highly relevant, are challenging statistical problems. Here we consider a versatile U-statistics-based approach for non-parametric clustering that allows for an unconventional way of solving these problems. In this paper we propose a statistical test to assess group homogeneity taking into account multiple testing issues and a clustering algorithm based on dissimilarities within and between groups that highly speeds up the homogeneity test. We also propose a test to verify classification significance of a sample in one of two groups. We present Monte Carlo simulations that evaluate size and power of the proposed tests under different scenarios. Finally, the methodology is applied to three different genetic data sets: global human genetic diversity, breast tumour gene expression and Dengue virus serotypes. These applications showcase this statistical framework's ability to answer diverse biological questions in the high dimension low sample size scenario while adapting to the specificities of the different datatypes.

Mixture of D-Vine copulas for chemical process monitoring

Complex dependence structures often exist in chemical process multivariate data. Although they are difficult to capture, vine copula shows good performance and stronger flexibility in depicting the highly nonlinear dependencies. To identify and fully understand the complex dependence patterns in multivariate data, a mixture of D-vine copulas (MDVC) is proposed. By using the expectation-maximization (EM) algorithm and stepwise semi-parametric (SSP) estimation for parameter estimation, the proposed model can depict the complex dependence structures in multivariate data. The highest density region (HDR) and the density quantile approach (DQA) are both used to construct the highest density region distance-based probability (HDRP) index to achieve a real-time process fault monitoring. The effectiveness and benefits of the proposed model are illustrated with a numerical example, the Tennessee Eastman (TE) benchmark process and a real acetic acid dehydration distillation system for fault detection. The results show that the proposed mixture of D-vine copulas can achieve good performance in chemical process fault monitoring.

Multilevel Gaussian graphical model for multilevel networks

Gaussian graphical models have become a popular tool to represent networks among variables such as genes. They use the conditional correlations from the joint distribution to describe the dependencies between gene pairs, and employ the precision matrix of the genes. Because of the sparse nature of the gene networks and small sample sizes in high dimensional genetic data, regularization approaches attracted much attention in aim at obtaining the shrinkage estimates of the precision matrix. However, existing methods have been focused on the Gaussian graphical model among genes; that is, they are only applicable to a single level Gaussian graphical model. It is known that pathways are not independent of each other because of shared genes and interactions among pathways. Developing multipathway analysis has been a challenging problem because of the complex dependence structure among pathways. By considering the dependency among pathways as well as the genes within each pathway, we propose a multilevel Gaussian graphical model (MGGM) in which one level describes the networks for genes and the other for pathways. We have developed a multilevel L1 penalized likelihood approach to achieve the sparseness on both levels. In addition, we have developed an iterative weighted graphical LASSO algorithm for MGGM. Our simulation results supported the advantages of our approach; our method estimated the network more accurately on the pathway level and sparser on the gene level. We also demonstrated the usefulness of our approach using a canine genes-pathways data set.

Autoregressive and cross-lagged model for bivariate non-commensurate outcomes.

Autoregressive and cross-lagged models have been widely used to understand the relationship between bivariate commensurate outcomes in social and behavioral sciences, but not much work has been carried out in modeling bivariate non-commensurate (e.g., mixed binary and continuous) outcomes simultaneously. We develop a likelihood-based methodology combining ordinary autoregressive and cross-lagged models with a shared subject-specific random effect in the mixed-model framework to model two correlated longitudinal non-commensurate outcomes. The estimates of the cross-lagged and the autoregressive effects from our model are shown to be consistent with smaller mean-squared error than the estimates from the univariate generalized linear models. Inclusion of the subject-specific random effects in the proposed model accounts for between-subject variability arising from the omitted and/or unobservable, but possibly explanatory, subject-level predictors. Our model is not restricted to the case with equal number of events per subject, and it can be extended to different types of bivariate outcomes. We apply our model to an ecological momentary assessment study with complex dependence and sampling data structures. Specifically, we study the dependence between the condom use and sexual satisfaction based on the data reported in a longitudinal study of sexually transmitted infections. We find negative cross-lagged effect between these two outcomes and positive autoregressive effect within each outcome. Copyright © 2017 John Wiley & Sons, Ltd.

Uniformly and strongly consistent estimation for the Hurst function of a Linear Multifractional Stable Motion

Since the middle of the 90s, multifractional processes have been introduced for overcoming some limitations of the classical Fractional Brownian Motion model. In their context, the Hurst parameter becomes a Hölder continuous function $H(\cdot)$ of the time variable $t$. Linear Multifractional Stable Motion (LMSM) is the most known one of them with heavy-tailed distributions. Generally speaking, global and local sample path roughness of a multifractional process are determined by values of its parameter $H(\cdot)$; therefore, since about two decades, several authors have been interested in their statistical estimation, starting from discrete variations of the process. Because of complex dependence structures of variations, in order to show consistency of estimators one has to face challenging problems. The main goal of our article is to introduce, in the setting of the symmetric $\alpha$-stable non-anticipative moving average LMSM, where $\alpha\in(1,2)$, a new strategy for dealing with such kind of problems. It can also be useful in other contexts. In contrast with previously developed strategies, this new one does not require to look for sharp estimates of covariances related to functionals of variations. Roughly speaking, it consists of expressing variations in such a way that they become independent random variables up to negligible remainders. Thanks to it, we obtain, an almost surely and $L^{p}(\Omega)$, $p\in(0,4]$, consistent estimator of the whole function $H(\cdot)$, which converges, uniformly in $t$, and even for some Hölder norms. Also, we obtain estimates for the rates of convergence. Such kind of strong consistency results in uniform and Hölder norms are rather unusual in the literature on statistical estimation of functions.

Examination and visualisation of the simplifying assumption for vine copulas in three dimensions

SummaryVine copulas are a highly flexible class of dependence models, which are based on the decomposition of the density into bivariate building blocks. For applications one usually makes the simplifying assumption that copulas of conditional distributions are independent of the variables on which they are conditioned. However this assumption has been criticised for being too restrictive. We examine both simplified and non‐simplified vine copulas in three dimensions and investigate conceptual differences. We show and compare contour surfaces of three‐dimensional vine copula models, which prove to be much more informative than the contour lines of the bivariate marginals. Our investigation shows that non‐simplified vine copulas can exhibit arbitrarily irregular shapes, whereas simplified vine copulas appear to be smooth extrapolations of their bivariate margins to three dimensions. In addition to a variety of constructed examples, we also investigate a three‐dimensional subset of the well‐known uranium data set and visually detect the fact that a non‐simplified vine copula is necessary to capture its complex dependence structure.

A Monte Carlo approach to quantifying discrepancies between intractable posterior distributions

ABSTRACTThe computational demand required to perform inference using Markov chain Monte Carlo methods often obstructs a Bayesian analysis. This may be a result of large datasets, complex dependence structures, or expensive computer models. In these instances, the posterior distribution is replaced by a computationally tractable approximation, and inference is based on this working model. However, the error that is introduced by this practice is not well studied. In this paper, we propose a methodology that allows one to examine the impact on statistical inference by quantifying the discrepancy between the intractable and working posterior distributions. This work provides a structure to analyse model approximations with regard to the reliability of inference and computational efficiency. We illustrate our approach through a spatial analysis of yearly total precipitation anomalies where covariance tapering approximations are used to alleviate the computational demand associated with inverting a large, dense covariance matrix.

Network autocorrelation models with egocentric data

Network autocorrelation models have been widely used for decades to model the joint distribution of the attributes of a network's actors. This class of models can estimate both the effect of individual characteristics as well as the network effect, or social influence, on some actor attribute of interest. Collecting data on the entire network, however, is very often infeasible or impossible if the network boundary is unknown or difficult to define. Obtaining egocentric network data overcomes these obstacles, but as of yet there has been no clear way to model this type of data and still appropriately capture the network effect on the actor attributes in a way that is compatible with a joint distribution on the full network data. This paper adapts the class of network autocorrelation models to handle egocentric data. The proposed methods thus incorporate the complex dependence structure of the data induced by the network rather than simply using ad hoc measures of the egos’ networks to model the mean structure, and can estimate the network effect on the actor attribute of interest. The vast quantities of unknown information about the network can be succinctly represented in such a way that only depends on the number of alters in the egocentric network data and not on the total number of actors in the network. Estimation is done within a Bayesian framework. A simulation study is performed to evaluate the estimation performance, and an egocentric data set is analyzed where the aim is to determine if there is a network effect on environmental mastery, an important aspect of psychological well-being.

Inventory management in food companies with statistically dependent demand

Purpose The objective of this paper is to propose a methodology based on random demand inventory models and dependence structures for a set of raw materials, referred to as “components”, used by food services that produce food rations referred to as “menus”. Design/methodology/approach The contribution margins of food services that produce menus are optimised using random dependent demand inventory models. The statistical dependence between the demand for components and/or menus is incorporated into the model through the multivariate Gaussian (or normal) distribution. The contribution margins are optimised by using probabilistic inventory models for each component and stochastic programming with a differential evolution algorithm. Findings When compared to the non-optimised system previously used by the company, the (average) expected contribution margin increases by 18.32 per cent when using a continuous review inventory model for groceries and uniperiodic models for perishable components (optimised system). Research limitations/implications The multivariate modeling can be improved by using (a) other non-Gaussian (marginal) univariate probability distributions, by means of the copula method that considers more complex statistical dependence structures; (b) time-dependence, through autoregressive time-series structures and moving average; (c) random modelling of lead-time; and (d) demands for components with values equal to zero using zero-inflated or adjusted probability distribution. Practical implications Professional management of the supply chain allows the users to register data concerning component identification, demand, and stock levels to subsequently be used with the proposed methodology, which must be implemented computationally. Originality/value The proposed multivariate methodology allows it to describe demand dependence structures through inventory models applicable to components used to produce menus in food services.

Spatial Pair-Copula Modeling of Grade in Ore Bodies: A Case Study

A real-world mining application of pair-copulas is presented to model the spatial distribution of metal grade in an ore body. Inaccurate estimation of metal grade in an ore reserve can lead to failure of a mining project. Conventional kriged models are the most commonly used models for estimating grade and other spatial variables. However, kriged models use the variogram or covariance function, which produces a single average value to represent the spatial dependence for a given distance. Kriged models also assume linear spatial dependence. In the application, spatial pair-copulas are used to appropriately model the non-linear spatial dependence present in the data. The spatial pair-copula model is adopted over other copula-based spatial models since it is better able to capture complex spatial dependence structures. The performance of the pair-copula model is shown to be favorable compared to a conventional lognormal kriged model.

Multivariate tests based on interpoint distances with application to magnetic resonance imaging.

The multivariate location problem is addressed. The most familiar method to address the problem is the Hotelling test. When the hypothesis of normal distributions holds, the Hotelling test is optimal. Unfortunately, in practice the distributions underlying the samples are generally unknown and without assuming normality the finite sample unbiasedness of the Hotelling test is not guaranteed. Moreover, high-dimensional data are increasingly encountered when analyzing medical and biological problems, and in these situations the Hotelling test performs poorly or cannot be computed. A test that is unbiased for non-normal data, for small sample sizes as well as for two-sided alternatives and that can be computed for high-dimensional data has been recently proposed and is based on the ranks of the interpoint Euclidean distances between observations. Five modifications of this test are proposed and compared to the original test and the Hotelling test. Unbiasedness and consistency of the tests are proven and the problem of power computation is addressed. It is shown that two of the modified interpoint distance-based tests are always more powerful than the original test. Particularly, the modified test based on the Tippett criterium is suggested when the assumption of normality is not tenable and/or in case of high-dimensional data with complex dependence structure which are typical in molecular biology and medical imaging. A practical application to a case-control study where functional magnetic resonance imaging is used is discussed.

Comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering.

Comparing large covariance matrices has important applications in modern genomics, where scientists are often interested in understanding whether relationships (e.g., dependencies or co-regulations) among a large number of genes vary between different biological states. We propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes. A distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices. Hence, the test is robust with respect to various complex dependence structures that frequently arise in genomics. We prove that the proposed procedure is asymptotically valid under weak moment conditions. As an interesting application, we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high-dimensional genomics data. Using an asthma gene expression dataset, we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets/pathways between the disease group and the control group, and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups. The proposed methods have been implemented in an R-package HDtest and are available on CRAN.

Multivariate probabilistic safety analysis of process facilities using the Copula Bayesian Network model

Integrated safety analysis of hazardous process facilities calls for an understanding of both stochastic and topological dependencies, going beyond traditional Bayesian Network (BN) analysis to study cause-effect relationships among major risk factors. This paper presents a novel model based on the Copula Bayesian Network (CBN) for multivariate safety analysis of process systems. The innovation of the proposed CBN model is in integrating the advantage of copula functions in modelling complex dependence structures with the cause-effect relationship reasoning of process variables using BNs. This offers a great flexibility in probabilistic analysis of individual risk factors while considering their uncertainty and stochastic dependence. Methods based on maximum likelihood evaluation and information theory are presented to learn the structure of CBN models. The superior performance of the CBN model and its advantages compared to traditional BN models are demonstrated by application to an offshore managed pressure drilling case study.

Multilevel modeling of insurance claims using copulas

In property-casualty insurance, claims management is featured with the modeling of a semi-continuous insurance cost associated with individual risk transfer. This practice is further complicated by the multilevel structure of the insurance claims data, where a contract often contains a group of policyholders, each policyholder is insured under multiple types of coverage, and the contract is repeatedly observed over time. The data hierarchy introduces a complex dependence structure among claims and leads to diversification in the insurer’s liability portfolio. To capture the unique features of policy-level insurance costs, we propose a copula regression for the multivariate longitudinal claims. In the model, the Tweedie double generalized linear model is employed to examine the semi-continuous claim cost of each coverage type, and a Gaussian copula is specified to accommodate the cross-sectional and temporal dependence among the multilevel claims. Estimation and inference is based on the composite likelihood approach and the properties of parameter estimates are investigated through simulation studies. When applied to a portfolio of personal automobile policies from a Canadian insurer, we show that the proposed copula model provides valuable insights to an insurer’s claims management process.

Exploring dependence between categorical variables: Benefits and limitations of using variable selection within Bayesian clustering in relation to log-linear modelling with interaction terms

This manuscript is concerned with relating two approaches that can be used to explore complex dependence structures between categorical variables, namely Bayesian partitioning of the covariate space incorporating a variable selection procedure that highlights the covariates that drive the clustering, and log-linear modelling with interaction terms. We derive theoretical results on this relation and discuss if they can be employed to assist log-linear model determination, demonstrating advantages and limitations with simulated and real data sets. The main advantage concerns sparse contingency tables. Inferences from clustering can potentially reduce the number of covariates considered and, subsequently, the number of competing log-linear models, making the exploration of the model space feasible. Variable selection within clustering can inform on marginal independence in general, thus allowing for a more efficient exploration of the log-linear model space. However, we show that the clustering structure is not informative on the existence of interactions in a consistent manner. This work is of interest to those who utilize log-linear models, as well as practitioners such as epidemiologists that use clustering models to reduce the dimensionality in the data and to reveal interesting patterns on how covariates combine.

Network Quantile Autoregression

It is a challenging task to understand the complex dependency structures in an ultra-high dimensional network, especially when one concentrates on the tail dependency. To tackle this problem, we consider a network quantile autoregression model (NQAR) to characterize the dynamic quantile behavior in a complex system. In particular, we relate responses to its connected nodes and node specific characteristics in a quantile autoregression process. A minimum contrast estimation approach for the NQAR model is introduced, and the asymptotic properties are studied. Finally, we demonstrate the usage of our model by investigating the financial contagions in the Chinese stock market accounting for shared ownership of companies.

Scalable Multivariate Time-Series Models for Climate Informatics

The increasing volume of climate data has created the need for scientists to develop scalable data analysis tools beyond traditional techniques. Climate data not only have a massive scale but also high dimension and complex dependency structures, making the analysis task extremely challenging. Climate informatics leverages advanced algorithmic tools from data science to solve problems in climate science. This article showcases how scalable multivariate time-series models can be developed for climate change attribution, spatiotemporal analysis, and extreme value time-series analysis.