Multivariate Count Time Series Research Articles

Modeling and inferences for bounded multivariate time series of counts

Modeling and inferences for bounded multivariate time series of counts

On the Validity of Granger Causality for Ecological Count Time Series

Knowledge of causal relationships is fundamental for understanding the dynamic mechanisms of ecological systems. To detect such relationships from multivariate time series, Granger causality, an idea first developed in econometrics, has been formulated in terms of vector autoregressive (VAR) models. Granger causality for count time series, often seen in ecology, has rarely been explored, and this may be due to the difficulty in estimating autoregressive models on multivariate count time series. The present research investigates the appropriateness of VAR-based Granger causality for ecological count time series by conducting a simulation study using several systems of different numbers of variables and time series lengths. VAR-based Granger causality for count time series (DVAR) seems to be estimated efficiently even for two counts in long time series. For all the studied time series lengths, DVAR for more than eight counts matches the Granger causality effects obtained by VAR on the continuous-valued time series well. The positive results, also in two ecological time series, suggest the use of VAR-based Granger causality for assessing causal relationships in real-world count time series even with few distinct integer values or many zeros.

Softplus negative binomial network autoregression

AbstractModelling multivariate time series of counts in a parsimonious way is a popular topic. In this paper, we consider an integer‐valued network autoregressive model with a non‐random neighbourhood structure, which uses negative binomial distribution as the conditional marginal distribution and the softplus function as the link function. The new model generalizes existing ones in the literature and has a great flexibility in modelling. Stationary conditions in cases of fixed dimension and increasing dimension are given. Parameters are estimated by maximizing the quasi‐likelihood function, and related asymptotic properties of the estimators are established. A simulation study is conducted to assess performances of the estimators, and a real data example is analysed to show superior performances of the proposed model compared with existing ones.

Multivariate threshold integer-valued autoregressive processes with explanatory variables

To capture the multivariate count time series showing piecewise phenomena, we introduce a class of first-order multivariate threshold integer-valued autoregressive process. The component of autoregressive coefficient is driven by explanatory variables via a logistic link function. Basic probabilistic and statistical properties of the model are discussed. The model parameters are estimated by means of conditional least squares and conditional maximum likelihood methods. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test of the model and existence test of explanatory variables are well addressed using a Wald-type test. Forecasting problem and model extension problem are also considered. Finally, some numerical results of the estimates and a real data example are presented.

Multivariate count time series segmentation with “sums and shares” and Poisson lognormal mixture models: a comparative study using pedestrian flows within a multimodal transport hub

Multivariate count time series segmentation with “sums and shares” and Poisson lognormal mixture models: a comparative study using pedestrian flows within a multimodal transport hub

Modeling and inference for multivariate time series of counts based on the INGARCH scheme

Modeling multivariate time series of counts using the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) scheme is proposed. The key idea is to model each component of the time series with a univariate INGARCH model, where the conditional distribution is modeled with a one-parameter exponential family distribution, and to use a (nonlinear) parametric function of all components to recursively produce the conditional means. It is shown that the proposed multivariate INGARCH (MINGARCH) model is strictly stationary and ergodic. For inference, the quasi-maximum likelihood estimator (QMLE) and the minimum density power divergence estimator (MDPDE) for robust estimation are adopted, and their consistency and asymptotic normality are verified. As an application, the change point test based on the QMLE and MDPDE is illustrated. The Monte Carlo simulation study and real data analysis using the number of weekly syphilis cases in the United States are conducted to confirm the validity of the proposed method.

Multivariate Count Time Series Modelling

Autoregressive models are reviewed for the analysis of multivariate count time series. A particular topic of interest which is discussed in detail is that of the choice of a suitable distribution for a vectors of count random variables. The focus is on three main approaches taken for multivariate count time series analysis: (a) integer autoregressive processes, (b) parameter-driven models and (c) observation-driven models. The aim is to highlight some recent methodological developments and propose some potentially useful research topics.

Bayesian modeling of multivariate time series of counts

AbstractIn this article, we present an overview of recent advances in Bayesian modeling and analysis of multivariate time series of counts. We discuss basic modeling strategies including integer valued autoregressive processes, multivariate Poisson time series and dynamic latent factor models. In so doing, we make a connection with univariate modeling frameworks such as dynamic generalized models, Poisson state space models with gamma evolution and present Bayesian approaches that extend these frameworks to multivariate setting. During our development, recent Bayesian approaches to the analysis of integer valued autoregressive processes and multivariate Poisson models are highlighted and concepts such as “decouple/recouple” and “common random environment” are presented. The role that these concepts play in Bayesian modeling and analysis of multivariate time series are discussed. Computational issues associated with Bayesian inference and forecasting from these models are also considered.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory Statistical Models > Time Series Models

Modeling Multivariate Count Time Series Data with a Vector Poisson Log-Normal Additive Model: Applications to Testing Treatment Effects in Single-Case Designs

In education and psychology, single-case designs (SCDs) have been used to detect treatment effects using time series data in the presence or absence of intervention. One popular design variant of SCDs is a multiple-baseline design for multiple outcomes, which often collects outcomes with some form of a count. A Poisson model is a natural choice for the count outcome. However, the assumption of the Poisson model that the outcome variable’s mean is equal to its variance is often violated in SCDs, as the variance is often larger than the mean (called overdispersion). In addition, when multiple outcomes are from the same participant, it is likely that they are correlated. In this paper, we present a vector Poisson log-normal additive (V-PLN-A) model to deal with (a) change processes (auto- and cross-correlations and data-driven trend) and (b) correlation and overdispersion in multivariate count time series. A multivariate normal distribution was adapted to account for correlation among multiple outcomes as well as possible overdispersion. The V-PLN-A model was applied to an educational intervention study to test treatment effects. Simulation study results showed that parameter recovery of the V-PLN-A model was satisfactory in a large number of timepoints using Bayesian analysis, and that ignoring change processes and overdispersion led to biased estimates of the treatment effects.

Endemic-epidemic models with discrete-time serial interval distributions for infectious disease prediction

Multivariate count time series models are an important tool for analyzing and predicting the spread of infectious disease. We consider the endemic-epidemic framework, a class of autoregressive models for infectious disease surveillance counts, and replace the default autoregression on counts from the previous time period with more flexible weighting schemes inspired by discrete-time serial interval distributions. We employ three different parametric formulations, each with an additional unknown weighting parameter estimated via a profile likelihood approach, and compare them to an unrestricted nonparametric approach. The new methods are illustrated in a univariate analysis of dengue fever incidence in San Juan, Puerto Rico, and a spatiotemporal study of viral gastroenteritis in the 12 districts of Berlin. We assess the predictive performance of the suggested models and several reference models at various forecast horizons. In both applications, the performance of the endemic-epidemic models is considerably improved by the proposed weighting schemes.

Multivariate count autoregression

We are studying linear and log-linear models for multivariate count time series data with Poisson marginals. For studying the properties of such processes we develop a novel conceptual framework which is based on copulas. Earlier contributions impose the copula on the joint distribution of the vector of counts by employing a continuous extension methodology. Instead we introduce a copula function on a vector of associated continuous random variables. This construction avoids conceptual difficulties related to the joint distribution of counts yet it keeps the properties of the Poisson process marginally. Furthermore, this construction can be employed for modeling multivariate count time series with other marginal count distributions. We employ Markov chain theory and the notion of weak dependence to study ergodicity and stationarity of the models we consider. Suitable estimating equations are suggested for estimating unknown model parameters. The large sample properties of the resulting estimators are studied in detail. The work concludes with some simulations and a real data example.

Flexible bivariate Poisson integer-valued GARCH model

Integer-valued time series models have been widely used, especially integer-valued autoregressive models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models. Recently, there has been a growing interest in multivariate count time series. However, existing models restrict the dependence structures imposed by the way they constructed. In this paper, we consider a class of flexible bivariate Poisson INGARCH(1,1) model whose dependence is established by a special multiplicative factor. Stationarity and ergodicity of the process are discussed. The maximization by parts algorithm and its modified version together with the alternative method by using R package Template Model Builder are employed to estimate the parameters of interest. The consistency and asymptotic normality for estimates are obtained, and the finite sample performance of estimators is given via simulations. A real data example is also provided to illustrate the model.

Modeling, simulation and inference for multivariate time series of counts using trawl processes

This article presents a new continuous-time modeling framework for multivariate time series of counts which have an infinitely divisible marginal distribution. The model is based on a mixed moving average process driven by Lévy noise, called a trawl process, where the serial correlation and the cross-sectional dependence are modeled independently of each other. Such processes can exhibit short or long memory. We derive a stochastic simulation algorithm and a statistical inference method for such processes. The new methodology is then applied to high frequency financial data, where we investigate the relationship between the number of limit order submissions and deletions in a limit order book.

Modelling Non-Stationary Multivariate Time Series of Counts via Common Factors

SummaryWe develop a new parameter-driven model for multivariate time series of counts. The time series is not necessarily stationary. We model the mean process as the product of modulating factors and unobserved stationary processes. The former characterizes the long-run movement in the data, whereas the latter is responsible for rapid fluctuations and other unknown or unavailable covariates. The unobserved stationary processes evolve independently of the past observed counts and might interact with each other. We express the multivariate unobserved stationary processes as a linear combination of possibly low dimensional factors that govern the contemporaneous and serial correlation within and across the observed counts. Regression coefficients in the modulating factors are estimated via pseudo-maximum-likelihood estimation, and identification of common factor(s) is carried out through eigenanalysis on a positive definite matrix that pertains to the autocovariance of the observed counts at non-zero lags. Theoretical validity of the two-step estimation procedure is documented. In particular, we establish consistency and asymptotic normality of the pseudo-maximum-likelihood estimator in the first step and the convergence rate of the second-step estimator. We also present an exhaustive simulation study to examine the finite sample performance of the estimators, and numerical results corroborate our theoretical findings. Finally, we illustrate the use of the proposed model through an application to the numbers of National Science Foundation fundings awarded to seven research universities from January 2001 to December 2012.

Multivariate integer-valued time series with flexible autocovariances and their application to major hurricane counts

This paper examines a bivariate count time series with some curious statistical features: Saffir–Simpson Category 3 and stronger annual hurricane counts in the North Atlantic and eastern Pacific Ocean Basins. As land and ocean temperatures on our planet warm, an intense climatological debate has arisen over whether hurricanes are becoming more numerous, or whether the strengths of the individual storms are increasing. Recent literature concludes that an increase in hurricane counts occurred in the Atlantic Basin circa 1994. This increase persisted through 2012; moreover, the 1994–2012 period was one of relative inactivity in the eastern Pacific Basin. When Atlantic activity eased in 2013, heavy activity in the eastern Pacific Basin commenced. When examined statistically, a Poisson white noise model for the annual severe hurricane counts is difficult to resoundingly reject. Yet, decadal cycles (longer term dependence) in the hurricane counts is plausible. This paper takes a statistical look at the issue, developing a stationary multivariate count time series model with Poisson marginal distributions and a flexible autocovariance structure. Our auto- and cross-correlations can be negative and have long-range dependence; features that most previous count models cannot achieve in tandem. Our model is new in the literature and is based on categorizing and super-positioning multivariate Gaussian time series. We derive the autocovariance function of the model and propose a method to estimate model parameters. In the end, we conclude that severe hurricane counts are indeed negatively correlated across the two ocean basins. Some evidence for long-range dependence is also presented; however, with only a 49-year record, this issue cannot be definitively judged without additional data.

Modelling, Simulation and Inference for Multivariate Time Series of Counts Using Trawl Processes

This article presents a new continuous-time modelling framework for multivariate time series of counts which have an infinitely divisible marginal distribution. The model is based on a mixed moving average process driven by Levy noise - called a trawl process - where the serial correlation and the cross-sectional dependence are modelled independently of each other. Such processes can exhibit short or long memory. We derive a stochastic simulation algorithm and a statistical inference method for such processes. The new methodology is then applied to high frequency financial data, where we investigate the relationship between the number of limit order submissions and deletions in a limit order book.

Multi‐stage multivariate modeling of temporal patterns in prescription counts for competing drugs in a therapeutic category

This article describes statistical analyses pertaining to marketing data from a large multinational pharmaceutical firm. We describe models for monthly new prescription counts that are written by physicians for the firm's focal drug and for competing drugs, as functions of physician‐specific and time‐varying predictors. Modeling patterns in discrete‐valued time series, and specifically time series of counts, based on large datasets, is the focus of much recent research attention. We first provide a brief overview of Bayesian approaches we have employed for modeling multivariate count time series using Markov Chain Monte Carlo methods. We then discuss a flexible level correlated model framework, which enables us to combine different marginal count distributions and to build a hierarchical model for the vector time series of counts, while accounting for the association among the components of the response vector, as well as possible overdispersion. We employ the integrated nested Laplace approximation (INLA) for fast approximate Bayesian modeling using the R‐INLA package (r‐inla.org). To enhance computational speed, we first build a model for each physician, use features of the estimated trends in the time‐varying parameters in order to cluster the physicians into groups, and fit aggregate models for all physicians within each cluster. Our three‐stage analysis can provide useful guidance to the pharmaceutical firm on their marketing actions. Copyright © 2017 John Wiley & Sons, Ltd.

Multivariate generalized Poisson geometric process model with scale mixtures of normal distributions

Abstract This paper proposes a new model named as the multivariate generalized Poisson log- t geometric process (MGPLTGP) model to study multivariate time-series of counts with overdispersion or underdispersion, non-monotone trends within each time-series and positive or negative correlation between pairs of time-series. This model assumes that the multivariate counts follow independent generalized Poisson distributions with an additional parameter to adjust for different degrees of dispersion including overdispersion and underdispersion. Their means after discounting the trend effect geometrically by ratio functions form latent stochastic processes and follow a multivariate log- t distribution with a flexible correlation structure to capture both positive correlation and negative correlation. By expressing the multivariate Student’s t -distribution in scale mixtures of normals, the model can be implemented through Markov chain Monte Carlo algorithms via the user-friendly WinBUGS software. The applicability of the MGPLTGP model is illustrated through an analysis of the possession and/or use of two illicit drugs, amphetamines and narcotics in New South Wales, Australia.

Hierarchical dynamic models for multivariate times series of counts

In several application areas, we see the need for accurate statistical modeling of multivariate time series of counts as a function of relevant covariates. In ecology, count responses on species abundance are observed over several time periods at several locations, and the covariates that influence the abundance may be location-specific and/or time-varying. This paper describes a Bayesian framework for estimation and prediction by assuming a multivariate Poisson sampling distribution for the count responses and by fitting a hierarchical dynamic model. Our modeling incorporates the temporal dependence as well as dependence between the components of the response vector.

Some properties of multivariate INAR(1) processes

INteger-valued AutoRegressive (INAR) processes are common choices for modeling non-negative discrete valued time series. In this framework and motivated by the frequent occurrence of multivariate count time series data in several different disciplines, a generalized specification of the bivariate INAR(1) (BINAR(1)) model is considered. In this new, full BINAR(1) process, dependence between the two series stems from two sources simultaneously. The main focus is on the specific parametric case that arises under the assumption of a bivariate Poisson distribution for the innovations of the process. As it is shown, such an assumption gives rise to a Hermite BINAR(1) process. The method of conditional maximum likelihood is suggested for the estimation of its unknown parameters. A short application on financial count data illustrates the model.