Statistical analysis of multivariate discrete-valued time series

Abstract This work gives an overview of statistical analysis for some models for multivariate discrete-valued (MDV) time series. We present observation-driven models and models based on higher-order Markov chains. Several extensions are highlighted including non-stationarity, network autoregressions, conditional non-linear autoregressive models, robust estimation, random fields and spatio-temporal models.

[1]  D. Brillinger Time series - data analysis and theory , 1981, Classics in applied mathematics.

[2]  Lionel Truquet,et al.  Multivariate time series models for mixed data , 2021, Bernoulli.

[3]  Richard A. Davis,et al.  Theory and Inference for a Class of Observation-driven Models with Application to Time Series of Counts , 2012, 1204.3915.

[4]  Lovekesh Vig,et al.  Long Short Term Memory Networks for Anomaly Detection in Time Series , 2015, ESANN.

[5]  Konstantinos Fokianos,et al.  Log-linear Poisson autoregression , 2011, J. Multivar. Anal..

[6]  Mia Hubert,et al.  Minimum covariance determinant and extensions , 2017, 1709.07045.

[7]  Wai-Ki Ching,et al.  On High-dimensional Markov Chain Models for Categorical Data Sequences with Applications , 2010 .

[8]  Yuriy Kharin,et al.  Robust estimation for Binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies , 2021, J. Multivar. Anal..

[9]  L. Held,et al.  Multivariate modelling of infectious disease surveillance data , 2008, Statistics in medicine.

[10]  Peter A. W. Lewis,et al.  Discrete Time Series Generated by Mixtures. I: Correlational and Runs Properties , 1978 .

[11]  Konstantinos Fokianos,et al.  On binary and categorical time series models with feedback , 2014, J. Multivar. Anal..

[12]  Yuriy S. Kharin,et al.  Robust Regressive Forecasting under Functional Distortions in a Model , 2002 .

[13]  Kishan G. Mehrotra,et al.  Forecasting the behavior of multivariate time series using neural networks , 1992, Neural Networks.

[14]  Aristidis K. Nikoloulopoulos On the estimation of normal copula discrete regression models using the continuous extension and simulated likelihood , 2013, 1304.0905.

[15]  Roland Fried,et al.  ROBUST FITTING OF INARCH MODELS , 2014 .

[16]  C. M. Andreassen Models and inference for correlated count data , 2013 .

[17]  S. Kocherlakota,et al.  Bivariate discrete distributions , 1992 .

[18]  Yuriy S. Kharin Robustness in Statistical Forecasting , 2016 .

[19]  Dimitris Kugiumtzis,et al.  Markov chain order estimation with conditional mutual information , 2013 .

[20]  Dag Tjøstheim,et al.  Poisson Autoregression , 2008 .

[21]  Nan-Jung Hsu,et al.  Subset selection for vector autoregressive processes using Lasso , 2008, Comput. Stat. Data Anal..

[22]  C. Weiß,et al.  An Introduction to Discrete-Valued Time Series , 2018 .

[23]  D. Tjøstheim,et al.  Asymptotic normality and parameter change test for bivariate Poisson INGARCH models , 2018 .

[24]  J. Doob Stochastic processes , 1953 .

[25]  K. Fokianos,et al.  On categorical time series models with covariates , 2017, Stochastic Processes and their Applications.

[26]  Andréas Heinen,et al.  Multivariate autoregressive modeling of time series count data using copulas , 2007 .

[27]  W. Dunsmuir,et al.  On autocorrelation in a Poisson regression model , 2000 .

[28]  Construction of balanced incomplete block designs with partial cycles of blocks , 1973 .

[29]  Shan Hu,et al.  Dynamic Models for Time Series of Counts with a Marketing Application , 2015 .

[30]  D. Karlis,et al.  Finite mixtures of multivariate Poisson distributions with application , 2007 .

[31]  B. Kedem,et al.  Regression Theory for Categorical Time Series , 2003 .

[32]  P. Pfeifer,et al.  A Three-Stage Iterative Procedure for Space-Time Modeling Phillip , 2012 .

[33]  E. Frees,et al.  Nonparametric Estimation of Copula Regression Models With Discrete Outcomes , 2020, Journal of the American Statistical Association.

[34]  Richard A. Davis,et al.  Theory and inference for a class of nonlinear models with application to time series of counts , 2016 .

[35]  Robust estimation for the covariance matrix of multi‐variate time series , 2011 .

[36]  Pradeep Ravikumar,et al.  A review of multivariate distributions for count data derived from the Poisson distribution , 2016, Wiley interdisciplinary reviews. Computational statistics.

[37]  P. Billingsley,et al.  Statistical Methods in Markov Chains , 1961 .

[38]  P. Pfeifer,et al.  A Three-Stage Iterative Procedure for Space-Time Modeling , 1980 .

[39]  Claudia Czado,et al.  Pair Copula Constructions for Multivariate Discrete Data , 2012 .

[40]  R. Lund,et al.  Count Time Series Modeling with Gaussian Copulas. , 2020 .

[41]  A. Agresti,et al.  Categorical Data Analysis , 1991, International Encyclopedia of Statistical Science.

[42]  Andrew Harvey,et al.  Time Series Models for Count or Qualitative Observations , 1989 .

[43]  R. Douc,et al.  The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains , 2015, 1509.09048.

[44]  Mike West,et al.  Bayesian forecasting of multivariate time series: scalability, structure uncertainty and decisions , 2019, Annals of the Institute of Statistical Mathematics.

[45]  L. Rüschendorf Copulas, Sklar’s Theorem, and Distributional Transform , 2013 .

[46]  M. C. Jones,et al.  Robust and efficient estimation by minimising a density power divergence , 1998 .

[47]  Dimitris Karlis,et al.  On composite likelihood estimation of a multivariate INAR(1) model , 2013 .

[48]  Yuriy S. Kharin Statistical Analysis of Discrete-valued Time Series by Parsimonious High-order Markov Chains , 2020 .

[49]  Michel Denuit,et al.  Constraints on concordance measures in bivariate discrete data , 2005 .

[50]  Heng Liu Some Models for Time Series of Counts , 2012 .

[51]  L. Onsager Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[52]  Nizar Bouguila,et al.  Flexible Distribution-Based Regression Models for Count Data: Application to Medical Diagnosis , 2020, Cybern. Syst..

[53]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[54]  Geir Drage Berentsen,et al.  Recognizing and visualizing copulas : an approach using local Gaussian approximation , 2014 .

[55]  Yuriy S. Kharin,et al.  Semibinomial conditionally nonlinear autoregressive models of discrete random sequences: probabilistic properties and statistical parameter estimation , 2020, Discrete Mathematics and Applications.

[56]  Heinz Koeppl,et al.  Optimal Kullback–Leibler Aggregation via Information Bottleneck , 2013, IEEE Transactions on Automatic Control.

[57]  Mohamed Alosh,et al.  An integer-valued pth-order autoregressive structure (INAR(p)) process , 1990, Journal of Applied Probability.

[58]  C. Genest,et al.  A Primer on Copulas for Count Data , 2007, ASTIN Bulletin.

[59]  P. Bühlmann,et al.  Variable Length Markov Chains: Methodology, Computing, and Software , 2004 .

[60]  Rainer Dahlhaus,et al.  A Likelihood Approximation for Locally Stationary Processes , 2000 .

[61]  Yuriy Kharin,et al.  Statistical forecasting of the dynamics of epidemiological indicators for COVID-19 incidence in the Republic of Belarus , 2020 .

[62]  H. Joe Multivariate models and dependence concepts , 1998 .

[63]  E. J. Martinez,et al.  Robust Estimation in Vector Autoregressive Moving‐Average Models , 1999 .

[64]  Leonhard Held,et al.  Power-law models for infectious disease spread , 2013, 1308.5115.

[65]  Konstantinos Fokianos,et al.  Multivariate count autoregression , 2017, Bernoulli.

[66]  Konstantinos Fokianos,et al.  On count time series prediction , 2015 .

[67]  K. Fokianos,et al.  Mallows’ quasi-likelihood estimation for log-linear Poisson autoregressions , 2016 .

[68]  B. Sturmfels Geometry of Higher-Order Markov Chains , 2012, 1207.1899.

[69]  N. L. Johnson,et al.  Discrete Multivariate Distributions , 1998 .

[70]  Mingyao Li,et al.  Joint Regression Analysis of Correlated Data Using Gaussian Copulas , 2009, Biometrics.

[71]  R. Lund,et al.  Multivariate integer-valued time series with flexible autocovariances and their application to major hurricane counts , 2018 .

[72]  Yuriy S. Kharin,et al.  Statistical analysis of conditionally binomial nonlinear regression time series with discrete regressors , 2020 .

[73]  Yuewen Liu,et al.  Network Vector Autoregression , 2016, SSRN Electronic Journal.

[74]  John Cristian Borges Gamboa,et al.  Deep Learning for Time-Series Analysis , 2017, ArXiv.

[75]  Leonhard Held,et al.  Endemic-epidemic models with discrete-time serial interval distributions for infectious disease prediction. , 2019 .

[76]  Brendan K. Beare,et al.  Vine Copula Specifications for Stationary Multivariate Markov Chains , 2015 .

[77]  Jaakko Astola,et al.  Compression-Based Methods of Statistical Analysis and Prediction of Time Series , 2016, Springer International Publishing.

[78]  Robert C. Jung,et al.  Dynamic Factor Models for Multivariate Count Data: An Application to Stock-Market Trading Activity , 2008 .

[79]  Almut E. D. Veraart Modeling, simulation and inference for multivariate time series of counts using trawl processes , 2019, J. Multivar. Anal..

[80]  V. Girardin,et al.  Kullback-Leibler Approach to CUSUM Quickest Detection Rule for Markovian Time Series , 2018, Sequential Analysis.

[81]  L. Fahrmeir,et al.  Multivariate statistical modelling based on generalized linear models , 1994 .

[82]  Alessio Farcomeni,et al.  Generalized Linear Mixed Models Based on Latent Markov Heterogeneity Structures , 2015 .

[83]  Benjamin Kedem,et al.  Regression models for time series analysis , 2002 .

[84]  Roland Fried,et al.  Robust estimation of (partial) autocorrelation , 2015 .

[85]  Helmut Ltkepohl,et al.  New Introduction to Multiple Time Series Analysis , 2007 .

[86]  Nalini Ravishanker,et al.  Count Time Series: A Methodological Review , 2021 .

[87]  E. J. Hannan,et al.  Multivariate time series analysis , 1973 .

[88]  João Nicolau A New Model for Multivariate Markov Chains , 2014 .

[89]  Jacek Jakubowski,et al.  Intricacies of dependence between components of multivariate Markov chains: weak Markov consistency and weak Markov copulae , 2011, 1105.2679.

[90]  E. A. Medved,et al.  Statistical Estimation of Parameters for Binary Conditionally Nonlinear Autoregressive Time Series , 2018 .

[91]  Konstantinos Fokianos,et al.  Robust estimation methods for a class of log-linear count time series models , 2016 .

[92]  Konstantinos Fokianos,et al.  Poisson Network Autoregression , 2021 .

[93]  Sangyeol Lee,et al.  Robust Estimation for Bivariate Poisson INGARCH Models , 2021, Entropy.

[94]  Claudia Czado,et al.  Predictive Model Assessment for Count Data , 2009, Biometrics.

[95]  Robustness of Sequential Testing of Hypotheses on Parameters of M-valued Random Sequences , 2013 .

[96]  A. Raftery,et al.  Estimation and Modelling Repeated Patterns in High Order Markov Chains with the Mixture Transition Distribution Model , 1994 .

[98]  Nora Muler,et al.  Robust estimation for vector autoregressive models , 2013, Comput. Stat. Data Anal..

[99]  Fukang Zhu,et al.  A new bivariate integer-valued GARCH model allowing for negative cross-correlation , 2018 .

[100]  I. Olkin,et al.  Families of Multivariate Distributions , 1988 .

[101]  Neil Davey,et al.  Time Series Prediction and Neural Networks , 2001, J. Intell. Robotic Syst..

[102]  W. Wu,et al.  Gaussian Approximation for High Dimensional Time Series , 2015, 1508.07036.

[103]  Garvesh Raskutti,et al.  Learning High-Dimensional Generalized Linear Autoregressive Models , 2019, IEEE Transactions on Information Theory.

[104]  Dimitris Karlis,et al.  Some properties of multivariate INAR(1) processes , 2013, Comput. Stat. Data Anal..

[105]  Masahito Hayashi,et al.  Information geometry approach to parameter estimation in Markov chains , 2016 .

[106]  S. Zeger,et al.  Markov regression models for time series: a quasi-likelihood approach. , 1988, Biometrics.

[107]  Alain Latour,et al.  The Multivariate Ginar(p) Process , 1997, Advances in Applied Probability.

[108]  M. Sklar Fonctions de repartition a n dimensions et leurs marges , 1959 .

[109]  Yuriy S. Kharin,et al.  A Markov chain of order s with r partial connections and statistical inference on its parameters , 2007 .

[110]  Properties of a class of binary ARMA models , 2009 .

[111]  Rob J Hyndman,et al.  Theory & Methods: Non‐Gaussian Conditional Linear AR(1) Models , 2000 .

[112]  B. Jørgensen,et al.  A state-space model for multivariate longitudinal count data , 1999 .

[113]  Christophe Croux,et al.  Robust Estimation of the Vector Autoregressive Model by a Least Trimmed Squares Procedure , 2008 .

[114]  Yuriy S. Kharin,et al.  Statistical estimation of parameters for binary Markov chain models with embeddings , 2013 .

[115]  Yurij S. Kharin,et al.  Statistical Analysis of Spatio-Temporal Data Based on Poisson Conditional Autoregressive Model , 2015, Informatica.

[116]  M. Eichler Graphical modelling of multivariate time series , 2006, math/0610654.

[117]  Michael N. Katehakis,et al.  A SUCCESSIVE LUMPING PROCEDURE FOR A CLASS OF MARKOV CHAINS , 2012, Probability in the Engineering and Informational Sciences.

[118]  Roland Fried,et al.  On robust estimation of negative binomial INARCH models , 2021, METRON.

[119]  Zinsou Max Debaly,et al.  Stationarity and Moment Properties of some Multivariate Count Autoregressions , 2019, 1909.11392.

[120]  Jin J Zhou,et al.  Regression Models for Multivariate Count Data , 2017, Journal of computational and graphical statistics : a joint publication of American Statistical Association, Institute of Mathematical Statistics, Interface Foundation of North America.

[121]  Alain Latour,et al.  Integer‐Valued GARCH Process , 2006 .

[122]  A. Raftery A model for high-order Markov chains , 1985 .