Some selected problems in discrete-valued time series analysis/ Subhankar Chattopadhyay

Thesis (Ph.D) - Indian Statistical Institute, 2024

Includes bibliography and List of Publication(s)/Pre-print(s)

Introduction -- A generalized Pegram’s operator based autoregressive process for modelling categorical time series -- A modified Pegram’s operator based autoregressive process for truncated counts -- Change-point analysis through INAR
process with application to some COVID-19 data -- Analysis of count time series through INAR process with zero-inflation and
seasonality -- Epilogue

Guided by Prof. Atanu Biswas

This thesis analyzes some discrete-valued time series problems, which are classified into two types: (i) categorical time series and (ii) count time series. In this thesis, we primarily use two well-known models in the context of discrete-valued time series research: (i) Pegram’s operator-based autoregressive (PAR) process, which can be used to analyze both categorical and count data; and (ii) the integer-valued autoregressive (INAR) process, which is used for modelling count time series data. In Chapter 1, we review literature on discrete-valued time series and provide brief descriptions of our research works. Chapter 2 discusses a study on categorical time series. In this chapter, we propose a generalized PAR (GPAR) process that utilizes a generalized kernel to overcome the limitation of the traditional PAR process, which solely provides weights for the same previous category. Chapter 3 consists of a study of time series with truncated counts. In this chapter, we propose a modified PAR (mPAR) process with a modified kernel to model truncated counts in order to avoid the aforementioned drawback of the traditional PAR process. In Chapter 4, we consider the problem of change- point analysis in count time series data using an INAR(1) process with time-varying covariates. We employ the Poisson INAR(1) (PINAR(1)) process with a time-varying smoothing covariate in this study. This model allows us to model both components of active cases at time-point t: (i) survival cases from the previous time-point, and (ii) the number of new cases (innovations) at time-point t. In Chapter 5, we analyze count time series data with zero-inflation and seasonality. To capture both of these features, we propose an INAR(1) process that employs zero-inflated Poisson innovations with seasonality. We investigate the distributional properties and h-step ahead forecasting of all proposed processes. We conduct extensive simulation experiments to explore the usefulness of the proposed processes. Finally, we analyze some real datasets to provide practical illustrations of our proposed methods. In Chapter 6, we summarize our findings and discuss potential future directions for these works.