## Time Series Modeling for Categorical Data

### Short Course at the 2nd Dortmund-Bielefeld Summer School on Modern Topics in Time Series Analysis, University of Bielefeld

by Prof. Dr. Christian H. Weiß

at 10.09.2019, 13:30 to 17:00.

**Literature:**

Book „An Introduction to Discrete-Valued Time Series“ (Further details …)

Slides „Time Series Modeling for Categorical Data“ and references therein.

### Contents:

After a brief introduction to the challenges regarding discrete-valued time series, we prepare our course on categorical time series by briefly reviewing Markov chains, maximum likelihood estimation, etc., and by learning some basics about time series of bounded counts.

Then we turn to the question of how to analyze categorical time series, where we have to distinguish between the case of an ordinal and a nominal range. This dinstinction affects the visual analysis of categorical time series, the characterization of marginal properties like location and dispersion, and the measurement of serial dependence.

Finally, we consider the modeling of categorical time series. Among others, we discuss types of parsimoniously parametrized Markov models, a family of discrete ARMA models, the class of Hidden-Markov models and several types of regression models for categorical time series.

Throughout the lecture, models and methods are illustrated with diverse real-data time series using the statistical software R.

## 1. An Introduction to Discrete-Valued Time Series

### Course for the XIII Summer School in Statistics UPC-UB 2019 at the Universitat Politècnica de Catalunya in Barcelona

by Prof. Dr. Christian H. Weiß

at 25.-27.06.2019, 9:00-13:00, and 28.06.2019, 8:00-11:00.

**Literature:**

Book „An Introduction to Discrete-Valued Time Series“ (Further details …)

Slides „An Introduction to Discrete-Valued Time Series“ and references therein.

### Inhalt/Beschreibung:

After a brief introduction to the challenges regarding discrete-valued time series, we prepare our course on discrete-valued time series by briefly reviewing Markov chains, maximum likelihood estimation, and common count data models. Concerning the latter, we study their stochastic properties, like dispersion properties and types of generating functions, the latter being useful tools when analyzing such count data models.

Then we shall turn to the case of time-dependent counts, a topic that has attracted a lot of research activity during the last years. We start with one of first approaches to model such count data time series, the INAR(1) model by McKenzie (1985). This model uses the binomial thinning operator by Steutel and van Harn (1978) and can be understood as a special type of branching process with immigration. Stochastic properties of this model are discussed in great detail, including some current research results in this area. In particular, this basic count data time series model serves as the base for introducing approaches for model identification, for model estimation and for checking model adequacy. In addition, the forecasting of such discrete-valued processes is discussed in detail.

Then we consider extensions of the INAR(1) model, to higher model orders (e.g., INARMA family) or by using different types of thinning operators (e.g., random coefficient thinning), discuss their properties and illustrate their application to real-data. We conclude the part on thinning-based models by addressing the binomial AR(1) model (and some of its extensions) for finite counts in some details.

Another popular approach for modeling stationary and ARMA-like processes of counts are the INGARCH models, which are particularly attractive for overdispersed counts. Results concerning the basic model with a conditional Poisson distribution are presented, but also generalizations with, e.g., a binomial or negative binomial conditional distribution are considered. Again, data applications are presented.

Closely related to the INGARCH models are more general regression models for count data time series. Regression models are particularly useful if being concerned with non-stationary count data processes, e.g., processes exhibiting seasonality. After having presented and illustrated diverse types of such regression models, we turn to the last class for count data models to be discussed in this lecture, the hidden Markov models. These parameter-driven models assume a latent state process and generate their counts according to the present state. Again, stochastic properties, model estimation & diagnostics as well as forecasting are considered.

Then we turn to the question of how to analyze categorical time series, where we have to distinguish between the case of an ordinal and a nominal range. This dinstinction affects the visual analysis of categorical time series, the characterization of marginal properties like location and dispersion, and the measurement of serial dependence.

Then, we consider the modeling of categorical time series. Among others, we discuss types of parsimoniously parametrized Markov models, a family of discrete ARMA models, the class of Hidden-Markov models and several types of regression models for categorical time series.

Finally, we consider the topic of statistically monitoring discrete-valued processes. The concept of control charts is introduced, computational issues related to control charts are discussed, and common types of control charts for discrete-valued processes are presented.

Throughout the lecture, models and methods are illustrated with diverse real-data time series using the statistical software R.

## A Short Course in Categorical Time Series Analysis

### Short Course at the Department of Quantitative and Computing Methods, Universidad Politécnica de Cartagena

by Prof. Dr. Christian H. Weiß

at 11.09.2018 and 12.09.2018, 16:00-18:00.

**Literature:**

Book „An Introduction to Discrete-Valued Time Series“ (Further details …)

Slides „A Short Course in Categorical Time Series Analysis“ and references therein.

### Contents:

After a brief introduction to the challenges regarding discrete-valued time series, we prepare our course on categorical time series by briefly reviewing Markov chains, maximum likelihood estimation, etc., and by learning some basics about time series of bounded counts.

Then we turn to the question of how to analyze categorical time series, where we have to distinguish between the case of an ordinal and a nominal range. This dinstinction affects the visual analysis of categorical time series, the characterization of marginal properties like location and dispersion, and the measurement of serial dependence.

Finally, we consider the modeling of categorical time series. Among others, we discuss types of parsimoniously parametrized Markov models, a family of discrete ARMA models, the class of Hidden-Markov models and several types of regression models for categorical time series.

Throughout the lecture, models and methods are illustrated with diverse real-data time series using the statistical software R.

## Introduction to Integer-Valued Time Series

### Tutorial at the DAGStat Tagung 2016, Georg-August-Universität Göttingen

by Prof. Dr. Christian H. Weiß

at 14.03.2016, 9:00-17:00.

**Literature:**

Slides „An Introduction to Integer-Valued Time Series“ and references therein.

Also see my poster presentation „SPC Methods for Time-Dependent Processes of Counts“ at DAGStat-2016 (Details).

### Inhalt/Beschreibung:

We start with a brief survey of common count data models and their stochastic properties. In particular, we discuss dispersion properties and types of generating functions, the latter being useful tools when analyzing such count data models.

Then we shall turn to the case of time-dependent counts, a topic that has attracted a lot of research activity during the last years. We start with one of first approaches to model such count data time series, the INAR(1) model by McKenzie (1985). This model uses the binomial thinning operator by Steutel and van Harn (1978) and can be understood as a special type of branching process with immigration. Stochastic properties of this model are discussed in great detail, including some current research results in this area. In particular, this basic count data time series model serves as the base for introducing approaches for model identification, for model estimation and for checking model adequacy. In addition, the forecasting of such discrete-valued processes is discussed in detail.

Then we consider extensions of the INAR(1) model, to higher model orders (e.g., INARMA family) or by using different types of thinning operators (e.g., random coefficient thinning), discuss their properties and illustrate their application to real-data. We conclude the part on thinning-based models by addressing the binomial AR(1) model (and some of its extensions) for finite counts in some details.

Another popular approach for modeling stationary and ARMA-like processes of counts are the INGARCH models, which are particularly attractive for overdispersed counts. Results concerning the basic model with a conditional Poisson distribution are presented, but also generalizations with, e.g., a binomial or negative binomial conditional distribution are considered. Again, data applications are presented.

Closely related to the INGARCH models are more general regression models for count data time series. Regression models are particularly useful if being concerned with non-stationary count data processes, e.g., processes exhibiting seasonality. After having presented and illustrated diverse types of such regression models, we turn to the last class for count data models to be discussed in this lecture, the hidden Markov models. These parameter-driven models assume a latent state process and generate their counts according to the present state. Again, stochastic properties, model estimation & diagnostics as well as forecasting are considered. The lecture concludes with a brief look at NDARMA models.

## Introduction to Integer-Valued Time Series Models

### Mini-course „Integer-Valued Time Series Models“, LMB trimesters, Université de Franche-Comté in Besançon, France

by Prof. Dr. Christian H. Weiß

at 30.06.2014

**Literature:**

Lecture notes „An Introduction to Integer-Valued Time Series Models“ and references therein.

### Inhalt/Beschreibung:

We start with a brief survey of common count data models and their stochastic properties. In particular, we discuss dispersion properties and types of generating functions, the latter being useful tools when analyzing such count data models.

Then we shall turn to the case of time-dependent counts, a topic that has attracted a lot of research activity during the last years. We start with one of first approaches to model such count data time series, the INAR(1) model by McKenzie (1985). This model uses the binomial thinning operator by Steutel and van Harn (1978) and can be understood as a special type of branching process with immigration. Stochastic properties of this model are discussed in great detail, including some current research results in this area. Also extensions of this model, to higher model orders or by using different types of thinning operators, are briefly considered. Finally, to conclude the part on thinning-based models, the binomial AR(1) model for finite counts is addressed in some details.

Another popular approach for modeling stationary processes of counts are the INGARCH models, which are particularly attractive for overdispersed counts. Results concerning the basic model with a conditional Poisson distribution are presented, but also generalizations with, e.g., a binomial or negative binomial conditional distribution are considered. The lecture concludes with a brief survey of further alternatives like NDARMA, Hidden Markov or regression models.

Letzte Änderung: 1. August 2019