# An Introduction to Discrete-Valued time Series - June 25th to 28th

**Date:**

June 25th to 28th. Moring: Tuesday to Thursday from 9.00 AM to 1.00 PM. Friday from 8.00 AM to 11.00 PM

**Classroom:**

PC2

**Instructor**

Prof. Dr. Christian H. Weiss

Education

1998 – 2004: Mathematics and Physics studies for teaching certificate (for secondary

schools) at the University of Würzburg; 1st Bavarian State Exam

(degree level) in Mathematics and Physics

1999 – 2004: Mathematics student at the University of Würzburg; Diplom in Mathematics

2002 – 2003: Studies abroad at the University of Helsinki, Finland

June 2009: Dr. rer. nat. at the Institute of Mathematics, University of Würzburg

Academic Positions

2004 – 2009: Lecturer at the University of Würzburg

2005 – 2009: Research Assistent at the Department of Mathematical Statistics, Institute

of Mathematics, at the University of Würzburg

2008 – 2009: Lecturer at the University of Cooperative Education in Bad Mergentheim

2009 – 2013: Akademischer Rat (permanent post) at the Department of Mathematics

at Darmstadt University of Technology

Since 2013: Professor at the Department of Mathematics and Statistics, Helmut

Schmidt University, Hamburg

Link to full CV, personal website, et cetera.

**Language**

English

**Description**

- Types of and challenges regarding discrete-valued time series
- Methods and models for count time series
- Methods and models for categorical time series
- Approaches for statistical process control
- Data illustrations with R

**Course goals**

- Broad knowledge & understanding about methods & models for discrete-valued time series
- Ability to implement these methods & models in statistical software (e.g., R), and to apply them to real-data examples

**Course contents**

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.

The course will follow the course instructors book

An Introduction to Discrete-Valued Time Series, John Wiley & Sons, Inc., Chichester, 2018.

**Targeted at**

Master and PhD students with a basic knowledge in time series analysis,

and a sound general education in Mathematics and Statistics

**Evaluation**

The evaluation method will be specified the first day of the course

**Computer class or student's laptop?**

Student's laptop

**Software requirements**

R (maybe some extra packages need to be installed during the course)