Categorical time series analysis and applications in statistical quality control

Categorical (nominal) time series occur in various fields of practice like computer science, biology, linguistics, and others. In spite of their practical relevance, there does not exist monographic literature covering the different aspects of categorical time series analysis, and the few published research articles on categorical time series appeared scattered over magazines from different scientific areas like statistics, computer science, biology and others. In fact, many of the standard tools of statistics cannot be applied to categorical time series: A repertoire of standard distributions does not exist, a visual analysis is problematic, not even elementary mathematical operations can be applied to categorical values. Typical techniques of cardinal time series analysis for seasonal adjustment or trend elimination cannot be used for categorical time series, it is not even clear how to define the terms 'trend' or 'season' in this case. The present text attempts to fully describe the discipline of categorical time series analysis in the time domain. It is a compilation of known and new results, it integrates concepts from previously isolated research areas into an overall structure. Chapter I discusses approaches to an exploratory analysis of a given categorical time series. Approaches for sequence comparison, string matching and for detecting patterns and regularities in categorical time series are reviewed. A procedure for sequential pattern analysis, based on iterated function systems, can even be applied to visually mining patterns. Chapter II introduces basic concepts of categorical time series analysis in the time domain. Forms of a weak stationarity for categorical processes are proposed, which are of practical relevance for categorical time series analysis and modeling. They are also helpful to define measures of serial dependence, which are important to identify a suitable process model for a given categorical time series. Such models for categorical processes are discussed in chapter III. After a review of elementary process models of Bernoulli- and Markov-type, advanced models for categorical processes are discussed and analyzed in great detail. Also the special case of a binary process is considered. Chapter IV is centered around approaches towards a statistical analysis of categorical time series. Characteristic features of categorical processes like patterns or runs are investigated as well as models for time series of counts, which may arise from an appropriate transformation of a categorical process. Chapter V shows how the results of chapters I to IV can be applied to design approaches for controlling a categorical process. After having reviewed important concepts from statistical process control in general, approaches to monitor the marginal distribution of a categorical process, to monitor categorical features of the process like runs and patterns, and approaches to control a serially dependent process of counts are considered. Chapter VI demonstrates the practical relevance of the theory developed within this text by a number of real-data examples. These examples illustrate the different aspects of and approaches to categorical time series analysis.