{"id":288,"date":"2017-10-05T13:28:29","date_gmt":"2017-10-05T11:28:29","guid":{"rendered":"https:\/\/www.hsu-hh.de\/mathstat\/?page_id=288"},"modified":"2026-02-17T12:42:04","modified_gmt":"2026-02-17T11:42:04","slug":"dissertation","status":"publish","type":"page","link":"https:\/\/www.hsu-hh.de\/mathstat\/weiss\/dissertation","title":{"rendered":"Dissertation"},"content":{"rendered":"\n

Categorical Time Series Analysis and Applications in Statistical Quality Control<\/h2>\n\n\n\n
\"Dissertation\"<\/div>
Christian H. Wei\u00df
Categorical Time Series Analysis<\/strong> and Applications in Statistical Quality Control<\/strong>
Dissertation (Fakult\u00e4t f\u00fcr Mathematik und Informatik der Universit\u00e4t W\u00fcrzburg), dissertation.de \u2013 Verlag<\/a>, Berlin, 2009.
552 Seiten, brosch\u00fcrt, \u20ac 63,00. ISBN<\/abbr> 978-3-86624-442-9

Dissertation’s Reference List \u00bb<\/a><\/div><\/div>\n\n\n\n

Summary<\/h3>\n\n\n\n

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.<\/p>\n\n\n\n

Contents<\/h3>\n\n\n\n
    \n
  1. Introduction<\/strong>
    \u00a0
    The Discipline of Categorical Time Series Analysis \u2666 Modeling Categorical Processes \u2666 Patterns in Categorical Time Series \u2666 Visually Analyzing Categorical Time Series \u2666 Statistical Control of Categorical Processes \u2666 Contributions of this Work \u2666 Structure of the Text
    \u00a0<\/li>\n\n\n\n
  2. Fields of Categorical Time Series Analysis<\/strong>
    \u00a0
    Intrusion Detection \u2666 Alarm Time Series in Telecommunication Networks \u2666 Event Pattern Analysis in Automation \u2666 Modeling Software Usage \u2666 Manufacturing Event Management \u2666 Biological Sequences \u2666 Musical Analysis \u2666 Speech and Text Recognition \u2666 Part-of-Speech Tagging
    \u00a0
    I. Exploratory Analysis of Categorical Time Series<\/li>\n\n\n\n
  3. \n
      \n
    1. \n
        \n
      1. Strings in Categorical Time Series<\/strong>
        \u00a0
        Categorical Time Series: Basic Terms and Notations \u2666 Suffix Tries and Suffix Trees \u2666 Similarities in the Range of a Categorical Time Series \u2666 Comparing Strings and Categorical Time Series \u2666 Expressing Similarity between Strings \u2666 The Dot Plot \u2013 Visual Sequence Comparison \u2666 String Matching
        \u00a0<\/li>\n\n\n\n
      2. Detecting Sequential Patterns<\/strong>
        \u00a0
        KDD and Data Mining \u2666 Association Rule Mining \u2666 Sequential Pattern Analysis \u2666 Sequential Pattern Analysis Based on Suffix Tries \u2666 Finding Frequent Patterns in a Categorical Time Series \u2666 Finding Rare Patterns in a Categorical Time Series
        \u00a0<\/li>\n\n\n\n
      3. Discovering Patterns in Categorical Time Series using IFS<\/strong>
        \u00a0
        Iterated Function Systems \u2666 Chaos Algorithm for Transformation of a Categorical Process \u2666 Pattern Discovery based on Fractal Time Series \u2666 Cube Transformations \u2666 Circle Transformations \u2666 Visual Tree Representations
        \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

        II. Foundations of Categorical Time Series Analysis<\/p>\n\n\n\n

          \n
        1. \n
            \n
          1. \n
              \n
            1. Statistical Properties of Categorical Random Variables<\/strong>
              \u00a0
              Measures of Location and Dispersion \u2666 Measures of Dependence \u2666 Basic Concepts \u2666 Proportional Reduction of Variation \u2666 Measures based on Pearson’s Chi\u00b2-Statistic \u2666 Further Measures of Dependence \u2666 Signed Dependence
              \u00a0<\/li>\n\n\n\n
            2. Statistical Properties of Categorical Processes<\/strong>
              \u00a0
              Stationarity of Categorical Processes \u2666 Concepts of Stationarity \u2666 The Rate Evolution Graph \u2666 Measures of Serial Dependence \u2666 Serial Dependence of Weakly Stationary Processes \u2666 Time Series Bitmaps \u2666 Periodicity of Categorical Processes \u2666 Trends in Categorical Processes \u2666 Categorical Features: Runs, Cycles, Patterns
              \u00a0<\/li>\n\n\n\n
            3. Elements of Categorical Time Series Analysis<\/strong>
              \u00a0
              Sequential Estimation of Probabilities \u2666 Marginal Probabilities \u2666 Conditional Probabilities \u2666 Smoothing Categorical Processes \u2666 Transforming Categorical Processes \u2666 Model Building: Identification, Estimation and Evaluation \u2666 Model Identification \u2666 Model Estimation \u2666 Model Evaluation and Selection \u2666 Forecasting Categorical Processes
              \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

              III. Modeling Categorical Time Series<\/p>\n\n\n\n

                \n
              1. \n
                  \n
                1. \n
                    \n
                  1. Fundamental Models for Categorical Time Series<\/strong>
                    \u00a0
                    Bernoulli and Markov Models \u2666 Elementary Properties of Ordinary Markov Chains \u2666 Asymptotic Properties of Ordinary Markov Chains \u2666 Estimation of Parameters and Model Choice \u2666 Variable Length Markov Models \u2666 Probabilistic Suffix Trees \u2666 Estimation of Parameters and Model Choice
                    \u00a0<\/li>\n\n\n\n
                  2. A Stochastic Model for Sequential Pattern Analysis<\/strong>
                    \u00a0
                    Rule Generation in Sequential Pattern Analysis \u2013 An Introduction \u2666 Rule Generation in Sequential Pattern Analysis \u2013 Implicit Assumptions \u2666 A Simple Model for Sequential Pattern Analysis \u2666 Model-Based Optimization of Sequential Pattern Analysis
                    \u00a0<\/li>\n\n\n\n
                  3. Advanced Models for Categorical Time Series<\/strong>
                    \u00a0
                    Mixture Transition Distribution Models \u2666 Properties of MTD(p) Models \u2666 Model Choice and Estimation of Parameters \u2666 A Generalization of MTD(p) Models \u2666 Infinite-Memory MTD Models \u2666 Discrete ARMA Models \u2666 Definition and Interpretation \u2666 Alternative Representations \u2666 Markov Chain Representation of NDARMA Models \u2666 Serial Dependence Structure \u2666 Model Identification and Estimation \u2666 Joint Distributions \u2666 Predicting NDARMA Processes \u2666 DAR(p) Models \u2666 DAR(p) Processes as Markov Chains \u2666 DMA(q) Models \u2666 Generalized Choice Models \u2666 The Backshift Process of NDARMA Models \u2666 Generalized Choice Models: Definition and Properties \u2666 Hidden Markov Models \u2666 State Space Models \u2666 Hidden Markov Models: Definition and Basic Properties \u2666 Hidden Markov Models: Model Estimation \u2666 Decoding the Hidden States \u2666 Regression Models \u2666 Introduction to Generalized Linear Models \u2666 Regression Models for Time Series
                    \u00a0<\/li>\n\n\n\n
                  4. Models for Binary Time Series<\/strong>
                    \u00a0
                    Basic Properties of Binary Processes \u2666 Serial Dependence and Stationarity of Binary Processes \u2666 Binarization of Categorical Processes \u2666 Binary Markov Processes \u2666 Properties of Binary Markov Chains \u2666 The Markov Binomial Distribution \u2666 Higher Order Binary Markov Processes \u2666 An ARMA Model for Binary Processes \u2666 The BinARMA(p, q) Model \u2666 The BinAR(p) Model \u2666 The BinMA(q) Model
                    \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

                    IV. Statistical Analysis of Categorical Time Series<\/p>\n\n\n\n

                      \n
                    1. \n
                        \n
                      1. \n
                          \n
                        1. Run Statistics for Categorical Processes<\/strong>
                          \u00a0
                          Runs and Run Length Distributions \u2666 Run Length Properties of Bernoulli and Markov Processes \u2666 Run Length Properties of NDARMA Processes \u2666 Run Length Properties of DAR(p) Processes \u2666 Run Length Properties of DMA(q) Processes \u2666 Cycles in Categorical Processes \u2666 Pattern Histograms \u2666 ARL Computation
                          \u00a0<\/li>\n\n\n\n
                        2. Patterns and Runs in Categorical Processes<\/strong>
                          \u00a0
                          Patterns in Markov Processes \u2666 Pattern Transitions \u2666 Distribution of Pattern Counts \u2666 Moments of Pattern Counts \u2666 Counting Runs in Markov Processes \u2666 Repeated Patterns
                          \u00a0<\/li>\n\n\n\n
                        3. Models for Time Series of Counts<\/strong>
                          \u00a0
                          NDARMA Models for Counts \u2666 Hidden Markov Models for Counts \u2666 Regression Models for Counts \u2666 The INGARCH Model \u2666 Thinning Operations \u2666 Binomial Thinning \u2666 Hypergeometric Thinning \u2666 Further Thinning Operations \u2666 The INAR(1) Model \u2666 Definition and Interpretation \u2666 Basic Properties \u2666 Regression Properties \u2666 Model Estimation \u2666 Joint Distributions \u2666 The Binomial AR(1) Model \u2666 BARMA Models for Binomial Marginals
                          \u00a0<\/li>\n\n\n\n
                        4. Advanced Integer-Valued ARMA Models<\/strong>
                          \u00a0
                          INMA(q) Models \u2666 Introduction to INMA(q) Models \u2666 Overall Process Distribution \u2666 INMA(q) \u2013 Independent Elements Model \u2666 INMA(q) \u2013 Changing States Model \u2666 INMA(q) \u2013 Lifetime Model \u2666 INMA(q) \u2013 Sale Model \u2666 INARMA Models with Poisson Marginals \u2666 Poisson INMA(q) Model \u2666 Poisson INAR(1) Model \u2666 Jumps in Poisson INARMA Processes \u2666 INAR(p) Models \u2666 Introduction to INAR(p) Models \u2666 INAR(p) \u2013 Moving Elements Model \u2666 INAR(p) \u2013 Independent Reproductions Model \u2666 INAR(p) \u2013 Lifetime Model \u2666 Combined INAR(p) Models \u2666 CINAR(p) \u2013 Identical Thinnings Model \u2666 CINAR(p) \u2013 Independent Thinnings Model \u2666 Binomial AR(p) Models \u2666 Binomial AR(p) \u2013 Identical Thinnings Model \u2666 Binomial AR(p) \u2013 Independent Thinnings Model
                          \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

                          V. Controlling Categorical Processes<\/p>\n\n\n\n

                            \n
                          1. \n
                              \n
                            1. \n
                                \n
                              1. An Introduction to Statistical Process Control<\/strong>
                                \u00a0
                                Terms and Aims of Statistical Process Control \u2666 A Brief Review of Standard Variables Control Charts \u2666 Shewhart Control Charts \u2666 Evaluating the Performance of Control Charts \u2666 EWMA Control Charts \u2666 CUSUM Control Charts \u2666 Hotelling\u2019s T2 Control Chart \u2666 Basic Concepts for Monitoring a Categorical Process
                                \u00a0<\/li>\n\n\n\n
                              2. Monitoring the Marginal Distribution of a Categorical Process<\/strong>
                                \u00a0
                                Monitoring Binary Processes \u2666 A Review of Sampling Approaches \u2666 Group Inspection of Dependent Binary Processes \u2666 Continuous Control: A Moving Average Approach \u2666 Continuous Control: An EWMA Approach \u2666 Monitoring Categorical Processes \u2666 Monitoring the Components of the Distribution p \u2666 Monitoring a Summarizing Statistic
                                \u00a0<\/li>\n\n\n\n
                              3. Monitoring Runs and Patterns in Categorical Processes<\/strong>
                                \u00a0
                                Monitoring Runs in Binary Processes \u2666 Monitoring Runs in Independent Binary Processes \u2666 Monitoring Runs in Dependent Binary Processes \u2666 Monitoring Runs in Categorical Processes \u2666 Monitoring Cycles in Categorical Processes \u2666 Monitoring Patterns in Categorical Processes \u2666 Monitoring the First Occurrence of a Critical Pattern \u2666 Continuous Monitoring of Patterns
                                \u00a0<\/li>\n\n\n\n
                              4. Controlling Processes of Counts<\/strong>
                                \u00a0
                                Controlling Processes of Independent Poisson Counts \u2666 Controlling INAR(1) Processes \u2666 Controlling INAR(1) Processes: Possible Approaches \u2666 Controlling INAR(1) Processes: ARL Performance \u2666 Controlling INAR(p) Processes \u2666 Controlling INMA(q) Processes \u2666 Controlling Binomial AR(1) Processes \u2666 Controlling Binomial AR(1) Processes: Possible Approaches \u2666 Controlling Binomial AR(1) Processes: ARL Performance
                                \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

                                VI. Applications<\/p>\n\n\n\n

                                  \n
                                1. \n
                                    \n
                                  1. \n
                                      \n
                                    1. Analysis of Shakespeare Data<\/strong>
                                      \u00a0
                                      Stationarity Analysis \u2666 Analysis of Serial Dependence Structure \u2666 Model-Based Analysis of Sequential Patterns
                                      \u00a0<\/li>\n\n\n\n
                                    2. Analysis of Diagnosis Data<\/strong>
                                      \u00a0
                                      Process Control with Estimated p0 \u2666 Process Control with Given p0 \u2666 Process Control without Knowledge on p0
                                      \u00a0<\/li>\n\n\n\n
                                    3. Analysis of Log-in Data<\/strong>
                                      \u00a0
                                      Model Building \u2666 Checking Model Adequacy \u2666 Control Charting in Phase II
                                      \u00a0<\/li>\n\n\n\n
                                    4. Analysis of Server Data<\/strong>
                                      \u00a0
                                      Stationarity Analysis of Categorical Time Series \u2666 Access Counts: Model Building \u2666 Analysis of User Activity \u2666 IP Counts per Minute: Model Building \u2666 IP Counts within Periods of 2 Minutes Length \u2666 Control Charting in Phase II
                                      \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\n\n\n

                                      VII. Appendix: Statistical Foundations<\/p>\n\n\n\n

                                        \n
                                      1. \n
                                          \n
                                        1. Probability Theory and Statistics: Basic Concepts<\/strong>
                                          \u00a0
                                          Probabilities in Discrete Sample Spaces \u2666 Random Variables \u2666 Moments of Discrete Random Variables \u2666 Generating Functions \u2666 Stochastic Processes \u2666 Likelihood Concepts \u2666 Cardinal ARMA Models \u2666 MA(q) Models \u2666 AR(p) Models
                                          \u00a0<\/li>\n\n\n\n
                                        2. Popular Discrete Distributions<\/strong>
                                          \u00a0
                                          The Binomial Distribution \u2666 The Poisson Distribution \u2666 The Negative Binomial Distribution \u2666 The Hypergeometric Distribution \u2666 The Multinomial Distribution \u2666 Generalized Binomial Distributions \u2666 Correlated Trials \u2666 Varying the Probability of Success \u2666 Varying the Number of Trials \u2666 The Quasi-Binomial Distribution \u2666 The Multivariate Poisson Distribution \u2666 The Generalized Poisson Distribution
                                          \u00a0<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

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