Ordinal Patterns: Entropy Concepts and Dependence Between Time Series

Since their introduction, ordinal patterns have proven to be a powerful tool not only in the context of dynamical systems, but also in time series analysis. Even though working with ordinal patterns leads to a loss of information, they bring many advantages which justify this loss. In this thesis, we contribute to ordinal pattern analysis in various ways.
With regard to the basics, we provide a comparative analysis of different representations of (multivariate generalizations of) ordinal patterns. Furthermore, we give a historical overview of the applications of ordinal patterns in data analysis and mathematical statistics. However, since there is already an extensive amount of literature available, we do not claim completeness.
Afterwards, we consider a specific measure of complexity in a time series or dynamical system, namely the symbolic correlation integral. We investigate it by providing limit theorems for an estimator of this quantity which is based on U-statistics under the assumption of short-range dependence. This also covers limit theorems for the Renyi-2 permutation entropy due to the close relation between these two. To this end, we slightly generalize existing limit theorems in the framework of approximating functionals. Afterwards we derive an estimator for the limit variance to lay the foundation for possible hypothesis tests.
Then, we turn our attention from the structure within a univariate time series to the structure between the components of a bivariate time series. Ordinal pattern dependence has been introduced in order to capture how strong the co-movement between two data sets or two time series is. Betken et al. (2021) aimed to show that ordinal pattern dependence fits into the axiomatic framework for multivariate measures of dependence between random vectors of the same dimension which had been proposed by Grothe et al. (2014). We reconsider the results by Betken et al. (2021). We show that there is an error with regard to the concordance ordering and that this cannot be verified in general for ordinal pattern dependence. Furthermore, we show that ordinal pattern dependence satisfies a modified set of axioms instead. In addition, we also consider ordinal pattern dependence in the context of supermodular ordering.
Finally, we prove general limit theorems for the distributions of multivariate ordinal patterns under the assumption of not only serial but also componentwise independence. We use our results to propose novel tests for cross-dependence. These include a test based on ordinal pattern dependence. We compare their performance with three competitors, namely classical Pearson’s and Spearman’s correlations and Chatterjee’s correlation coefficient. To this end, we conduct a comprehensive simulation study. Two real-world data examples complete this thesis.