Ordinal patterns: Different representations and their application in the context of dependence

Since the seminal paper by Bandt and Pompe, so called ordinal patterns have been used extensively in contexts of data analysis, dynamical systems as well as time series analysis and mathematical statistics. Ordinal patterns are defined as the description of the order of the values in a data set (vector) of length d. Due to this simple definition, there are various ways on how to encode ordinal patterns. Sometimes authors pick a certain representation, telling their readers, why it is useful in the context they have in mind. Most of the time, however, one gets the impression, that the representation was chosen randomly or only because ‘others have used it before’.

Therefore, here we describe and analyze different approaches to represent ordinal patterns. All of these can be found in the literature. The most important representations (plus sub-classes) are compared in terms of their applicability in different contexts, namely, we consider digital implementation, inverse patterns and ties between values. Thereafter, we consider a simulation study with regard to dependence between time series using so called multivariate ordinal patterns in order to demonstrate the use of ordinal pattern representations in practice. At the end we provide a guideline on which occasions which representation should be used.