Ordinal Patterns and Ordinal Pattern Dependence

Ordinal patterns describe the order structure of data points over a small time horizon. Using a moving window approach we reduce the complexity of a time series by analyzing the sequence of ordinal patterns instead of the original data. We present limit theorems for ordinal pattern probabilities and tests for structural breaks in the short-range dependent as well as in the long-range dependent setting. In the long-range dependent case, we investigate the ordinal information of a subordinated Gaussian process with a non-summable autocovariance function. We establish the asymptotic behavior of different estimators for ordinal pattern probabilities by using a multivariate Hermite decomposition.

Ordinal pattern dependence is a new way of measuring the degree of dependence between time series. Since it only relies on the ordinal structure of the data, it is robust against monotone transformations and measurement errors. This method has proved to be useful already in the context of hydrological, financial as well as medical data. Using this concept it is possible to analyze whether the dependence structure between two time series changes over time.