Ordinal-Pattern-Dependence: Limit Theorems and Structural Breaks in the Long Range Dependent Case with Applications to Hydrology, Medicine and Finance

• Applicant: Professor Dr. Alexander Schnurr
• Subject Area: Mathematics
• Term: since 2016
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 320834150

The origin of ordinal patterns is the theory of dynamical systems. One considers n consecutive data points only using their relative positions. This ordinal information is encoded in an archetype structure. 2014 I have introduced a method which uses ordinal patterns to measure the degree of dependence between time series: if the same patterns do appear at the same instants of time in both time series very often, this is called positive ordinal pattern dependence. In the analysis of negative dependence, reflected patterns are counted.Before my project was started, we were only able to analyze short-range dependent time series. In particular in hydrology, data sets are well-known to be long-range dependent. Hence, it was important to prove limit theorems in this framework. We obtained results of this kind and showed that the limit distribution depends on the interplay between the Hurst exponent and properties of the estimator under consideration. In hydrology our methods are used in order to plan gauge networks and to define homogeneous groups. However, we are facing also new problems: in hydrology often categorial data shows up (like flood alert levels in the range 0 to 5). In these data sets ties appear quite often. By now, these are considered as ‚monotone increasing‘. Hence, co-movement is systematically overestimated. The only solution is to consider explicitly pattern with ties and built a new theory dealing with this concept. This makes it also possible to deal with high frequency financial data. Here, also several ties do appear. Limit theorems and tests for structural breaks for these generalized patterns are main objectives for the next years. In addition, multivariate generalizations for ordinal patterns will be introduced. Using two-dimensional space as index set instead of one-dimensional time, allows us to analyze climate data on the earth’s surface. Since there are other dependence measures as well, we analyze the interplay between these and ordinal pattern dependence. The relation to Kendall's Tau, Spearman's Rho and classical correlation is quite well understood, however, these concepts have multivariate counterparts, which should be analyzed in the same way. Furthermore, copulas and distance correlation will be considered.In addition to our theoretical results, we have produced a package written in GNU R which is published on CRAN. Other researchers can use it free of charge. Every new theoretical result will be implemented in the package.

• DFG Programme: Research Grants