Generalized ordinal patterns allowing for ties and their applications in hydrology

When using ordinal patterns, which describe the ordinal structure within a data vector, the problem of ties appears permanently. So far, model classes were used which do not allow for ties; randomization has been another attempt to overcome this problem. Often, time periods with constant values even have been counted as times of monotone increase. However, ties can contain valuable information which is disregarded by all of these approaches. To overcome this, a new approach is proposed: it explicitly allows for ties and, hence, considers more patterns than before. Ties are no longer seen as nuisance, but the information they carry is taken into account explicitly. Limit theorems in the new framework are provided, both, for a single time series and for the dependence between two time series. The methods are applied to hydrological data sets. In hydrology, it is common to distinguish five flood classes (plus ‘absence of flood’). Considering data vectors of these classes at a certain gauge in a river basin, one will usually encounter several ties. Co-monotonic behavior between the data sets of two gauges (increasing, constant, decreasing) can be detected by the method as well as spatial patterns. Thus, it helps to analyze the strength of dependence between different gauges in an intuitive way. This knowledge can be used to assess risk and to plan future construction projects.