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Generalized Ordinal Patterns Allowing for Ties and Their Applications in Hydrology

Alexander Schnurr, Svenja Fischer

Abstract

When using ordinal patterns, which describe the ordinal structure within a data vector, the problem of ties appeared 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. 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 to carry valuable information. 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 used on hydrological data sets. 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 asses risk and to plan future construction projects.

Generalized Ordinal Patterns Allowing for Ties and Their Applications in Hydrology

Abstract

When using ordinal patterns, which describe the ordinal structure within a data vector, the problem of ties appeared 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. 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 to carry valuable information. 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 used on hydrological data sets. 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 asses risk and to plan future construction projects.
Paper Structure (12 sections, 4 theorems, 51 equations, 6 figures, 4 tables)

This paper contains 12 sections, 4 theorems, 51 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Generalized Ordinal Pattern Dependence
  4. Generalized Average Weighted Ordinal Pattern Dependence
  5. Natural Estimators for the Ordinal Pattern Coefficient and the Total Score
  6. Limit Theorems
  7. Pairwise comparison in an SRD regime
  8. Limit Theorems in a Multivariate i.i.d. Setting
  9. Application and Simulation: Temporal and Spatial Patterns in Flood Classification
  10. The Temporal Approach
  11. The Spatial Approach
  12. Summary and Conclusions

Key Result

Proposition 2.2

The mapping $d_f:T_n\times T_n\to \mathbb{R}$ is a metric.

Figures (6)

  • Figure 1: The 13 generalized patterns of length 3
  • Figure 2: The first Fubini numbers
  • Figure 3: Two categorial data sets with partially co-monotonic behavior
  • Figure 4: The temporal approach (left) and the spatial approach (right)
  • Figure 5: Map of the Mulde River basin in Saxony, Germany and its location in Central Europe.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • ...and 1 more