Page 366 - Contributed Paper Session (CPS) - Volume 7

CPS2133 Alexander Schnurr et al.

Space dependent ordinal pattern probabilities in
time series
Ines Muenker, Alexander Schnurr
Siegen University
Abstract
We consider the ordinal information of consecutive points in a given data
set. This information is encoded in a permutation which is called ordinal
pattern. Applications of ordinal patterns include statistical estimation of the
Hurst parameter in long-range dependent time series, calculation of the
Kolmogorov-Sinai entropy in dynamical systems as well as tests for structural
breaks. Recently it has been shown that the probabilities of ordinal patterns
in stationary time series are different, if – instead of the whole data set – only
extremal events are considered. Here, we analyze in an empirical study,
whether (and how) the current area of the data set influences the appearance
of certain patterns. It turns out that in the discharge data we consider, we
indeed find different pattern frequencies for different level sets.

Keywords
Order structure; model free data analysis; permutation; long-range
dependence; hydrology

1. Introduction
Ordinal patterns were invented by Bandt and Pompe (2002) in order to
analyze the chaotic behavior of dynamical systems. They have also been used
to analyze – mostly model-free – data from biology, medicine, finance and
hydrology (cf. Bandt and Shiha (2007), Keller et al. (2007) and Keller and Sinn
(2005)).
The concept works as follows: for consecutive data points, there are !
possibilities how they can be ordered (if ties are excluded). These possibilities
are called ordinal patterns. We encode them by writing down a permutation
as follows: first the index of the data point with the highest value, then the
index of the data point with the second highest value and so on. The
mathematical definition can be found in Section 2. In Figure 1 some of the 24
patterns of length 4 are showcased.

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