Page 367 - Contributed Paper Session (CPS) - Volume 7

CPS2133 Alexander Schnurr et al.
Over the last decade ordinal patterns have been used in different areas of
statistics: Sinn and Keller (2011) have used their relative frequencies in order
to estimate the Hurst parameter of a fractional Brownian motion. Sinn et al.
(2012) have described a method to detect change points in a given data set.
Schnurr (2014) and Schnurr and Dehling (2017) have analyzed the dependence
structure between two time series by means of ordinal patterns. The
relationship between ordinal pattern dependence and other kinds of
dependence has been analyzed in Schnurr and Muenker (2017).
Recently Oesting and Schnurr (2019) have tackled the question which
patterns can be found in clusters of extremal events. This is the starting point
of the present study. We ask ourselves the question whether the probabilities
of the patterns which appear depend on the current state of the process. To
this end we separate the state space of a stationary time series into blocks
which are analyzed one-by-one.
The paper is organized as follows: the ordinal pattern analysis is described
in Section 2 along with some statistical issues related to our data example. The
latter one is described in Section 3, where the reader finds in addition the
results of our study. The discussion and outlook in Section 4 round out the
paper.

2. Methodology
Our method of choice is the so called ordinal pattern analysis. Let us begin with
the formal definition of ordinal patterns (in this section we follow closely Schnurr
and Muenker (2017)): let ∈ ℕ = {1,2,3, … } be a positive integer and = (/, 0, …
1
, 1) ∈ ℝ a vector. Furthermore, let 1 denote the space of permutations of length
, that is,
1
1 ≔ { ∈ ℕ : 1 ≤ 9 ≤ and 9 ≠ ; whenever ≠ }.
The ordinal pattern of is the unique permutation
∏() = ( , … , ) ∈

1,

such that

The latter is to make a decision in the case of ties. Dealing with real world
data it is sometimes more appropriate to put a small noise on the data set in
order to get rid of the ties. Otherwise, following (ii) one overestimates the
probabilities of certain patterns. In the data set we consider in Section 3 it is
most likely that the ties we encounter are due to rounding.
One advantage of ordinal patterns is that the whole ordinal information,
that is, the up-and-down behaviour is kept, while the metrical information is
not considered. Hence, the method is stable under monotone transformations

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