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CPS2133 Alexander Schnurr et al.
and robust under measurement errors. In particular in the context of
hydrological data it is an additional advantage that the method is robust with
respect to some kinds of structural breaks like ‘shift of mean’. Some other
methods (autocorrelation) are sensitive to these shifts which appear e.g. in
discharge data, if a dam is built. The variety of applications we have mentioned
above shows that the ordinal information is sufficient for several situations.
We analyze the probabilities of ordinal patterns respectively their relative
frequencies in given data sets of length . To this end we use a moving
window approach. That is, for every ∈ n we use the estimator:

For the probability

Here, and in the following we tacitly assume that the model we are
considering in the background satisfies standard assumptions like being
stationary as well as some ergodicity condition (cf. Section 4) in order to
ensure that ̂ converges to π.
In our considerations always denotes the length of the data set while
is the length of the pattern. In our empirical study, we have chosen = 3. There
is always a trade-off between the information contained in the pattern and the
number of parameters under consideration: since for data points we have to
consider ! parameters (the probability of each pattern), it is better to stick
with some small number although a larger results in a more detailed
knowledge of the ordinal structure of the process. In theoretical papers the
authors even let tend to infinity. This is mostly done in the context of
dynamical systems (cf. Bandt et al. (2002)).

3. Result
We are considering daily discharge data from the river Rhine measured in
th
st
Cologne from November 1 1816 to October 30 2013 which was extracted
from the webpage of GRDC (Global Runoff Data Center).
Since the winter months December, January and February are known to
fulfil the demanded property of stationarity better than the all year long data
we will consider them additionally. So after extracting the winter months for
each of the 197 years, we have transformed the two data sets (all-year data
and only-winter data) to standard-normal distributions. It is a standing
assumption in the literature that afterwards we can assume that the process
(̃ ), = 1,2, … is Gaussian (Chilès and Delfiner(1999)). This yields the nice

property that the increment process we will consider later on is also Gaussian.
Since ordinal patterns are not affected by monotone transformations this

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