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CPS2133 Alexander Schnurr et al.
-1.8 the patterns (3,1,2) and (1,3,2) are getting a relative frequency of 0.2712
resp. 0.2627, while (2,1,3) and (2,3,1) only get 0.0254 resp. 0.0454. When the
smoke has settled this phenomenon can easily be explained in the geometric
interpretation of these ordinal pattern. In the case where the data values are
below a certain threshold one would expect the smallest entry of a data vector
with length =3 to be the second entry, and this is exactly what the patterns
(3,1,2) and (1,3,2) describe. Hence it is only natural that in the other extremal
event, namely when one only considers data values above a certain threshold
(in our case 1.8) then one observes the same phenomenon but only for the
respective space reversion of the patterns discussed above. This can be seen
in the last column of Table 1 concerning the winter data above.

4. Discussion and Conclusion
The most important result – which opens a new perspective on ordinal
patterns – is, that we indeed find the phenomenon of state-space-dependence
of ordinal pattern frequencies in a real-world data set. One advantage of the
method provided here is that it is (at least in the first step) model free. In order
to prove limit theorems, one has to make some assumptions on the theoretical
model in the background, but at first one does not have to care for the model.
However, let us shortly comment on the theoretical background of the pilot
study described above:
Since ordinal patterns of ( ) are uniquely determined by the increment
∈ℕ 0
process () where = − −1 , = 1,2, … we can now define the
∈ℕ

mapping (cf. Keller and Sinn (2011))
̃
Π(2, … . , n) ≔ Π(0, 2, 2 + 3, … , 2+. . . +n) = Π(1, … , n)
and hence rewrite the estimator above to

Betken and Wendler (2019+) have shown that there is a large class of long-
range dependent processes which satisfy a slightly technical condition such
that the increment process of these processes is shortrange dependent. This
basically means that the autocorrelations of this process are summable. The
estimator for the Hurst parameter yields =0.35 for the increment process of
the all-year data which confirms the theoretical result above. We can now
apply the results of Arcones (1994), Theorem 4 and hence get asymptotic
normality of our estimator. Let us mention that Keller and Sinn (2011) provide
an estimator for probabilities of ordinal patterns that has better statistical
properties than the one we have used here. However, in order to use this
estimator, one needs that the probability of each pattern remains unchanged

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