Page 372 - Contributed Paper Session (CPS) - Volume 7

CPS2133 Alexander Schnurr et al.
if the pattern is reversed in space and/or time. In our case that means we
should get almost equal relative frequencies for all patterns in /={(1,2,3),
(3,2,1)} as well as in 0={(1,3,2), (2,3,1), (3,1,2), (2,1,3)}. Unfortunately, it is easily
seen in the tables above that this assumption is not fulfilled in the empirical
results here. Anyway, it is interesting to mention that if we study the relative
frequencies of ordinal patterns in the integrated process of the all-year data
this assumption is satisfied (as one can see in Table 3).

Add data
{3,2,1} 48,35%
{1,2,3} 47,73%
{2,1,3} 0,96%
{3,1,2}
1,05%
{2,3,1} 1,00%
{1,3,2} 0,92%
Table 3 : relative frequencies of ordinal patterns in the integrated all-year data

So if one is interested in the distributions of the ordinal patterns of the
integrated process (which are not the same distributions as in the integrated
process of the original data set before transformation), one could apply the
improved estimator mentioned above

and would get asymptotic normality here, too, since the estimated Hurst
parameter of the all-year data (which plays the role of the increment process
here) is smaller than 0.75. Concerning the integrated process we could not
conclude asymptotic normality of the estimator ̂ by now, because in this

case the increment process would be long-range dependent and the
asymptotic behaviour in this case is ongoing research.
Finally, let us sketch some applications: In the context of model selection
ordinal patterns can be very helpful. A good model should (at least) match the
ordinal structure of the data sets under consideration. In a continuous-time
Markovian setting, our state-space dependent analysis can be used to decide
whether Levy processes are a useful model (homogeneous in space) or more
complicated models are appropriate (like Feller processes). In the future we
would like to understand the theoretical background better. Another long
time goal of the method presented here is to predict the length of extremal
events or the remaining time within one ‘regime’ by analyzing the
encountered patterns.

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