Nichtlineare Methoden zur Quantifizierung von Abhängigkeiten und ...

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Abstract
In order to determine the relation between two stochastic processes information
theory offers an appropriate framework in which of the relationships can be interpreted
in terms of information. The dependence can be measured with the mutual
information, giving the amount of information which both processes share, i.e.
the degree of similarities. Mutual information can also give hints for the coupling
direction, however, due to serial correlation in time the results might be misleading.
Additionally, only non-coupled systems can be distinguished from coupled
systems. To determine the coupling directions, the dynamics of the processes have
to be taken into account which leads to the transfer entropy. By considering the
past, transfer entropy measures the direct impact which the driving process has
on the future state of the driven process by excluding any influence due to the
serial correlations. Based on information theory, coupling strength is quantified
as the amount of effective information transmission from one process to the other.
Thus, transfer entropy allows to distinguish between unidirectional coupling and
bidirectional coupling.
While values for mutual information and transfer entropy can be easily archived
for processes with discrete state space, their estimations from finite data
sets are difficult. Partitioning the state space, mutual information and transfer
entropy of the discretised processes converge to the corresponding values of the
continuous processes if the partitions are refined. Furthermore, mutual information
shows monotonically increasing convergence and thus can be used to reject the
assumption of both processes being independent. For transfer entropy no similar
monotonic convergence seems to hold. Kernel estimators represent an alternative
approach in order to estimate information theoretical quantities. They are easy
to implement and the bias of the estimators due to serial correlations in the data
sets can be suppressed easily too.
A special class of stochastic processes are point processes. Here, the discrete
times at which an event occurs are of interest. Again mutual information can
be used to quantify dependence between two point processes but this time the
increments, i.e. the number of events within a certain time interval, have to be
considered. A weaker measure for dependence is the covariance of the increments
leading in a special case to the number of coincidences. Considering increments,
coupling directions can be determined with the transfer entropy as well. Unfortunately,
due to the large bias in the estimators the exact values of the information
transmissions cannot reliablely be given. When using increments, the time scale
on which dependence is detected is given by the length of the time intervals. As
an alternative method inter-event intervals and cross-event intervals are introduced
which are ordered to one discrete time index congruently. By calculating
the mutual information between these event intervals dependence between point
processes is detectable without choosing a certain time scale.