Nichtlineare Methoden zur Quantifizierung von AbhÃ¤ngigkeiten und Kopplungen zwischen stochastischen Prozessen basierend auf Informationstheorie
Kaiser
Andreas
Kaiser, Andreas
aut
2003-01-08
2003-01-14
2014-08-12
de
In order to determine the relation between two stochastic processes
information theory offers an appropriate framework in which 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.
<p>
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.
<p>
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.
urn:nbn:de:hbz:468-20030038
2014-08-12T08:33:54.236Z
2014-08-12T09:40:33.869Z
published
Diss
fb08/diss2002/kaiser