The aim of this work was the development of the Granger causality index
(GCI)to detect interactions between components of a multivariate biomedical
process. The starting point of methodological developments was Granger's
prediction concept. The principle states that a random process X influences
another random process Y if the prediction of Y is improved by adding the
information from the observations of X. For implementation of the GCI, all
theoretical models are suitable which provide a prediction error. In this
work, a time-variant GCI approach based on time-variant multivariate
autoregressive (MVAR) models and a regime-dependent GCI based on
self-exciting multivariate threshold autoregressive (MSETAR) models were
developed. MVAR and MSETAR models represent extensions of the classic
linear AR-approach. By means of time-variant MVAR models, multidimensional
processes with dynamic properties can be investigated. In contrast, MSETAR
models can be applied to analyze processes with nonlinear properties. A
focus of this work therefore was the development and application of a
multivariate GCI using both approaches.
Confidence intervals for the GCI and significance statements were achieved
by Bootstrap methods and surrogate procedures.
The time-variant GCI was applied to EEG as well as fMRI data. The EEG data
were recorded during an attention paradigm (Stroop task) and during a laser
stimulation, respectively. The fMRI data were registered during a motor
experiment (self-paced finger tapping). Using the time-variant GCI
approach, conclusions about the temporal development of directed
interactions between the investigated areas could be established. In
contrast to the bivariate approach, the multivariate or partial GCI
analysis between two signal components was able to eliminate the influence
of signal components which are involved but not included in the model. 
The combined application of regime-dependent and time-variant GCI
contributed fundamentally to the extent of information on directed
interactions between the BOLD responses of a fMRI signal, a significant and
complementary aid in the interpretation of the results.