Considering Temporal Structures in Independent Component Analysis

In this work we show how temporal structures in time series can be used in the framework of independent component analysis assuming the signals arise from Markov chains with finite order. Taking into account the past of the underlying processes, by using time embedding vectors, not only instantaneous independent but also uncoupled sources can be found. As a result signals which are gaussian distributed at each time can be decomposed as long as the time embedding vectors are non-gaussian. Using the model of independent time embedding vectors, we derive an algorithm (FastTeICA) which is similar to the well known FastICA algorithm introduced by Hyvärinen (1999). A weakening of the strict assumption of independent embedding vectors which still takes into account the dynamics of the processes for signal decomposition can be achieved by assuming independent increments, i.e. the change of state of the processes. Both approaches, independent states and independent dynamics, are a special case of the independence of the time embedding vectors.