Predicting high-dimensional heterogeneous time series employing generalized local states

Kurzfassung

Tremendous advances in predicting the behavior of complex systems have been made in recent years by applying machine learning. For high-dimensional systems, these machine learning methods often suffer from the curse of dimensionality meaning that the number of nodes of the network has to be considerably larger than the dimensionality of the input data rendering the training unfeasible with a naive approach. With a parallel prediction scheme based on local states (LS), however, the forecasting of high-dimensional chaotic spatiotemporal systems of arbitrarily large extent becomes possible. The definition of LS relies on defining spatial local neighborhoods, thus the knowledge of the position of the time series in space is a necessary prerequisite for defining LS. Yet, the similarity of time series can also be defined in a much more general way by deducing a distance measure and thus a local neighborhood from the correlations among the time series. We employ this approach to define generalized local states (GLS) for the prediction of high-dimensional systems with which some of the shortcomings of the LS approach can be overcome. First, GLS can make excellent predictions in the case of mixed systems, where LS are doomed to fail. In our examples GLS is even able to infer the different origins of a set of heterogeneous time series, for which the generating processes are unknown. Second, prediction of high-dimensional systems remains feasible when no spatial information is available. This is more and more the typical case in real world applications, when analyzing such heterogeneous data sets like remote sensing data, financial data, social media data, etc.