Time series analysis in reconstructed state spaces and the predictability of extreme events

Times series analysis aims at the extraction of dynamical features of a time dependent phenomenon from observed data. This is a nontrivial task, if the corresponding phenomenon is complex in time and space. Dynamical structures unfold themselves in appropriate vector valued state spaces. The embedding procedure for the reconstruction of such state spaces from data will be reviewed, and analysis in reconstructed spaces will be illustrated by several experimental data sets. As data sources, we consider determinstic systems as well as nonlinear stochastic processes, also tolerating nonstationarity due to slow parameter variations. As an application, the prediction of turbulent gusts in surface wind will be presented. These are extreme events in a clearly complex system, which motivates us to stress the issue of extreme events on a more general level.