Introduction to Nonlinear Time Series Analysis
We start by an introduction on the goals of time series analysis. In order to explain why the word “nonlinear” appears at all, we refer to a time series as one realization of some general stochastic process, which, as Mathematics tells us, is uniquely defined by knowing all its n-point joint probability distributions. While Gaussian processes, which can be fully captured by linear processes, are completely characterized by their second order two-point statistics, going beyond implies to analyse nonlinear properties of the time evolution. We recall the classical example of stochastic resonance in order to highlight the potential consequences of non-linearity. We then study deterministic chaoticsystems as the paradigm of nonlinear dynamics, and discuss time series analysis tools for these. We interpret these in a wider context, and discuss several applications, including the forecasting method by analogues (E. Lorenz). From information theory, we come to tools for the detection of couling and causal influences when observing multivariate time series, needed for network reconstruction. We end with a specific class of model identification problems, namely inferring stochastic differential equations from data.