Examples of analysis of stochastic processes based on time series data

Time series data from stochastic processes described by the Langevin equation are analyzed. Analysis is based on estimation of the deterministic and random terms of the Langevin equation from data. The terms are presented as fields and inspected visually. Forming a model of the process, the terms are also used to reconstruct the deterministic and stochastic process trajectories. The deterministic term is approximated by an analytical ansatz. The equations obtained by the approximation are used to generate deterministic process trajectories and to study their linear stability. Influence of measurement noise on the estimates is also discussed.