Langevin-package: An R package for stochastic data analysis

In Langevin: Langevin Analysis in One and Two Dimensions

Description

The Langevin package provides functions to estimate drift and diffusion functions from data sets.

Details

This package was developed by the research group Turbulence, Wind energy and Stochastics (TWiSt) at the Carl von Ossietzky University of Oldenburg (Germany).

Mathematical Background

A wide range of dynamic systems can be described by a stochastic differential equation, the Langevin equation. The time derivative of the system trajectory \dot{X}(t) can be expressed as a sum of a deterministic part D^{(1)} and the product of a stochastic force Γ(t) and a weight coefficient D^{(2)}. The stochastic force Γ(t) is δ-correlated Gaussian white noise.

For stationary continuous Markov processes Siegert et al. and Friedrich et al. developed a method to reconstruct drift D^{(1)} and diffusion D^{(2)} directly from measured data.

\dot{X}(t) = D^{(1)}(X(t),t) + √{D^{(2)}(X(t),t)}\,Γ(t)\quad \mathrm{with}

D^{(n)}(x,t) = \lim_{τ \rightarrow 0} \frac{1}{τ} M^{(n)}(x,t,τ)\quad \mathrm{and}

M^{(n)}(x,t,τ) = \frac{1}{n!} \langle (X(t+τ) - x)^n \rangle |_{X(t) = x}

The Langevin equation should be interpreted in the way that for every time t_i where the system meets an arbitrary but fixed point x in phase space, X(t_i+τ) is defined by the deterministic function D^{(1)}(x) and the stochastic function √{D^{(2)}(x)}Γ(t_i). Both, D^{(1)}(x) and D^{(2)}(x) are constant for fixed x.

One can integrate drift and diffusion numerically over small intervals. If the system is at time t in the state x = X(t) the drift can be calculated for small τ by averaging over the difference of the system state at t+τ and the state at t. The average has to be taken over the whole ensemble or in the stationary case over all t = t_i with X(t_i) = x. Diffusion can be calculated analogously.

Author(s)

Philip Rinn

References

A review of the Langevin method can be found at:

Friedrich, R.; et al. (2011) Approaching Complexity by Stochastic Methods: From Biological Systems to Turbulence. Physics Reports, 506(5), 87–162.

For further reading:

Risken, H. (1996) The Fokker-Planck equation. Springer.

Siegert, S.; et al. (1998) Analysis of data sets of stochastic systems. Phys. Lett. A.

Friedrich, R.; et al. (2000) Extracting model equations from experimental data. Phys. Lett. A.

Honisch, C.; Friedrich, R. (2011). Estimation of Kramers-Moyal coefficients at low sampling rates.. Physical Review E, 83(6), 066701.

Langevin documentation built on Oct. 19, 2021, 5:06 p.m.