Analysis of time series from stochastic processes

Abstract

Analysis of time series from stochastic processes governed by a Langevin equation is discussed. Several applications for the analysis are proposed based on estimates of drift and diffusion coefficients of the Fokker-Planck equation. The coefficients are estimated directly from a time series. The applications are illustrated by examples employing various synthetic time series and experimental time series from metal cutting.

New Role for Nonlinear Dynamics and Chaos in Integrated Semiconductor Laser Technology

“…Unlike Ref. [18], where the fundamental frequencies are on the order of kHz, we could not acquire a large amount of data points per period of oscillation since our fundamental frequencies were in the 1 GHz range. This is why we used polynomial interpolation and only focus on the deterministic content of the dynamics.…”

“…[18,19], stochastically equivalent to the evolution equation of _X t that is denoted above. Roughly speaking, for a periodic motion such distribution would resemble a ''tube,'' centered around the integral curves ofD .…”

“…Parallel to the procedure outlined in Ref. [18], we calculate the average deterministic drift vectorD. Unlike Ref.…”

“…The method of Ref. [18] was improved by 6 orders of magnitude, in order to analyze dynamics on a GHz scale. Using experimental data alone, we reconstructed the drift vectorD , which contains all relevant information of the deterministic part of the dynamics.…”

Communication: Microsecond peptide dynamics from nanosecond trajectories: A Langevin approach

“…The data-driven Langevin equation (dLE) represents an alternative approach to construct a low-dimensional dynamical model from MD data. [17][18][19][20][21][22][23][24] Based on a given time series, the dLE estimates the drift and the diffusion field of the dynamics, which are then employed to reproduce the main features of the original time series. As the method requires only local information, the input data need not to be Boltzmann weighted in order to warrant that the Langevin model yields correct Boltzmann-distributed results.…”