Identification methods for nonlinear stochastic systems

Abstract

Model identifications based on orbit tracking methods are here extended to stochastic differential equations. In the present approach, deterministic and statistical features are introduced via the time evolution of ensemble averages and variances. The aforementioned quantities are shown to follow deterministic equations, which are explicitly written within a linear as well as a weakly nonlinear approximation. Based on such equations and the observed time series, a cost function is defined. Its minimization by simulated annealing or backpropagation algorithms then yields a set of best-fit parameters. This procedure is successfully applied for various sampling time intervals, on a stochastic Lorenz system.

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“…Although no general method exists for inferring the parameters of stochastic nonlinear dynamical models from measurements, various schemes have been proposed [33,20,22,35,19,42,48,45]. An important numerical technique, suggested in [43,20], is based on estimating drift and diffusion coefficients at a number of points in the phase space of the dynamical system.…”

“…The Bayesian method was used for parameter estimation in maps in the presence of dynamical [34] and weak measurement [35] noise and was applied to inference of continuous systems in [19].…”

“…the right-hand side of (1.17), is assumed known, and the unknown values of the associated coefficients are then estimated from data. This is the approach reported in [19], where the diffusion matrix is taken in the formD = τ…”

“…(Note that this is already more ambitious than what was done in [19], since we are attempting to estimate all model coefficients, including those that are equal to ±1. )…”

“…To make a fair comparison we use the same number of data points as in [19]. As an indication of the inference accuracy, we quote in Table 1.1 results for data simulated with the standard Lorenz parameter set and two values of dynamical noise intensity for weak and strong cases.…”