Parameter Estimation via Fokker-Planck Type Residual: Application to Linear Stationary Random Vibration

TL;DR

This paper introduces a novel parameter estimation method for stochastic differential equations using Fokker-Planck residuals, which does not require explicit PDF forms and is applicable to complex random vibration systems.

Contribution

The paper presents a versatile, PDF-independent approach for estimating parameters in SDE models based on Fokker-Planck residuals, applicable even with complex noise and nonlinearities.

Findings

FPE residuals approach zero at true parameter values

Method successfully applied to two random vibration systems

Demonstrates robustness without explicit PDF derivation

Abstract

In this study, we propose a new method that is useful for estimating unknown parameter values of stochastic differential equation (SDE) models, based on probability density function (PDF) data measured from random dynamical systems. As our method does not require explicit description of PDF, it can be applied to the SDE models even when their PDFs are hardly derived in explicit forms due to multiplicative-noise terms, nonlinear terms, and so on. Therefore, our method is expected to provide a versatile tool to dynamically parameterize measured PDF data. In our proposed method, it is assumed that a measured PDF is obtained from a random dynamical system whose structure is described by a known SDE model with unknown parameter values. With the help of It\^o calculus, the Fokker-Planck equation (FPE) is derived from the SDE model. The measured PDF and a candidate of parameter values are substituted into the FPE to calculate a FPE residual. Our method is applied to two random vibration systems. Their FPE residuals tend to zero as the parameter values tend to exact values, showing that our proposed FPE residual can be utilized for unknown parameter estimation of SDE models.