Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Stochastic Occupation Kernel Method for System Identification

Michael Wells, Kamel Lahouel, Bruno Jedynak

TL;DR

This work addresses nonparametric identification of stochastic differential equations from snapshot data by learning both drift and diffusion in an RKHS framework. It introduces a two-step method: first estimate the drift by matching the expected trajectory using occupation kernels, then estimate the diffusion-squared through Itô's isometry and a convex optimization over a positive semidefinite kernel parameter, effectively reduced to a finite-dimensional SDP via the representer theorem. The main contributions are deriving a practical finite-dimensional formulation for drift and diffusion estimation, handling multiple initial conditions, and solving the diffusion problem via a dual SDP. The results on synthetic 1D SDEs show accurate drift and diffusion fits, indicating the method's potential for data-driven, nonparametric SDE identification.

Abstract

The method of occupation kernels has been used to learn ordinary differential equations from data in a non-parametric way. We propose a two-step method for learning the drift and diffusion of a stochastic differential equation given snapshots of the process. In the first step, we learn the drift by applying the occupation kernel algorithm to the expected value of the process. In the second step, we learn the diffusion given the drift using a semi-definite program. Specifically, we learn the diffusion squared as a non-negative function in a RKHS associated with the square of a kernel. We present examples and simulations.

The Stochastic Occupation Kernel Method for System Identification

TL;DR

This work addresses nonparametric identification of stochastic differential equations from snapshot data by learning both drift and diffusion in an RKHS framework. It introduces a two-step method: first estimate the drift by matching the expected trajectory using occupation kernels, then estimate the diffusion-squared through Itô's isometry and a convex optimization over a positive semidefinite kernel parameter, effectively reduced to a finite-dimensional SDP via the representer theorem. The main contributions are deriving a practical finite-dimensional formulation for drift and diffusion estimation, handling multiple initial conditions, and solving the diffusion problem via a dual SDP. The results on synthetic 1D SDEs show accurate drift and diffusion fits, indicating the method's potential for data-driven, nonparametric SDE identification.

Abstract

The method of occupation kernels has been used to learn ordinary differential equations from data in a non-parametric way. We propose a two-step method for learning the drift and diffusion of a stochastic differential equation given snapshots of the process. In the first step, we learn the drift by applying the occupation kernel algorithm to the expected value of the process. In the second step, we learn the diffusion given the drift using a semi-definite program. Specifically, we learn the diffusion squared as a non-negative function in a RKHS associated with the square of a kernel. We present examples and simulations.
Paper Structure (7 sections, 39 equations, 3 figures)

This paper contains 7 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Description of the method
  3. Problem setup
  4. Estimating the drift
  5. Estimating the diffusion
  6. Experiment
  7. Conclusion

Figures (3)

  • Figure 1: One of the samples of the dataset we used.
  • Figure 2: Fits of the drift on three samples of the same random dataset. The predictions are in blue. The true function is in orange.
  • Figure 3: Fits of the diffusion on three samples of the same random dataset. The predictions are in blue. The true function is in orange.