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Data-Driven Approximation of Stationary Nonlinear Filters with Optimal Transport Maps

Mohammad Al-Jarrah, Bamdad Hosseini, Amirhossein Taghvaei

TL;DR

A data-driven nonlinear filtering algorithm for the case when the state and observation processes are stationary, referred to as optimal transport data-driven filter (OT-DDF), is evaluated for its significant computational efficiency during the online stage while maintaining the flexibility and accuracy of OT methods in nonlinear filtering.

Abstract

The nonlinear filtering problem is concerned with finding the conditional probability distribution (posterior) of the state of a stochastic dynamical system, given a history of partial and noisy observations. This paper presents a data-driven nonlinear filtering algorithm for the case when the state and observation processes are stationary. The posterior is approximated as the push-forward of an optimal transport (OT) map from a given distribution, that is easy to sample from, to the posterior conditioned on a truncated observation window. The OT map is obtained as the solution to a stochastic optimization problem that is solved offline using recorded trajectory data from the state and observations. An error analysis of the algorithm is presented under the stationarity and filter stability assumptions, which decomposes the error into two parts related to the truncation window during training and the error due to the optimization procedure. The performance of the proposed method, referred to as optimal transport data-driven filter (OT-DDF), is evaluated for several numerical examples, highlighting its significant computational efficiency during the online stage while maintaining the flexibility and accuracy of OT methods in nonlinear filtering.

Data-Driven Approximation of Stationary Nonlinear Filters with Optimal Transport Maps

TL;DR

A data-driven nonlinear filtering algorithm for the case when the state and observation processes are stationary, referred to as optimal transport data-driven filter (OT-DDF), is evaluated for its significant computational efficiency during the online stage while maintaining the flexibility and accuracy of OT methods in nonlinear filtering.

Abstract

The nonlinear filtering problem is concerned with finding the conditional probability distribution (posterior) of the state of a stochastic dynamical system, given a history of partial and noisy observations. This paper presents a data-driven nonlinear filtering algorithm for the case when the state and observation processes are stationary. The posterior is approximated as the push-forward of an optimal transport (OT) map from a given distribution, that is easy to sample from, to the posterior conditioned on a truncated observation window. The OT map is obtained as the solution to a stochastic optimization problem that is solved offline using recorded trajectory data from the state and observations. An error analysis of the algorithm is presented under the stationarity and filter stability assumptions, which decomposes the error into two parts related to the truncation window during training and the error due to the optimization procedure. The performance of the proposed method, referred to as optimal transport data-driven filter (OT-DDF), is evaluated for several numerical examples, highlighting its significant computational efficiency during the online stage while maintaining the flexibility and accuracy of OT methods in nonlinear filtering.
Paper Structure (14 sections, 3 theorems, 28 equations, 3 figures, 1 table, 1 algorithm)

This paper contains 14 sections, 3 theorems, 28 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Mathematical setup
  4. Modelling assumptions
  5. Objective
  6. The Optimal transport Data-Driven Filter
  7. OT formulation for conditioning
  8. Proposed methodology
  9. Error analysis
  10. Numerics
  11. The numerical algorithm
  12. Linear dynamics with linear and quadratic observation models
  13. Lorenz 63
  14. Discussion

Key Result

Theorem 1

If $\eta_X$ has a finite second moment and it is absolutely continuous with respect to the Lebesgue measure, then the max-min problem eq:OT-formulation-min-max admits a unique optimal pair $(\overline f,\overline T)$, modulo additive constant shifts for $\overline{f}$, and the relationship eq:OT-for

Figures (3)

  • Figure 1: Numerical results for the linear dynamic example with linear observation function. The left column shows the trajectory of the second component of the particles along with the trajectory of the true state, where $w=50$ for the OT-DDF method. The right column shows the MSE in estimating the state as a function of time in the upper corner and as a function of the window size $w$ in the lower corner.
  • Figure 2: Numerical results for the linear dynamic example with quadratic observation function. The left column shows the trajectory of the second component of the particles along with the trajectory of the true state, where $w=5$ for the OT-DDF method. The right column shows the MMD distance, with respect to the true posterior, as a function of time in the upper corner and as a function of the window size $w$ in the lower corner.
  • Figure 3: Numerical results for the Lorenz 63 example. The left column shows the trajectory of one of the unobserved components of the particles along with the trajectory of the true state, where $w=50$ for the OT-DDF method (the other components exhibit similar behavior). The right column shows the MSE, in estimating the true state, as a function of time in the upper corner and as a function of the window size $w$ in the lower corner.

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Lemma 1
  • Proposition 1
  • Remark 4
  • proof : Proof of Proposition \ref{['prop:mean-field']}