[Paper Review] Stochastic state estimation via incremental iterative sparse polynomial chaos based Bayesian-Gauss-Newton-Markov-Kalman filter

Bojana Rosić|arXiv (Cornell University)|Sep 12, 2019

Target Tracking and Data Fusion in Sensor Networks29 references5 citations

TL;DR

This paper proposes a novel incremental iterative Gauss-Newton-Markov-Kalman filter that uses sparse polynomial chaos expansions and Bayesian estimation to improve stochastic state estimation in nonlinear, chaotic systems. By reformulating the filter in an incremental, predictor-corrector form with pseudo-measurements and time-adaptive polynomial chaos, it achieves robust, sampling-free state estimation even under high noise and large time steps, as validated on the Lorenz-1984 system.

ABSTRACT

In this paper is proposed a novel incremental iterative Gauss-Newton-Markov-Kalman filter method for state estimation of dynamic models given noisy measurements. The mathematical formulation of the proposed filter is based on the construction of an optimal nonlinear map between the observable and parameter (state) spaces via a convergent sequence of linear maps obtained by successive linearisation of the observation operator in a Gauss-Newton-like form. To allow automatic linearisation of the dynamical system in a sparse form, the smoother is designed in a hierarchical setting such that the forward map and its linearised counterpart are estimated in a Bayesian manner given a forecasted data set. To improve the algorithm convergence, the smoother is further reformulated in its incremental form in which the current and intermediate states are assimilated before the initial one, and the corresponding posterior estimates are taken as pseudo-measurements. As the latter ones are random variables, and not deterministic any more, the novel stochastic iterative filter is designed to take this into account. To correct the bias in the posterior outcome, the procedure is built in a predictor-corrector form in which the predictor phase is used to assimilate noisy measurement data, whereas the corrector phase is constructed to correct the mean bias. The resulting filter is further discretised via time-adapting sparse polynomial chaos expansions obtained either via modified Gram-Schmidt orthogonalisation or by a carefully chosen nonlinear mapping, both of which are estimated in a Bayesian manner by promoting the sparsity of the outcomes. The time adaptive basis with non-Gaussian arguments is further mapped to the polynomial chaos one by a suitably chosen isoprobabilistic transformation. Finally, the proposed method is tested on a chaotic nonlinear Lorenz 1984 system.

Motivation & Objective

• To address the limitations of traditional Kalman filters in highly nonlinear and chaotic systems with noisy measurements.
• To overcome sampling errors and slow convergence of Monte Carlo methods in Bayesian state estimation.
• To develop a deterministic, sampling-free alternative to particle filters and ensemble methods for nonlinear state estimation.
• To improve convergence and accuracy in state estimation by using incremental pseudo-time stepping and Bayesian learning of polynomial chaos expansions.
• To enable robust initial state recovery in chaotic systems like Lorenz-84 under high measurement noise and infrequent observations.

Proposed method

• The filter constructs an optimal nonlinear map between observation and state space via successive linearisation in a Gauss-Newton-like form.
• It uses Bayesian relevance vector machines to estimate the forward map and its linearised counterpart in a hierarchical, sparse polynomial chaos framework.
• The incremental form assimilates current and intermediate states as pseudo-measurements, treating them as random variables to enable stochastic filtering.
• A predictor-corrector structure is employed: the predictor assimilates noisy measurements, while the corrector reduces mean bias in posterior estimates.
• Time-adaptive sparse polynomial chaos expansions are generated via modified Gram-Schmidt or nonlinear isoprobabilistic mapping, with basis functions estimated in a Bayesian manner.
• The method discretises the update equation using polynomial chaos expansions in standard normal variables, enabling efficient, exact computation without sampling.

Experimental results

Research questions

• RQ1 Can a Gauss-Newton-based iterative filter be made robust to high nonlinearity and measurement noise in chaotic systems?
• RQ2 How can the convergence and accuracy of deterministic filters be improved in nonlinear state estimation without relying on sampling?
• RQ3 To what extent can incremental pseudo-time stepping with Bayesian polynomial chaos improve posterior estimation in state estimation problems?
• RQ4 Can a fully deterministic, sampling-free method outperform traditional particle filters in terms of robustness and computational efficiency for chaotic systems?
• RQ5 How does the use of sparse, time-adaptive polynomial chaos with Bayesian learning affect the accuracy of conditional expectation approximation in state estimation?

Key findings

• The proposed filter successfully recovers the initial state of the chaotic Lorenz-1984 system even under high measurement noise and large time steps.
• The method achieves robust performance without sampling, avoiding sampling errors inherent in Monte Carlo-based approaches.
• The use of incremental pseudo-time stepping significantly improves convergence and stability of the state estimation process.
• The Bayesian estimation of polynomial chaos coefficients ensures sparsity and adaptivity, maintaining accuracy even at low measurement frequencies.
• The time-adaptive basis functions, generated via modified Gram-Schmidt or nonlinear mapping, preserve estimation accuracy across varying observation intervals.
• The predictor-corrector structure effectively reduces mean bias in posterior estimates, enhancing overall accuracy of the state reconstruction.