Nested smoothing algorithms for inference and tracking of heterogeneous multi-scale state-space systems

TL;DR

This paper introduces a nested filtering approach for Bayesian inference in multi-scale systems with different time-scales, enabling joint estimation of static parameters and dynamic states, demonstrated on a stochastic Lorenz 96 model.

Contribution

It develops a novel recursive nested filtering methodology combining SMC and UKF for multi-scale systems, addressing the challenge of joint parameter and state estimation.

Findings

Effective in estimating parameters and states in a three-scale Lorenz model

Combines SMC and UKF for efficient multi-scale inference

Demonstrates applicability to complex stochastic systems

Abstract

Multi-scale problems, where variables of interest evolve in different time-scales and live in different state-spaces, can be found in many fields of science. Here, we introduce a new recursive methodology for Bayesian inference that aims at estimating the static parameters and tracking the dynamic variables of these kind of systems. Although the proposed approach works in rather general multi-scale systems, for clarity we analyze the case of a heterogeneous multi-scale model with 3 time-scales (static parameters, slow dynamic state variables and fast dynamic state variables). The proposed scheme, based on nested filtering methodology of P\'erez-Vieites et al. (2018), combines three intertwined layers of filtering techniques that approximate recursively the joint posterior probability distribution of the parameters and both sets of dynamic state variables given a sequence of partial and noisy observations. We explore the use of sequential Monte Carlo schemes in the first and second layers while we use an unscented Kalman filter to obtain a Gaussian approximation of the posterior probability distribution of the fast variables in the third layer. Some numerical results are presented for a stochastic two-scale Lorenz 96 model with unknown parameters.