Cluster Sampling Filters for Non-Gaussian Data Assimilation

Abstract

This paper presents a fully non-Gaussian filter for sequential data assimilation. The filter is named the "cluster sampling filter", and works by directly sampling the posterior distribution following a Markov Chain Monte-Carlo (MCMC) approach, while the prior distribution is approximated using a Gaussian Mixture Model (GMM). Specifically, a clustering step is introduced after the forecast phase of the filter, and the prior density function is estimated by fitting a GMM to the prior ensemble. Using the data likelihood function, the posterior density is then formulated as a mixture density, and is sampled following an MCMC approach. Four versions of the proposed filter, namely C MCMC, C HMC, MC-C HMC, and MC-C HMC are presented. C MCMC uses a Gaussian proposal density to sample the posterior, and C HMC is an extension to the Hamiltonian Monte-Carlo (HMC) sampling filter. MC-C MCMC and MC-C HMC are multi-chain versions of the cluster sampling filters C MCMC and C HMC respectively. The multi-chain versions are proposed to guarantee that samples are taken from the vicinities of all probability modes of the formulated posterior. The new methodologies are tested using a simple one-dimensional example, and a quasi-geostrophic (QG) model with double-gyre wind forcing and bi-harmonic friction. Numerical results demonstrate the usefulness of using GMMs to relax the Gaussian prior assumption especially in the HMC filtering paradigm.

A Robust Non-Gaussian Data Assimilation Method for Highly Non-Linear Models

“…For this reason, one prefers to use a different model to describe the prior error distribution. A Gaussian Mixture Model is frequently used to relax the Gaussian assumption over the forecast distribution [20][21][22][23].…”

Efficient Formulation and Implementation of Data Assimilation Methods

“…Thus, alternatives to EnKF formulations are a must under such circumstances, and therefore, sampling methods based on Markov chain Monte Carlo (MCMC) methods can be exploited to successfully sample from posterior error distributions. In "Cluster Sampling Filters for Non-Gaussian Data Assimilation" [14], Attia et al propose filters which account for non-Gaussian errors in prior and observations. Furthermore, the convergence of MCMC is sped up by using Verlet integrators.…”

Medical Image Retrieval based on the Parallelization of the Cluster Sampling Algorithm

“…Following the strategy described in [1], starting with a synthetic prior ensemble generated from a GMM with N c = 5. A GMM approximation of the true prior probability distribution is constructed using the EM algorithm.…”

“…Cluster sampling filters (C HMC, and MC-C HMC) [1] are developed as extension of the Hamiltonian Monte-Carlo (HMC) sampling filter presented in [3] where the true (unknown) prior distribution is approximate using a Gaussian mixture model (GMM). Given the current computational power, it is natural to try to run Monte-Carlo simulations in parallel.…”

“…Usually, derivative-based iterative minimization algorithms are employed to solve the problem described by (1). The derivative of the objective function T (H(x)) w.r.t the parameter ]x is given by:…”