A revised implicit equal‐weights particle filter

Abstract

Particle filters are fully nonlinear data assimilation methods and as such are highly relevant. While the standard particle filter degenerates for high‐dimensional systems, recent developments have opened the way for new particle filters that can be used in such systems. The implicit equal‐weights particle filter (IEWPF) is an efficient approach that avoids filter degeneracy because it gives equal particle weights by construction. The method uses implicit sampling, whereby auxiliary vectors drawn from a proposal distribution undergo a transformation before they are added to each particle. In the original formulation of the IEWPF, the proposal distribution has a gap, causing all but one particle to have an inaccessible region in state space. We show that this leads to a systematic bias in the predictions and we modify the proposal distribution to eliminate the gap. We achieved this by using a two‐stage proposal method, where a single variance parameter is tuned to obtain adequate statistical coverage properties of the predictive distribution. We discuss the properties of the implicit mapping from an auxiliary random vector to the state vector, keeping in mind the aim of avoiding particle resampling. The revised filter is tested on linear and weakly nonlinear dynamical models in low‐dimensional and moderately high‐dimensional settings, demonstrating the success of the new methodology in removing the bias.

Efficient nonlinear data assimilation using synchronization in a particle filter

“…A brief explanation of the modified IEWPF scheme (Skauvold et al . ) is as follows. For each particle two samples are drawn from a Gaussian distributed proposal q ( ξ , η ) = q ( ξ | η ) q ( η ), such that ξ is perpendicular to η .…”

“…), but with the extension advocated in Skauvold et al . (). This filter does not suffer from degeneracy by construction and allows for unbiased mean and covariance.…”

Massively parallel implicit equal-weights particle filter for ocean drift trajectory forecasting

“…Using (A.2), the expression for the weights from (10) The essence of the IEWPF is that in order to ensure a significant weight for all particles, α i is chosen so that all weights become equal to a target weight, w n i = w t ar get for i = 1, ..., N e , leading to a nonlinear equation for each α i . See [28] or the appendix of Skauvold et al [30] for further details.…”

“…which can be solved numerically for α i by, e.g., the Newton method, as illustrated by Skauvold et al [30]. Here, we use that c i = w t arget − c i = max j=1,..., N e {c j } − c i , (A.10)…”

“…To overcome the above-mentioned challenges, a revised two-stage IEWPF scheme has been proposed [30], which explores the complete proposal density and does not underestimate the posterior variance. The new update equation is ψ n i = ψ n,a i + β 1/2 P 1/2 ν i + α 1/2 i P 1/2 ξ i , (A.…”