Exploring Practical Estimates of the Ensemble Size Necessary for Particle Filters

Abstract

Particle filtering methods for data assimilation may suffer from the ''curse of dimensionality,'' where the required ensemble size grows rapidly as the dimension increases. It would, therefore, be useful to know a priori whether a particle filter is feasible to implement in a given system. Previous work provides an asymptotic relation between the necessary ensemble size and an exponential function of t 2 , a statistic that depends on observationspace quantities and that is related to the system dimension when the number of observations is large; for linear, Gaussian systems, the statistic t 2 can be computed from eigenvalues of an appropriately normalized covariance matrix. Tests with a low-dimensional system show that these asymptotic results remain useful when the system is nonlinear, with either the standard or optimal proposal implementation of the particle filter. This study explores approximations to the covariance matrices that facilitate computation in high-dimensional systems, as well as different methods to estimate the accumulated system noise covariance for the optimal proposal. Since t 2 may be approximated using an ensemble from a simpler data assimilation scheme, such as the ensemble Kalman filter, the asymptotic relations thus allow an estimate of the ensemble size required for a particle filter before its implementation. Finally, the improved performance of particle filters with the optimal proposal, relative to those using the standard proposal, in the same low-dimensional system is demonstrated.

Importance Sampling: Intrinsic Dimension and Computational Cost

“…Secondly, it can provide insight for filters initialized close to stationarity [24]. As in [93], [92], [95] we cast the analysis of importance sampling in joint space and consider as target µ := P u|y 1 , with u := (v 0 , v 1 ) and with the standard and optimal proposals defined in subsection 4.1.…”

“…Varying the parameters related to these three features may cause a breakdown of absolute continuity. The deterioration of importance sampling in large nominal dimensional limits has been widely investigated [9], [12], [94], [95], [93], [92]. In particular, the key role of the intrinsic dimension, rather than the nominal one, in explaining this deterioration was studied in [9].…”

Scientific Challenges of Convective-Scale Numerical Weather Prediction

“…A problem with PFs is that the number of particles required for accurate performance grows exponentially as the system's dimension increases (Bocquet et al 2010). Choosing the importance proposal densities that give a larger overlap with the conditional density may delay the filter collapse, or even prevent it (Slivinski and Snyder 2016). Hybrid EnKF-PF methods are promising alternative approaches to this problem (Chustagulprom et al 2016).…”

Efficient Assimilation of Simulated Observations in a High-Dimensional Geophysical System Using a Localized Particle Filter

“…The filter's stability for large applications can be improved by sampling from a proposal distribution conditioned on the previous model state and most recent observations. Recent studies by Slivinski and Snyder (2016) and Snyder et al (2015), however, suggest the weight collapse for high-dimensional systems may still be inevitable, even when the proposal distribution is nearly optimal. Despite this work, methods such as the equivalent-weights PF (van Leeuwen 2010; Ades and van Leeuwen 2015) have shown some skill in high-dimensional systems with relatively small numbers of particles.…”