Implicit particle filters for data assimilation

Abstract

Implicit particle filters for data assimilation update the particles by first choosing probabilities and then looking for particle locations that assume them, guiding the particles one by one to the high probability domain. We provide a detailed description of these filters, with illustrative examples, together with new, more general, methods for solving the algebraic equations and with a new algorithm for parameter identification.

A Survey of Implicit Particle Filters for Data Assimilation

“…The samples we find thus depend on the map ξ j → X n+1 j we chose to solve (5). To obtain a high probability sample X n+1 j , we chose maps that satisfy the following conditions (see [26] for detailed explanation): the map should be (i) one-to-one and onto with probability one (so that the whole sample space is covered); (ii) smooth near the high-probability region of ξ (so that the weights do not vary unduly from particle to particle); and (iii) there should be an easy way to evaluate the Jacobians J = det ∂ X n+1 j /∂ ξ j (for efficient implementation).…”

“…This idea was presented in [26], and is related to the quadratic expansion construction in [18] (see Section 4.2 for more details). To find a suitable quadratic equation, expand F j to second order accuracy around its minimum:…”

“…Specifically, both method require the solution of an optimization problem (to search for the high probability regions in the target pdf), and sampling from a multivariate Gaussian density. However, the implicit particle filter avoids the difficulties which arise in the optimal importance function filter from the need to compute p(b n+1 |X n j ) in (26). Since the computation of this term can be expensive, the implicit particle filter seems to be more efficient and easier to implement.…”

“…The reason for the large difference in the number of particles required for these two filters is as follows. The large gap between observations implies that the SIR importance function and the target density become nearly mutually singular [19,20,26]. The "unguided" SIR particles are therefore very likely to become unlikely, and only very few of them carry a significant weight.…”

“…In what follows, we explain how implicit particle filters [25][26][27][28][29] tackle this problem. The basic idea is to first look for regions of high probability in the target density and then focus the particles onto these regions, so that only particles with significant weights are generated and the number of particles required remains manageable even if the state dimension is large.…”