2019/02/19 by Amirhossein Taghvaei, Taghvaei, Amirhossein, Prashant G. Mehta +3 · 1 citation
Computer Science · Environmental Science · #FOS: Mathematics #Hydrological Forecasting Using AI #Hydrology and Drought Analysis #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Probability (math.PR) #Target Tracking and Data Fusion in Sensor Networks
paper · pdf · doi:10.48550/arxiv.1902.07263
openalex publication_date 2019/02/19 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Feedback particle filter (FPF) is a numerical algorithm to approximate the\nsolution of the nonlinear filtering problem in continuous-time settings. In any\nnumerical implementation of the FPF algorithm, the main challenge is to\nnumerically approximate the so-called gain function. A numerical algorithm for\ngain function approximation is the subject of this paper. The exact gain\nfunction is the solution of a Poisson equation involving a probability-weighted\nLaplacian \Δ_\ρ. The numerical problem is to approximate this solution\nusing em only finitely many particles sampled from the probability\ndistribution \ρ. A diffusion map-based algorithm was proposed by the\nauthors in a prior work to solve this problem. The algorithm is named as such\nbecause it involves, as an intermediate step, a diffusion map approximation of\nthe exact semigroup e\Δ_\ρ. The original contribution of this paper\nis to carry out a rigorous error analysis of the diffusion map-based algorithm.\nThe error is shown to include two components: bias and variance. The bias\nresults from the diffusion map approximation of the exact semigroup. The\nvariance arises because of finite sample size. Scalings and upper bounds are\nderived for bias and variance. These bounds are then illustrated with numerical\nexperiments that serve to emphasize the effects of problem dimension and sample\nsize. The proposed algorithm is applied to two filtering examples and\ncomparisons provided with the sequential importance resampling (SIR) particle\nfilter.\n