Feedback particle filter for collective inference

Jin-Won Kim; Amirhossein Taghvaei; Yongxin Chen; Prashant G. Mehta

doi:10.3934/fods.2021018

Article Contents

2021, Volume 3, Issue 3: 543-561. Doi: 10.3934/fods.2021018

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Feedback particle filter for collective inference

1.
Coordinated Science Laboratory, University of Illinois, Urbana-Champaign, 1308 W. Main St., Urbana, IL 61801, USA
2.
Department of Mechanical and Aerospace Engineering, University of California, Irvine, 4100 Calit2 Building, Irvine, CA 92697-2800, USA
3.
School of Aerospace Engineering, Georgia Institute of Technology, Guggenheim 448B, Atlanta, GA 30332, USA

^* Corresponding author: Prashant G. Mehta
^* Corresponding author: Prashant G. Mehta

Received: February 2021

Revised: June 2021

Early access: August 2021

Published: September 2021

Kim and Mehta are supported in part by the C3.ai Digital Transformation Institute sponsored by C3.ai Inc. and the Microsoft Corporation, and in part by the National Science Foundation grant NSF 1761622. Chen is supported by the NSF 2008513

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Abstract

The purpose of this paper is to describe the feedback particle filter algorithm for problems where there are a large number ($ M $) of non-interacting agents (targets) with a large number ($ M $) of non-agent specific observations (measurements) that originate from these agents. In its basic form, the problem is characterized by data association uncertainty whereby the association between the observations and agents must be deduced in addition to the agent state. In this paper, the large-$ M $ limit is interpreted as a problem of collective inference. This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states. A feedback particle filter (FPF) algorithm for this problem is presented and illustrated via numerical simulations. Results are presented for the Euclidean and the finite state-space cases, both in continuous-time settings. The classical FPF algorithm is shown to be the special case (with $ M = 1 $) of these more general results. The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large $ M $.

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Mathematics Subject Classification: Primary: 60G35, 62M20; Secondary: 94A12.

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Figure 1. Normalized error for mean (blue circle) and variance (orange circle) with the KF and CKF algorithms. The KF algorithms were run as $ M $ independent Kalman filters with fully known data associations

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Figure 2. Normalized error for mean (blue circle) and variance (orange circle) with the CKF and FPF algorithms. The number of agents is fixed to $ M = 30 $ for this simulation

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