# American Institute of Mathematical Sciences

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

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: 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

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 $M$.

Citation: Jin-Won Kim, Amirhossein Taghvaei, Yongxin Chen, Prashant G. Mehta. Feedback particle filter for collective inference. Foundations of Data Science, doi: 10.3934/fods.2021018
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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
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