Marginalized Particle Flow Filter

Abstract

As an alternative to the Kalman filter and the particle filter, the particle flow filter has recently attracted interest for solving the curse of dimensionality of the particle filter. Compared with the particle filter, the particle flow filter can obtain a better performance in high-dimensional state spaces with fewer samples. However, for some unobservable state dimensions, the flow operation wastes computational resources. In this paper, we propose a marginalized particle flow filter to handle the unobservable sub-state estimation. In contrast to the standard particle flow filter, we only migrate those observable dimensions of each particle according to homotopy theory and estimate the unobservable dimensions using the Kalman filter. The proposed algorithm can enhance the estimation quality of the unobservable state space and reduce the runtime of the particle flow filter. We evaluate the performance of the proposed algorithm through a multi-target tracking simulation.

Appendix

In the appendices below, we provide the proofs of Lemma 2 in this article. This procedure consists of two parts. The first part is a Kalman update using the artificial measurement. Moreover, the second step consists of a time update using the result from the first step. Let us start by Eq. (20):

$$\begin{aligned} \begin{aligned} x_{k+1,b}&= F^{b,b}x_{k,b} + F^{b,a}x_{k,a} + G_{k}^{bb}w_{k,b}\\ {\hat{z}}_{k}&= F^{a,b}x_{k,b} + G_{k}^{aa}w_{k,a} \end{aligned} \end{aligned}$$

where ${\hat{z}}_{k} = x_{k+1,a} - F^{a,a}x_{k,a}$ is a “virtual” measurement. Analogously to the update of Kalman filter, we obtain

$$\begin{aligned} {\hat{x}}_{k|k,b}^{*}&= x_{k|k,b} + K_k \left( {\hat{z}}_{k} - F^{a,b}x_{k,b} \right) \\ P_{k|k,b}^{*}&= P_{k|k,b} - K_k S_k K_k^T \\ K_k&= P_{k|k,b} \left( F^{ab}\right) ^T S_k^{-1} \\ S_k&= F^{ab} P_{k|k,b} \left( F^{ab}\right) ^T + G_{k}^{aa}Q_{k,a}{G_{k}^{aa}}^T \end{aligned}$$

The time update is to compute

$$\begin{aligned} x_{k+1|k+1,b}&= F^{bb} {\hat{x}}_{k|k,b}^{*} + F^{ab}x_{k|k,a} \\&= F^{bb} x_{k|k,b} + F^{ab}x_{k|k,a} + F^{bb} K_k\left( {\hat{z}}_{k} - F^{a,b}x_{k,b}\right) \\&= F^{bb} x_{k|k,b} + F^{ab}x_{k|k,a} + L_k\left( {\hat{z}}_{k} - F^{a,b}x_{k,b}\right) \\ P_{k+1|k,b}&= F^{bb} P_{k|k,b}^{*} \left( F^{bb}\right) ^T + G^{bb}Q_{k}^{bb}{(G_k^{bb})}^T \\&= F^{bb}\left( P_{k|k,b} - K_k S_k K_k^T\right) (F^{bb})^T + G^{bb}Q_{k}^{bb}{(G_k^{bb})}^T \\&= F^{bb} P_{k|k,b} \left( F^{bb}\right) ^T + G^{bb}Q_{k}^{bb}{\left( G_k^{bb}\right) }^T - F^{bb} K_k S_k K_k^T (F^{bb})^T \\&= F^{bb} P_{k|k,b} \left( F^{bb}\right) ^T + G^{bb}Q_{k}^{bb}{\left( G_k^{bb}\right) }^T - L_k N_k L_k^T \end{aligned}$$

where

$$\begin{aligned} N_k&= S_k = F^{ab} P_{k|k,b} \left( F^{ab}\right) ^T + G_{k}^{aa}Q_{k,a}{G_{k}^{aa}}^T \\ L_k&= F^{bb} K_k = F^{bb} P_{k|k,b} \left( F^{ab}\right) ^T N_k^{-1} \end{aligned}$$