Second-order multi-object filtering with target interaction using determinantal point processes

Abstract

The probability hypothesis density (PHD) filter, which is used for multi-target tracking based on sensor measurements, relies on the propagation of the first-order moment, or intensity function, of a point process. This algorithm assumes that targets behave independently, an hypothesis which may not hold in practice due to potential target interactions. In this paper, we construct a second-order PHD filter based on determinantal point processes which are able to model repulsion between targets. Such processes are characterized by their first- and second-order moments, which allows the algorithm to propagate variance and covariance information in addition to first-order target count estimates. Our approach relies on posterior moment formulas for the estimation of a general hidden point process after a thinning operation and a superposition with a Poisson point process, and on suitable approximation formulas in the determinantal point process setting. The repulsive properties of determinantal point processes apply to the modeling of negative correlation between distinct measurement domains. Monte Carlo simulations with correlation estimates are provided.

Appendix: Janossy density approximation

Since the corrector terms $l^{(1)}_{z_{1:m}} ( x )$, $l^{(1)}_{z_{1:m}} (x;z)$, $l^{(2)}_{z_{1:m}} (x,y)$, $l^{(2)}_{z_{1:m}} (x,y;z)$, $l^{(2)}_{z_{1:m}} (x,y;z,z')$ in (3.15), (3.19) and the kernel update formula (5.8) have no closed form expression in the determinantal setting, we propose to use the Janossy density approximations

$$\begin{aligned} j^{(n)}_\Phi (x_1,\ldots , x_{r-1},x,x_{r+1},\ldots , x_n) \simeq J_\Phi (x,x) j^{(n-1)}_\Phi (x_1,\ldots , x_{r-1},x_{r+1},\ldots , x_n)\nonumber \\ \end{aligned}$$

(A.1)

$n\ge 1$, which corresponds to a (Poisson) first-order approximation, and

$$\begin{aligned}&j^{(n)}_\Phi (x_1,\ldots , x_{r-1},x,x_{r+1},\ldots , x_{p-1},y,x_{p+1},\ldots , x_n) \nonumber \\&\quad \simeq (J_\Phi (x,x)J_\Phi (y,y)- ( J_\Phi (x,y))^2 ) j^{(n-2)}_\Phi (x_1,\ldots , {\hat{x}}_r , \ldots , {\hat{x}}_p ,\ldots , x_n),\nonumber \\ \end{aligned}$$

(A.2)

$n\ge 2$, which corresponds to a second-order (determinant) approximation, obtained from (4.8) by assuming that the off-diagonal entries $J_\Phi (x_i,x_j)$, $i\not = j$, are small.

This Janossy approximation is specially relevant to $\alpha$-determinantal Ginibre point processes (GPP) which approximate a Poisson point process when $\alpha \in [-1,0)$ tends to 0, see Shirai and Takahashi [30].

Proposition A.1

Under (A.1) we have the first-order Poisson approximations $l^{(1)}_{z_{1:m}} (x) \simeq J_\Phi (x,x)$, $m\ge 0$, and

$$\begin{aligned} l^{(1)}_{z_{1:m}} (x;z) \simeq \frac{ J_\Phi (x,x) }{ l_c(z) + \int _\Lambda J_\Phi (u,u) {\tilde{l}}_d (z | u) \nu ( \hbox {d}u) }, \end{aligned}$$

$z\in z_{1:m}$, $x\in \Lambda$, $m\ge 1$.

Proof

By (3.16) and (A.1) we have

$$\begin{aligned} \Upsilon ^{(1)}_{z_{1:m}} (x)= & {} \displaystyle \sum _{S\subset \{1,\ldots , m\}} \sum _{p\ge |S|} \frac{q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p+1)}_\Phi (x_{1:p} , x ) \prod _{i \in S} {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\\simeq & {} J_\Phi (x,x) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S} {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\= & {} J_\Phi (x,x) j^{(m)}_\Xi (z_1,\ldots , z_m) , \end{aligned}$$

(A.3)

by (3.9), which yields the approximation $l^{(1)}_{z_{1:m}} ( x ) \simeq J_\Phi (x,x)$. On the other hand, for $r=1,\ldots ,m$, using again (A.1) and (3.9) we have

$$\begin{aligned}&j^{(m)}_\Xi (z_1,\ldots , z_m) \nonumber \\&\quad = \displaystyle \frac{\partial _{\delta _{z_1}} }{\partial g} \cdots \frac{\partial _{\delta _{z_m}} }{\partial g} {{{\mathcal {G}}}}_{\Phi , \Xi } ( \mathbf{1},g)_{\mid g=0} \nonumber \\&\quad = \displaystyle \sum _{p\ge 0} \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S|\le p \end{array}} \frac{ q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p)}_\Phi (y_{1:p}) \prod _{i \in S} {\tilde{l}}_d (z_i | y_i) \nu (dy_{1:p} ) \nonumber \\&\quad \simeq l_c(z_r) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \nonumber \\&\qquad \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&\qquad + \int _\Lambda J_\Phi (x_r,x_r) {\tilde{l}}_d (z_r | x_r) \nu ( \hbox {d}x_r) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S| \le p+1 , r\in S \end{array}} \frac{q_d^{p+1-|S|}}{(p+1-|S|)!} \prod _{j \notin S} l_c(z_j)\nonumber \\&\qquad \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S \setminus \{ r \} } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&\quad = l_c(z_r) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j)\nonumber \\&\qquad \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p}) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&\qquad + \int _\Lambda J_\Phi (u,u) {\tilde{l}}_d (z_r | u) \nu ( \hbox {d}u) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j)\nonumber \\&\qquad \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S \setminus \{ r \} } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&\quad = \left( l_c(z_r) + \int _\Lambda J_\Phi (u,u) {\tilde{l}}_d (z_r | u) \nu ( \hbox {d}u) \right) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \nonumber \\&\qquad \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&= \left( l_c(z_r) + \int _\Lambda J_\Phi (u,u) {\tilde{l}}_d (z_r | u) \nu ( \hbox {d}u) \right) j^{(m-1)}_\Xi (z_1,\ldots , z_{r-1},z_{r+1}, \ldots , z_m). \end{aligned}$$

(A.4)

We conclude by taking $z_r=z$ and noting that by (3.15) and (A.3)–(A.4) we have

$$\begin{aligned} l^{(1)}_{z_{1:m}} (x;z) = \frac{ \Upsilon ^{(1)}_{z_{1:m}\! \setminus z} (x) }{j^{(m)}_\Xi (z_{1:m})} \simeq J_\Phi (x,x) \frac{ j^{(m-1)}_\Xi ( z_{1:m} \! \setminus z ) }{j^{(m)}_\Xi (z_{1:m})}. \end{aligned}$$

Proposition A.2

Under (A.1)–(A.2), we have the second-order approximations

$$\begin{aligned}&l^{(2)}_{z_{1:m}} (x,y) \simeq J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2,\\&\quad l^{(2)}_{z_{1:m}} (x,y;z) \simeq \frac{ J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 }{ l_c(z) + \int _\Lambda J_\Phi (u,u) {\tilde{l}}_d (z | u) \nu ( \hbox {d}u) } , \end{aligned}$$

$z \in z_{1:m}$, $x,y\in \Lambda$, $m\ge 1$, and

$$\begin{aligned} l^{(2)}_{z_{1:m}} (x,y;z,z') \simeq \frac{ J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu ( \hbox {d}u) \nu ( \hbox {d}v) } , \end{aligned}$$

$z, z' \in z_{1:m}$, $z\not = z'$, $x,y\in \Lambda$, $m\ge 2$, where

$$\begin{aligned} s_c(z) : = l_c(z) + \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z | v)\nu (\hbox {d}v), \quad z\in \Lambda . \end{aligned}$$

(A.5)

Proof

By (3.21) and (A.2) we have

$$\begin{aligned}&\Upsilon ^{(2)}_{z_{1:m}} (x,y)\nonumber \\&\quad = \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p+2)}_\Phi (x_{1:p}, x, y ) \prod _{i \in S} {\tilde{l}}_d (z_i | x_i) \nu (\hbox {d}x_{1:p} ) \nonumber \\&\quad = (J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \nonumber \\&\qquad \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S} {\tilde{l}}_d (z_i | x_i) \nu (\hbox {d}x_{1:p} )\nonumber \\&\quad = ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) j^{(m)}_\Xi ( z_{1:m}), \end{aligned}$$

(A.6)

and for $r,u=1,\ldots ,m$, using (A.1)–(A.2) and (3.9) we find

$$\begin{aligned}&j^{(m)}_\Xi (z_{1:m}) \\&\quad = \displaystyle \sum _{n\ge 0} \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S|\le n \end{array}} \prod _{j \notin S} l_c(z_j) \frac{ q_d^{n-|S|}}{(n-|S|)!} \int _{\Lambda ^n} j^{(n)}_\Phi (y_{1:n}) \prod _{i \in S} {\tilde{l}}_d (z_i | y_i) \nu (dy_{1:n} )\\&\quad \simeq l_c(z_r) l_c(z_u) \sum _{p\ge 0} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r , u \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} \prod _{j \notin S} l_c(z_j) \int _{\Lambda ^p} j^{(p)}_\Phi (x_{1:p} ) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p})\\&\qquad + l_c(z_r) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_u | v)\nu (\hbox {d}v) \\&\qquad \times \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r \} \\ |S| \le p+1, u \in S \end{array}} \frac{q_d^{p+1-|S|}}{(p+1-|S|)!} j^{(p)}_\Phi (x_{1:p} ) \prod _{j \notin S } l_c(z_j) \prod _{i \in S \setminus \{ u\} } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \\&\qquad + l_c(z_u) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_r | v)\nu (\hbox {d}v) \\&\qquad \times \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ u \} \\ |S| \le p+1, r \in S \end{array}} \frac{q_d^{p+1-|S|}}{(p+1-|S|)!} j^{(p)}_\Phi (x_{1:p}) \prod _{j \notin S } l_c(z_j) \prod _{i \in S \setminus \{ r \} } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \\&\qquad + \int _\Lambda ( J_\Phi (x_r,x_r)J_\Phi (x_u,x_u)-J_\Phi (x_r,x_u)^2 ) {\tilde{l}}_d (z_r | x_r) {\tilde{l}}_d (z_u | x_u) \nu ( \hbox {d}x_r) \nu ( \hbox {d}x_u) \\&\qquad \times \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \\ |S| \le p+2 , r\in S \end{array}} \frac{q_d^{p+2-|S|}}{(p+2-|S|)!} j^{(p)}_\Phi (x_{1:p} ) \prod _{j \notin S} l_c(z_j) \prod _{i \in S \setminus \{ r , u \} } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \\&\quad = l_c(z_r) l_c(z_u) \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r , u\} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} j^{(p)}_\Phi (x_{1:p} ) \prod _{j \notin S} l_c(z_j) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \\&\qquad + l_c(z_r) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_u | v)\nu (\hbox {d}v) \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r , u \} \\ |S| \le p, u \in S \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} j^{(p)}_\Phi (x_{1:p} )\\&\qquad \prod _{j \notin S } l_c(z_j) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \\&\qquad + l_c(z_u) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_r | v)\nu (\hbox {d}v) \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r, u \} \\ |S| \le p, r \in S \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} j^{(p)}_\Phi (x_{1:p} )\\&\qquad \prod _{j \notin S } l_c(z_j) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \\ \end{aligned}$$

$$\begin{aligned}&\qquad + \int _{\Lambda ^2} ( J_\Phi (u,u)J_\Phi (v,v)-J_\Phi (u,v)^2 ) {\tilde{l}}_d (z_r | u) {\tilde{l}}_d (z_u | v) \nu ( \hbox {d}u) \nu ( \hbox {d}v) \nonumber \\&\qquad \times \sum _{p\ge 0} \int _{\Lambda ^p} \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r , u \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} j^{(p)}_\Phi (x_{1:p} ) \prod _{j \notin S} l_c(z_j) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&\quad = \left( l_c(z_r) l_c(z_u) + l_c(z_r) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_u | v)\nu (\hbox {d}v) + l_c(z_u) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_r | v)\nu (\hbox {d}v) \right. \nonumber \\&\left. \qquad + \int _{\Lambda ^2} (J_\Phi (u,u)J_\Phi (v,v)-J_\Phi (u,v)^2 ) {\tilde{l}}_d (z_r | u) {\tilde{l}}_d (z_u | v) \nu ( \hbox {d}u) \nu ( \hbox {d}v) \right) \nonumber \\&\qquad \times \sum _{p\ge 0} \int _{\Lambda ^p} \displaystyle \sum _{\begin{array}{c} S\subset \{1,\ldots , m\} \setminus \{ r , u \} \\ |S| \le p \end{array}} \frac{q_d^{p-|S|}}{(p-|S|)!} j^{(p)}_\Phi (x_{1:p} ) \prod _{j \notin S} l_c(z_j) \prod _{i \in S } {\tilde{l}}_d (z_i | x_i) \nu ( \hbox {d}x_{1:p}) \nonumber \\&\quad = \left( l_c(z_r) l_c(z_u) + l_c(z_r) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_u | v)\nu (\hbox {d}v) \right. \nonumber \\&\left. \qquad + l_c(z_u) \int _\Lambda J_\Phi (v,v) {\tilde{l}}_d (z_r | v)\nu (\hbox {d}v) \right. \nonumber \\&\qquad \left. + \int _{\Lambda ^2} ( J_\Phi (u,u)J_\Phi (v,v)-J_\Phi (u,v)^2 ) {\tilde{l}}_d (z_r | u) {\tilde{l}}_d (z_u | v) \nu ( \hbox {d}u) \nu ( \hbox {d}v) \right) \nonumber \\&\qquad \times j^{(m-2)}_\Xi (z_1,\ldots , z_{r-1},z_{r+1}, \ldots , z_{u-1},z_{u+1}, \ldots , z_m) \nonumber \\&\quad = \left( s_c(z_r) s_c(z_u) - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z_r | u) {\tilde{l}}_d (z_u | v) \nu ( \hbox {d}u) \nu ( \hbox {d}v) \right) \nonumber \\&\qquad \times j^{(m-2)}_\Xi (z_1,\ldots , z_{r-1},z_{r+1}, \ldots , z_{u-1},z_{u+1}, \ldots , z_m) . \end{aligned}$$

(A.7)

We conclude by taking $(z_r,z_u)=(z,z')$ and noting that by (3.20) and (A.6)–(A.7) we have

$$\begin{aligned} l^{(2)}_{z_{1:m}} (x,y,z,z')= & {} \frac{\Upsilon ^{(2)}_{z_{1:m}\! \setminus \{ z,z'\}} (x,y)}{j^{(m)}_\Xi (z_{1:m})} \\\simeq & {} ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) \frac{ j^{(m-2)}_\Xi ( z_{1:m}\! \setminus \{ z,z' \} ) }{j^{(m)}_\Xi (z_{1:m})} \\\simeq & {} \frac{ ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu ( \hbox {d}u) \nu ( \hbox {d}v) } , \end{aligned}$$

$z, z' \in z_{1:m}$, $z\not = z'$, $m\ge 2$. Similarly, by (3.19) and (A.4), (A.6) we also have

$$\begin{aligned} l^{(2)}_{z_{1:m}} (x,y;z)= & {} \frac{\Upsilon ^{(2)}_{z_{1:m}\! \setminus z } (x,y)}{j^{(m)}_\Xi (z_{1:m})} \\\simeq & {} ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) \frac{ j^{(m-1)}_\Xi ( z_{1:m}\! \setminus z ) }{j^{(m)}_\Xi (z_{1:m})} \\\simeq & {} \frac{ J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 }{l_c(z) + \int _\Lambda J_\Phi (u,u) {\tilde{l}}_d (z | u) \nu ( \hbox {d}u) }, \qquad z \in z_{1:m}, \quad m\ge 1. \end{aligned}$$

As a consequence of (3.18) and Proposition A.2, the second-order conditional factorial moment density of $\Phi$ given that $\Xi =z_{1:m}=(z_1,\ldots , z_m)$ will be approximated as

$$\begin{aligned}&\rho ^{(2)}_{\Phi \mid \Xi =z_{1:m}} (x,y) \simeq q_d^2 ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) \nonumber \\&\qquad + q_d \sum _{z\in z_{1:m}} \frac{ ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) \big ( {\tilde{l}}_d (z | x) + {\tilde{l}}_d (z | y) \big )}{ s_c(z) } \nonumber \\&\qquad + \sum _{\begin{array}{c} z,z'\in z_{1:m} \\ z \not = z' \end{array}} \frac{ ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y) }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu (\hbox {d}u) \nu (\hbox {d}v) }, \end{aligned}$$

(A.8)

$m\ge 0$, with $\rho ^{(2)}_{\Phi \mid \Xi =z_{1:m}} (x,x):=0$, $x\in \Lambda$.

Proposition A.3

The (approximate) kernel update formula is given by

$$\begin{aligned} K_{\Phi \mid \Xi =z_{1:m}} (x,y)^2\simeq & {} q^2_d J_\Phi (x,y)^2 + q_d J_\Phi (x,y)^2 \sum _{z\in z_{1:m}} \frac{ \big ( {\tilde{l}}_d (z | x) + {\tilde{l}}_d (z | y) \big ) }{ s_c ( z ) } \displaystyle \\&\displaystyle + J_\Phi ( x , x ) J_\Phi ( y , y ) \sum _{z,z'\in z_{1:m}} \frac{ {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y)}{ s_c(z) s_c(z') } \\&+ \sum _{\begin{array}{c} z,z'\in z_{1:m} \\ z \not = z' \end{array}} \frac{ (J_\Phi (x,y)^2 - J_\Phi (x,x)J_\Phi (y,y) ) {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y) }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu (\hbox {d}u) \nu (\hbox {d}v) } , \end{aligned}$$

$m\ge 0$, $x,y\in \Lambda$.

Proof

By (3.14) and Proposition A.1, we have the approximation

$$\begin{aligned} \mu ^{(1)}_{\Phi \mid \Xi =z_{1:m}} (x)= & {} q_d l^{(1)}_{z_{1:m}} ( x ) + \sum _{z\in z_{1:m}} {\tilde{l}}_d(x\mid z) l^{(1)}_{z_{1:m}} (x;z) \nonumber \\\simeq & {} q_d J_\Phi ( x , x ) + \sum _{z\in z_{1:m}} \frac{ J_\Phi ( x , x ) {\tilde{l}}_d (z | x)}{ l_c (z ) + \int _{\Lambda } {\tilde{l}}_d (z | u ) J_\Phi ( u,u) \nu ( \hbox {d}u ) } , \qquad m\ge 0,\nonumber \\ \end{aligned}$$

(A.9)

hence by (A.8) and (A.9), we find

$$\begin{aligned}&\rho ^{(2)}_{\Phi \mid \Xi =z_{1:m}} (x,y) - \mu ^{(1)}_{\Phi , \Xi =z_{1:m}} (x) \mu ^{(1)}_{\Phi , \Xi =z_{1:m}} (y) \\&\quad \simeq - q_d J_\Phi ( x , x )J_\Phi ( y , y ) \bigg ( q_d + \sum _{z\in z_{1:m}} \frac{ {\tilde{l}}_d (z | x) + {\tilde{l}}_d (z | y)}{ s_c(z) } \bigg )\\&\qquad - J_\Phi ( x , x )J_\Phi ( y , y ) \sum _{z,z'\in z_{1:m}} \frac{ {\tilde{l}}_d (z | x){\tilde{l}}_d (z' | y)}{ s_c (z ) s_c (z' ) } \\&\quad + q_d^2 ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) + q_d ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 )\\&\qquad \sum _{z\in z_{1:m}} \frac{ {\tilde{l}}_d (z | x) + {\tilde{l}}_d (z | y)}{ s_c(z) } \\&\quad + \sum _{\begin{array}{c} z,z'\in z_{1:m} \\ z\not = z' \end{array}} \frac{ ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y) }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu (\hbox {d}u) \nu (\hbox {d}v) } \\&= - q^2_d J_\Phi (x,y)^2 - q_d J_\Phi (x,y)^2 \sum _{z\in z_{1:m}} \frac{ {\tilde{l}}_d (z | x) + {\tilde{l}}_d (z | y) }{ s_c(z) } \\&\quad - J_\Phi ( x , x ) J_\Phi ( y , y ) \sum _{z,z'\in z_{1:m}} \frac{ {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y)}{ s_c (z ) s_c (z' ) } \\&\qquad + \sum _{\begin{array}{c} z,z'\in z_{1:m} \\ z\not = z' \end{array}} \frac{ ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y) }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu (\hbox {d}u) \nu (\hbox {d}v) } , \end{aligned}$$

$m\ge 0$, and we conclude by (4.4), i.e.

$$\begin{aligned} ( K_{\Phi \mid \Xi =z_{1:m}} (x,y) )^2 = \mu ^{(1)}_{\Phi \mid \Xi =z_{1:m}} (x) \mu ^{(1)}_{\Phi \mid \Xi =z_{1:m}} (y) - \rho ^{(2)}_{\Phi \mid \Xi =z_{1:m}} (x,y) \end{aligned}$$

and (A.10).

The next result, which provides an approximation formula for the posterior covariance of Proposition 3.4, is a consequence of Proposition A.3 and (A.9).

Corollary A.4

Under (A.1)–(A.2) the posterior covariance of $\Phi$ given that $\Xi =z_{1:m}=(z_1,\ldots , z_m)$ is approximated as

$$\begin{aligned}&{ c^{(2)}_{\Phi \mid \Xi =z_{1:m}} (A,B) \simeq q_d \int _{A\cap B} J_\Phi ( x , x ) \nu ( dx ) - q^2_d \int _{A\times B} J_\Phi (x,y)^2 \nu ( dx ) \nu ( dy ) } \nonumber \\&\qquad - q_d \sum _{z\in z_{1:m}} \frac{ 1 }{ s_c (z ) } \int _{A\times B} J_\Phi (x,y)^2 \big ( {\tilde{l}}_d (z | x) + {\tilde{l}}_d (z | y) \big ) \nu ( dx ) \nu ( dy ) \displaystyle \nonumber \\&\qquad \displaystyle + \sum _{z\in z_{1:m}} \frac{1}{ s_c (z ) } \bigg ( \int _{A\cap B} {\tilde{l}}_d (z | x) J_\Phi ( x , x ) \nu (\hbox {d}x)\nonumber \\&\qquad - \frac{ \int _A {\tilde{l}}_d (z | x) J_\Phi ( x , x ) \nu (dx ) \int _B {\tilde{l}}_d (z | y) J_\Phi ( y , y ) \nu (\hbox {d}y) }{ s_c (z ) } \bigg ) \nonumber \\&\qquad + \sum _{\begin{array}{c} z,z'\in z_{1:m} \\ z\not = z' \end{array}} \frac{ \int _{\Lambda ^2} ( J_\Phi (x,x)J_\Phi (y,y)-J_\Phi (x,y)^2 ) {\tilde{l}}_d (z | x) {\tilde{l}}_d (z' | y) \nu (\hbox {d}x) \nu (\hbox {d}y) }{ s_c(z) s_c(z') - \int _{\Lambda ^2} J_\Phi (u,v)^2 {\tilde{l}}_d (z | u) {\tilde{l}}_d (z' | v) \nu (\hbox {d}u) \nu (\hbox {d}v) }, \quad m \ge 0.\nonumber \\ \end{aligned}$$

(A.10)

1.1 Conclusion

Our observations have shown that the performance of the multi-target tracking PPP-based standard PHD filter is degraded in the presence of target interaction such as repulsion. To address this issue, we have constructed a second-order DPP-based PHD filter based on Determinantal Point Processes which are able to model repulsion between targets, and can propagate variance and covariance information in addition to first-order target count estimates. We have derived posterior moment formulas for the estimation of DPPs after thinning and superposition with a Poisson Point Process (PPP), based on suitable approximation formulas. Our numerical experiments include an assessment of the spooky effect on disjoint domains, with negative correlation estimates which are consistent with the nature of DPPs. We have also compared the robustness and performance recovery of the DPP and PPP-PHD filters when subjected to sudden changes in target numbers.