Calibrating the Gaussian multi-target tracking model

Abstract

We present novel batch and online (sequential) versions of the expectation–maximisation (EM) algorithm for inferring the static parameters of a multiple target tracking (MTT) model. Online EM is of particular interest as it is a more practical method for long data sets since in batch EM, or a full Bayesian approach, a complete browse of the data is required between successive parameter updates. Online EM is also suited to MTT applications that demand real-time processing of the data. Performance is assessed in numerical examples using simulated data for various scenarios. For batch estimation our method significantly outperforms an existing gradient based maximum likelihood technique, which we show to be significantly biased.

Appendices

Appendix 1: Kalman filter updates

• $(\mu _{t | t-1}, \Sigma _{t | t-1})= \text {KP}(\mu _{t-1 | t-1}, \Sigma _{t -1 | t-1})$:

  $$\begin{aligned} \mu _{t | t-1}&= F \mu _{t-1 | t-1} \\ \Sigma _{t | t-1}&= F \Sigma _{t -1 | t-1} F^{T} + W. \end{aligned}$$

• $(B_{t}, b_{t}, \Sigma _{t-1|t}) = \text {BT}(\mu _{t-1 | t-1}, \Sigma _{t-1 | t-1})$:

  $$\begin{aligned} B_{t}&= \Sigma _{t-1 | t} F^{T} (F \Sigma _{t-1 | t} F^{T} + W)^{-1} \\ b_{t}&= (I_{d_{x} \times d_{x}} - B_{t} F ) \mu _{t-1 | t-1} \\ \Sigma _{t-1 | t}&= (I_{d_{x} \times d_{x}} - B_{t} F ) \Sigma _{t-1 | t-1}. \end{aligned}$$

• $(\mu _{t | t}, \Sigma _{t | t}) = \text {KF}(\mu _{t | t-1}, \Sigma _{t | t-1}, c^{d}_{t}, y_{t})$:

  $$\begin{aligned} \mu _{t | t}&= \left\{ \begin{array}{l@{\quad }l} \mu _{t | t-1} + \Sigma _{t | t-1} G^{T} \Gamma _{t}^{-1} \epsilon _{t}, &{} c_{t}^{d} = 1, \\ \mu _{t | t-1}, &{} c_{t}^{d} = 0. \end{array}\right. \\ \Sigma _{t | t}&= \left\{ \begin{array}{l@{\quad }l} \Sigma _{t | t-1} - \Sigma _{t | t-1} G^{T} \Gamma _{t}^{-1} G \Sigma _{t | t-1}, &{} c_{t}^{d} = 1 \\ \Sigma _{t | t-1} , &{} c_{t}^{d} = 0. \end{array}\right. \end{aligned}$$

  where $\Gamma _{t} = G \Sigma _{t | t-1} G^{T} + V$ and $\epsilon _{t} = y_{t} - G \mu _{t | t-1}$.

Appendix 2: Recursive updates for sufficient statistics in a GLSSM

We present the update rules for the recursion variables of the intermediate functions for the sufficient statistics in (21),

$$\begin{aligned} P_{m, t, ij}, \;q_{m, t, ij}, \;r_{m, t, ij}, \quad 1 \le m \le 7, \end{aligned}$$

where $i, j = 1, \ldots , d_{x}$ for $m = 1, 3, 4, 5, 7$; $i = 1, \ldots , d_{x}, j = 1, \ldots , d_{y}$ for $m = 2$; and $i = 1, \ldots , d_{x}$, $j = 1$ for $m = 6$. All $P_{m, t, ij}$’s, $q_{m, t, ij}$’s and $r_{m, t, ij}$’s are $d_{x} \times d_{x}$ matrices, $d_{x} \times 1$ vectors and scalars, respectively. To avoid repetition, we present the recursions with averaging by the step-sizes $\{ \gamma _{t} \}_{t \ge 1}$. For the calculations of Sect. 3.2.1 and 3.2.2 and Algorithm 1 replace every instance of $(1-\gamma _{t})$ and $\gamma _{t}$ below by the constant $1$. For Sect. 3.3 and Algorithm 2, no change needed.

Start with the initial values $P_{m, 1, ij} = 0_{d_{x} \times d_{x}}$, $q_{m, 1, ij} = 0_{d_{x} \times 1}$, and $r_{m, 1, ij} = 0$ for all $m$ except $P_{1, 1, ij} = \gamma _{1} c_{1}^{d} e_{i}e_{j}^{T}$, $P_{7, 1, ij} = \gamma _{1} e_{i}e_{j}^{T}$, $q_{2, 1, ij} = c_{1}^{d} y_{1}(j) e_{i}$, and $q_{6, 1, i1} = \gamma _{1} e_{i}$. At time $t$, update

$$\begin{aligned} P_{1, t, ij}&= (1 - \gamma _{t}) B_{t}^{T} P_{1, t-1, ij} B_{t} + \gamma _{t} c^{d}_{t} e_{i}e_{j}^{T} \\ q_{1, t, ij}&= (1 \!-\! \gamma _{t}) \left[ B_{t}^{T} q_{1, t-1, ij} \!+\! B_{t}^{T} \left( P_{1, t-1, ij} + P_{1, t-1, ij}^{T} \right) b_{t} \right] \\ r_{1, t, ij}&= (1 - \gamma _{t}) \big [ r_{1, t-1, ij} + \text {tr} \left( P_{1, t-1, ij} \Sigma _{t-1 | t} \right) + q_{1, t-1, ij}^{T} b_{t} \\&+ b_{t}^{T} P_{1, t-1, ij} b_{t} \big ] \\ P_{2, t, ij}&= 0_{d_{x} \times d_{x}} \\ q_{2, t, ij}&= (1 - \gamma _{t}) B_{t}^{T} q_{2, t-1, ij} + \gamma _{t} c_{t}^{d} y_{t}(j) e_{i} \\ r_{2, t, ij}&= (1- \gamma _{t}) \left[ r_{2, t-1, ij} + q_{2, t-1, ij}^{T} b_{t} \right] \\ P_{3, t, ij}&= B_{t}^{T} \left[ (1 - \gamma _{t}) P_{3, t-1, ij} + \gamma _{t} e_{i}e_{j}^{T} \right] B_{t} \\ q_{3, t, ij}&= (1 - \gamma _{t}) B_{t}^{T} \left[ q_{3, t-1, ij} +\left( P_{3, t-1, ij} + P_{3, t-1, ij}^{T} \right) b_{t} \right] \\&+ \gamma _{t} B_{t}^{T} \left( e_{i}e_{j}^{T} + e_{j}e_{i}^{T} \right) b_{t} \\ r_{3, t, ij}&= (1 - \gamma _{t}) \big [ r_{3, t-1, ij} + \text {tr}(P_{3, t-1, ij} \Sigma _{t-1 | t}) + q_{3, t-1, ij}^{T} b_{t} \\&+ b_{t}^{T} P_{3, t-1,ij} b_{t} \big ] + \gamma _{t} \left( e_{j}^{T} \Sigma _{t-1|t} e_{i} + b_{t}^{T} e_{i} e_{j}^{T} b_{t} \right) \\ P_{4, t, ij}&= (1 - \gamma _{t}) B_{t}^{T} P_{4, t-1, ij} B_{t} + \gamma _{t} e_{i}e_{j}^{T} \\ q_{4, t, ij}&= (1 - \gamma _{t}) \left[ B_{t}^{T} q_{4, t-1, ij} + B_{t}^{T} \left( P_{4, t-1, ij} + P_{4, t-1, ij}^{T} \right) b_{t} \right] \\ r_{4, t, ij}&= (1- \gamma _{t}) \big [ r_{4, t-1, ij} + \text {tr} \left( P_{4, t-1, ij} \Sigma _{t-1 | t} \right) + q_{4, t-1, ij}^{T} b_{t} \\&+ b_{t}^{T} P_{4, t-1, ij} b_{t} \big ] \end{aligned}$$

$$\begin{aligned} P_{5, t, ij}&= (1 -\gamma _{t}) B_{t}^{T} P_{5, t-1, ij} B_{t} + \gamma _{t} e_{i}e_{j}^{T} B_{t} \\ q_{5, t, ij}&= (1 - \gamma _{t}) \left[ B_{t}^{T} q_{5, t-1, ij} + B_{t}^{T} \left( P_{5, t-1, ij} + P_{5, t-1, ij}^{T} \right) b_{t} \right] \\&+ \gamma _{t} e_{j} b_{t}^{T} e_{i} \\ r_{5, t, ij}&= (1 - \gamma _{t}) \big [ r_{5, t-1, ij} + \text {tr} \left( P_{5, t-1, ij} \Sigma _{t-1 | t} \right) + q_{5, t-1, ij}^{T} b_{t} \\&+ b_{t}^{T} P_{5, t-1, ij} b_{t} \big ] \\ P_{6, t, i1}&= 0_{d_{x} \times d_{x}} \\ q_{6, t, i1}&= (1 - \gamma _{t}) B_{t}^{T} q_{6, t-1, i1} \\ r_{6, t, i1}&= (1 - \gamma _{t}) \left[ r_{6, t-1, i1} + q_{6, t-1, i1}^{T} b_{t} \right] \\ P_{7, t, ij}&= (1 - \gamma _{t}) B_{t}^{T} P_{7, t-1, ij} B_{t} \\ q_{7, t, ij}&= (1 - \gamma _{t}) \left[ B_{t}^{T} q_{7, t-1, ij} + B_{t}^{T} \left( P_{7, t-1, ij} + P_{7, t-1, ij}^{T} \right) b_{t} \right] \\ r_{7, t, ij}&= (1 - \gamma _{t}) \big [ r_{7, t-1, ij} + \text {tr} \left( P_{7, t-1, ij} \Sigma _{t-1 | t} \right) + q_{7, t-1, ij}^{T} b_{t} \\&+ b_{t}^{T} P_{7, t-1, ij} b_{t} \big ] \end{aligned}$$

Appendix 3: SMC algorithm for MTT

An SMC algorithm is mainly characterised by how the particles are proposed, hence we present the proposal scheme of our SMC algorithm. This L-best assignments approach has appeared previously in the literature in similar ways, e.g. see Cox and Miller (1995); Ng et al. (2005) We exclude the superscripts for particle numbers from the notation for simplicity. Assume that $z_{1:t-1}$ is the ancestor of the particle of interest with weight $w_{t-1}$. We sample $z_{t} = (k_{t}^{b}, c_{t}^{s}, c_{t}^{d}, k_{t}^{f}, a_{t} )$ and calculate its weight by performing the following steps:

• Birth-death move: Sample $k_{t}^{b} \sim \mathcal {P}(\lambda _{b})$ and $c_{t}^{s}(j) \sim \text {Bernoulli}(p_{s})$ for $j = 1, \ldots , k_{t-1}^{x}$. Set $k_{t}^{s} = \sum _{j = 1}^{k_{t-1}^{x}} c_{t}^{s}$ and construct the $k_{t}^{s} \times 1$ vector $i_{t}^{s}$ from $c_{t}^{s}$. Set $k_{t}^{x} = k_{t}^{s} + k_{t}^{b}$.

• Prediction moments for state and observation: For $1 \le j \le k_{t}^{s}$, set

  $$\begin{aligned} (\mu _{t | t-1, j}, \Sigma _{t | t-1, j}) = \text {KP}(\mu _{t-1 | t-1, i_{t}^{s}(j)}, \Sigma _{t-1 | t-1, i_{t}^{s}(j)}); \end{aligned}$$

  for $k^{s}_{t}+ 1 \le j \le k_{t}^{x}$, set $( \mu _{t|t-1, j}, \Sigma _{t | t-1, j}) = ( \mu _{b}, \Sigma _{b} )$. Also, for $j = 1, \ldots , k_{t}^{x}$, calculate

  $$\begin{aligned} (\mu _{t, j}^{y}, \Sigma _{t, j}^{y} ) = ( G \mu _{t|t-1, j}, G \Sigma _{t | t-1, j} G^{T} + V ). \end{aligned}$$

• Detection and target-observation association Define the $k^{x}_{t} \times ( k^{y}_{t} + k^{x}_{t} )$ matrix $D_{t}$ as

  $$\begin{aligned} D_{t}(i, j) = \left\{ \begin{array}{l@{\quad }l} \log [ p_{d} \mathcal {N}(y_{t, i}; \mu _{t, j}^{y}, \Sigma _{t, j}^{y}) ] &{} \text { if } j \le k^{y}_{t}, \\ \log [ (1- p_{d}) \lambda _{f} / | \mathcal {Y} | ] &{} \text { if } i = j - k^{y}_{t}, \\ - \infty &{} \text { otherwise. } \end{array}\right. \end{aligned}$$

  and an assignment is a one-to-one mapping $\alpha _{t} : \{ 1, \ldots , k^{x}_{t} \} \rightarrow \{ 1, \ldots , k^{y}_{t} + k^{x}_{t} \}$. The cost of the assignment, up to an identical additive constant for each $\alpha _{t}$ is

  $$\begin{aligned} d(D_{t}, \alpha _{t}) = \sum _{j = 1}^{k_{t}^{d}} D_{t}(j, \alpha _{t}(j)). \end{aligned}$$

  Find the set $\mathcal {A}_{L} = \{ \alpha _{t, 1}, \ldots , \alpha _{t, L} \}$ of $L$ assignments producing the highest assignment scores. The set $\mathcal {A}_{L}$ can be found using the Murty’s assignment ranking algorithm (Murty 1968). Sample $\alpha _{t} = \alpha _{t, j}$ from $\mathcal {A}_{L}$ with probability

  $$\begin{aligned} \frac{\exp [d(D_{t}, \alpha _{t, j})]}{\sum _{j^{\prime } = 1}^{L} \exp [d(D_{t}, \alpha _{t, j^{\prime }})]}, \quad j = 1, \ldots , L \end{aligned}$$

  Given $\alpha _{t}$, one can infer $c^{d}_{t}$ (hence $i^{d}_{t}$), $k^{d}_{t}$, $k^{f}_{t}$ and the association $a_{t}$ as follows: for $1 \le k \le k_{t}^{x}$,

  $$\begin{aligned} c^{d}_{t}(k) = \left\{ \begin{array}{ll} 1 &{} \text { if } \alpha _{t}(k) \le k^{y}_{t}, \\ 0 &{} \text { if } \alpha _{t}(k) > k^{y}_{t}. \end{array}\right. \end{aligned}$$

  Then $k^{d}_{t} = \sum _{j = 1}^{k^{x}_{t}} c^{d}_{t}(k)$, $k^{f}_{t} = k^{y}_{t} - k^{d}_{t}$, $i^{d}_{t}$ is constructed from $c^{d}_{t}$, and finally

  $$\begin{aligned} a_{t}(k) = \alpha _{t}(i^{d}_{t}(k)), \qquad k = 1, \ldots , k^{d}_{t}. \end{aligned}$$

• Filtering moments for state: For $1 \le k \le k^{x}_{t}$ calculate

  $$\begin{aligned} (\mu _{t | t, k}, \Sigma _{t | t, k}) = \text {KF}(\mu _{t | t-1, k}, \Sigma _{t | t-1, k}, c^{d}_{t}(k), y_{t, a'_{t}(k)} ). \end{aligned}$$

  where $a'_{t}(k) = a_{t}( \sum _{j = 1}^{k} c_{t}^{d}(j))$.

• Reweighting: After sampling $z_{t} = (k_{t}^{b}, c_{t}^{s}, c_{t}^{d}, k_{t}^{f}, a_{t} )$ from $q_{\theta }(z_{t} | z_{1:t-1}, \mathbf {y}_{t})$, we calculate the weight of the particle, which becomes for this sampling scheme as

  $$\begin{aligned} w_{t} \propto w_{t-1} \lambda _{f}^{- k^{x}_{t}} \sum _{j = 1}^{L} \exp [d(D_{t}, \alpha _{t, j})]. \end{aligned}$$