Algorithm 1 Construction of the prior distribution of DDP-STP

(i) Available parameters at time step $(k-1)$

– Object state parameter ${\mathbf{x}}_{\ell ,k-1}$, ℓ = $1,\dots ,N_{k-1}$, set ${\mathcal{X}}_{N_{k-1},k-1}$

– Cluster parameter ${\mathit{\theta }}_{\ell ,k-1}$, ℓ = $1,\dots ,N_{k-1}$, for ℓth object, set ${\mathsf{\Theta }}_{N_{k-1},k-1}$

– Parameter of unique cluster ${\mathit{\theta }}_{l,k-1}^{🟉}$, l = $1,\dots ,D_{k-1}$, set ${\mathsf{\Theta }}_{D_{k-1},k-1}^{🟉}$

– Cluster label indicator $c_{l,k-1}$, l = $1,\dots ,D_{k-1}$ and set ${\mathcal{C}}_{D_{k-1},k-1}$

– Cardinality of lth unique cluster $q_{l,k-1}^{🟉}$, l = $1,\dots ,D_{k-1}$

(ii) Transitioning from time step $(k$−$1)$ to k

– Draw object survival indicator ${\mathrm{s}}_{\ell ,k\mid k-1}$∼$\mathrm{Bernoulli}\left({\mathrm{P}}_{\ell ,k\mid k-1}\right)$, ℓ = $1,\dots ,N_{k-1}$

– If ${\mathrm{s}}_{\ell ,k\mid k-1}$ = 1, the ℓth object survives; if ${\mathrm{s}}_{\ell ,k\mid k-1}$ = 0, it leaves the scene

– Compute number of transitioned objects $N_{k\mid k-1}$ = ${\displaystyle \sum _{\ell =1}^{N_{k-1}}{\mathrm{s}}_{\ell ,k\mid k-1}}$

– Denote cardinality of lth cluster, l = $1,\dots ,D_{k-1}$, after transitioning by $q_{l,k\mid k-1}$

– If $q_{l,k\mid k-1}$≥ 1, cluster survival indicator ${\lambda }_{l,k\mid k-1}$ = 1; if $q_{l,k\mid k-1}$ = 0, ${\lambda }_{l,k\mid k-1}$ = 0

– Compute number of unique clusters to $D_{k\mid k-1}$ = ${\displaystyle \sum _{l=1}^{D_{k-1}}{\lambda }_{l,k\mid k-1}}$

– Denote cardinality of lth transitioned cluster by $q_{l,k\mid k-1}^{🟉}$, l = $1,\dots ,D_{k\mid k-1}$

– Denote parameter of transitioned cluster by ${\mathit{\theta }}_{l,k\mid k-1}^{🟉}$, l = $1,\dots ,D_{k\mid k-1}$

(iii) Current time step k

for ℓ = 1 to $D_{k\mid k-1}$ do

if Case 1 (on page 6) then

Draw ${\mathbf{x}}_{\ell ,k}$ from the prior PDF in (6) with probability $P_k^{\left(1\right)}$ in (5)

else if Case 2 (on page 6) then

Draw ${\mathit{\theta }}_{\ell ,k}$ from $p({\mathit{\theta }}_{\ell ,k}\mid {\mathit{\theta }}_{\ell ,k-1}^{🟉})$

Draw ${\mathbf{x}}_{\ell ,k}$ from the prior PDF in (8) with probability $P_k^{\left(2\right)}$ in (7)

else if Case 3 (on page 7) then

Draw ${\mathit{\theta }}_{\ell ,k}\sim G_0$ following $\mathrm{DP}(\alpha ,G_0)$

Draw ${\mathbf{x}}_{\ell ,k}$ from the PDF in (10) with probability $P_k^{\left(3\right)}$ in (9)

end if

end for

Update number of objects $N_k$ using $N_{k\mid k-1}$ and number of new objects under Case 3

Update lth unique cluster cardinality $q_{l,k}^{🟉}$ and parameter ${\mathit{\theta }}_{l,k}^{🟉}$

return ${\mathcal{X}}_{N_k,k}$, ${\mathsf{\Theta }}_{N_k,k}$