Consider a single target tracking model where a moving object (or target) is observed when it traverses in a surveillance region. We define the target state and the noisy observation at time t to be the random variables Xt ∈ X ⊂ Rdx and Yt ∈ Y ⊂ Rdy
respectively. The statistical model most commonly used for the evolution of individual targets {Xt, Yt}t≥1 is the hidden Markov model (HMM). In a HMM, it is assumed that
{Xt}t≥1is a hidden Markov process with initial and transition probability densities µψand
fψ, respectively, and{Yt}t≥1 is the observation process with the conditional observation
density gψ, i.e. X1 ∼ µψ(·), Xt|(X1:t−1= x1:t−1)∼ fψ(·|xt−1) Yt| {Xi = xi}i≥1,{Yi = yi}i6=t ∼ gψ(·|xt). (5.1)
Here the densities µψ, fψ and gψ are parametrised by a real valued vector ψ ∈ Ψ ⊂ Rdψ.
In this work, we consider a specific type of HMM, the Gaussian linear state-space model (GLSSM), which can be specified as
µψ(x) = N (x; µb, Σb), fψ(x′|x) = N (x′; F x, W ), gψ(y|x) = N (y; Gx, V ). (5.2)
whereN (x; µ, Σ) denotes the probability density function for the multivariate normal dis- tribution with mean µ and covariance Σ. In this case, ψ parametrizes (µb, Σb, F, G, W, V ).
In a MTT model, the state and the observation at each time (t≥ 1) are random finite sets, Xt = Xt,1, Xt,2, . . . , Xt,Kx
t
and Yt =Yt,1, Yt,2, . . . , Yt,Kty
. Here each element of Xt is the state of an individual target and elements of Yt are the distinct measurements
of these targets at time t. The number of targets Kx
t under surveillance changes over
time due to targets entering and leaving the surveillance region X . Xt evolves to Xt+1
state transition density fψ in (5.2), otherwise it dies. The random deletion and Markov
motion happens independently for all the elements of Xt. In addition to the surviving
targets, new targets are created. The number of new targets created per time follows a Poisson distribution with mean λb and each of their states is initiated independently
according to the initial density µψ in (5.2). Now Xt+1is defined to be the superposition of
the states of the surviving and evolved targets from time t and the newly born targets at time t + 1. The points of Xt are observed through the following model: with probability
pd, each point of Xt generates a noisy observation in the observation space Y through
the observation density gψ in (5.2). This happens independently for each point of Xt. In
addition to these target generated observations, false measurements are also generated. The number of false measurements collected at each time follows a Poisson distribution with mean λf and their values are uniform overY. Ytis the superposition of observations
originating from the detected targets and these false measurements.
A series of random variables, which are essential for the statistical analysis to follow are now defined. Let Cs
t be a Kt−1x × 1 vector of 1’s and 0’s where 1’s indicate survivals
and 0’s indicate deaths of targets at time t. More clearly, for i = 1, . . . , Kt−1x ,
Cts(i) =
1 i’th target at time t− 1 survives to time t
0 i’th target at time t− 1 does not survive to time t.
The number of surviving targets at time t is Ks
t =
PKx t−1
i=1 Cts(i). We also define the Kts×1
vector Is
t containing the indices of surviving targets at time t,
Its(i) = min ( k : k X j=1 Cts(j) = i ) , i = 1, . . . , Kts. Note that Is
t(i) denotes the ancestor of target i from time t− 1, i.e. Xt−1,Is
t(i) evolves
to Xt,i for i = 1, . . . , Kts. Denoting the number of ‘births’ at time n as Ktb, we have
Kx
t = Kts + Ktb. Note that according to these definitions, the surviving targets from
time t− 1 are re-labeled as Xt,1, . . . , Xt,Ks
t, and the newly born targets are denoted as
Xt,Ks
t+1, . . . , Xt,Ktx. Next, given K
x
t targets we define Ctd to be a Ktx× 1 vector of 1’s and
0’s where 1’s indicate detections and 0’s indicate non-detections. For i = 1, . . . , Kx t,
Ctd(i) =
1 i’th target at time t is detected at time t 0 i’th target at time t is not detected at time t. Therefore, the number of detected targets at time t is Kd
t =
PKx t
also define the Kd
t × 1 vector Itd showing the indices of the detected targets,
Itd(i) = min ( k : k X j=1 Ctd(j) = i ) , i = 1, . . . , Ktd. Id
t(i) denotes the label of the i-th detected target at time t. So the detected targets at
time t are Xt,Id
t(1), . . . , Xt,Itd(Ktd). Finally, defining the number of false measurements at
time t as Ktf, we have Kty = Kd
t + K
f
t and the association from the detected targets to
the observations can be represented by a one-to-one mapping At:{1, . . . , Ktd} → {1, . . . , K
y t}
where at time t the i’th detected target is target Id
t(i) with state value Xt,Id
t(i) and
generates Yt,At(i). We assume that Atis uniform over the set of all K
y t!/K
f
t! possible one-
to-one mappings. To summarise, we give the list of the random variables in the MTT model introduced in this section as well as a sample realisation of them in Figure 5.1.
The main difficulty in an MTT problem is that in general we do not know birth-death times of targets, whether they are detected or not, and which observation point in Yt is
associated to which detected target in Xt. Let
Zt=
Cts, Ctd, Ktb, Ktf, At
be the collection of the just mentioned unknown random variables at time t, and θ = (ψ, ps, pd, λb, λf)∈ Θ = Ψ × [0, 1]2× [0, ∞)2
be the vector of the MTT model parameters. We can write the joint likelihood of all the random variables of the MTT model up to time n given θ as
pθ(z1:n, x1:n, y1:n) = pθ(z1:n)pθ(x1:n|z1:n)pθ(y1:n|x1:n, z1:n) where pθ(z1:n) = n Y t=1 pkst s (1− ps)k x t−1−kstPO(kb t; λb)p kd t d (1− pd)k x t−ktdPO(kf t; λf) ktf! kyt! ! (5.3) pθ(x1:n|z1:n) = n Y t=1 ks t Y j=1 fψ(xt,j|xt−1,is t(j)) kx t Y j=ks t+1 µψ(xt,j) (5.4) pθ(y1:n|x1:n, z1:n) = n Y t=1 |Y|−kft kdt Y j=1 gψ(yt,at(j)|xt,id t(j)) (5.5)
The list of the variables in the MTT model Xt,k, Yt,k: k’th target and k’th observation at time t.
Xt={X1, . . . , XKx
t}, Yt={Yt,1, . . . , Yt,Kty}: Sets of targets and observations at time t.
Ktb, Ktf: Numbers of newborn targets and false measurements at time t
Kts, Ktd: Numbers of targets survived from time t− 1 to time t and detected at time t. Ktx, Kty: Numbers of alive targets, observations at time t. Kx
t = Kts+ Ktb, Kty = Ktd+ Ktf.
Cts: Kt−1x × 1 vector of 0’s and 1’s indicating survivals from time t − 1 to time t. Ctd: Kx
t × 1 vector of 0’s and 1’s indicating detections at time t.
Its: Kts× 1 vector of indices of surviving targets from time t − 1 to time t. Itd: Ktd× 1 vector of indices of detected targets at time t.
At:{1, . . . , Ktd} → {1, . . . , Kty}: Association from detected targets to observations at time t.
Zt= (Cts, Ctd, Ktb, Ktf, At) X1,1 X2,1 X3,1 X4,1 X5,1 Y1,4 Y3,3 Y5,3 X1,2 X2,2 X3,2 X4,2 X5,2 Y1,1 Y2,1 Y3,5 Y4,1 Y5,2 X1,3 X2,3 X3,3 X4,3 X5,3 Y1,2 Y2,3 Y3,4 Y4,2 Y5,1 Y1,3 X2,4 Y3,1 X4,4 X5,4 Y1,5 Y2,2 Y3,2 Y4,3 Y5,4 Cs 1:5 = ([ ] , [1, 1, 1] , [1, 0, 1, 1] , [0, 1, 1] , [1, 1, 1, 1]); I1:5s = ([ ] , [1, 2, 3] , [1, 3, 4] , [2, 3] , [1, 2, 3, 4]); Cd 1:5 = ([1, 1, 0] , [0, 1, 1, 1] , [1, 1, 1] , [0, 1, 1, 0] , [1, 1, 1, 1]); I1:5d = ([1, 2] , [2, 3, 4] , [1, 2, 3] , [2, 3] , [1, 2, 3, 4]); Kts = (0, 3, 3, 2, 4); K b 1:5 = (3, 1, 0, 2, 0); K d t = (2, 3, 3, 2, 4); K f 1:5 = (3, 0, 2, 1, 0), A1:5 = ([4, 1] , [1, 3, 2] , [3, 5, 4] , [1, 2] , [3, 2, 1, 4]).
Figure 5.1: Top: The list of the random variables in the MTT model. Bottom: A realisation for an MTT model: States of a targets are connected with arrows. Also, observations generated from targets are connected to those targets with arrows. Mis- detected targets are highlighted with shadows, and observations from false measurements are coloured with grey.
Here PO(k; λ) denotes the probability mass function of the Poisson distribution with mean λ, |Y| is the volume (w.r.t. the Lebesgue measure) of Y and the term kft!/kyt! in
(5.3) corresponds to the law of At. The marginal likelihood of the observation sequence
y1:n is
pθ(y1:n) = Eθ[pθ(y1:n|X1:n, Z1:n)] . (5.6)
The main aim of this work is, given Y1:n= y1:n, to estimate the static parameter θ∗ where
we assume the data is generated by some true but unknown θ∗ ∈ Θ. Our main contribu-
MLE of θ∗:
θML= arg max
θ∈Θ pθ(y1:n).