Approximate Bayesian algorithms for multiple target tracking with binary sensors

Sensor networks are systems made of many small and simple sensors deployed over an area in an attempt to sense events of interest within that particular area. In general, the sensors have limited capacities in terms of say range, precision, etc. The ultimate information level for a sensor is a binary one, referring to its output. However, it is important to make a distinction according to the nature of this binary information. Actually, it can be related to a $0-1$ information (non-detection or detection) or to relative $\{+,-\}$ motion information. For example, if the sensors are getting sound levels, instead of using the real sound level (which may cause confusion between loud near objects and quieter close objects), the sensor may simply report whether the Doppler frequency is suddenly changing, which can be easily translated in whether the target is getting closer or moving away. Moreover, low-power sensors with limited computation and communication capabilities can only perform binary detection. We could also cite video sensors, with the intuitive reasoning: the target is getting closer if its size is increasing. The need to use that kind of sensor networks leads to the development of a model for target tracking in binary sensor networks.

The very first work on this kind of binary directional sensors has been made by Aslam et al [1]. In their article, the authors provide a modified version of the particle filter to address the issue. They use the geometrical properties implied by such sensors to build a pseudo-likelihood and use it in the filter. However, the approach is limited by the same geometrical constraints the use, and while the velocity can be estimated with a reasonable accuracy, the position estimation is not improved. In [2, 3], we developed a new approach for the estimation of both the velocity and the position by learning the target’s behaviour. We then use this knowledge to improve the estimation of the position, which leads to a final improvement of both estimation.
Several issues appear as soon as someone wants to perform a tracking of multiple targets with a sensor network, even if the sensor measurements are perfect. Indeed, tracking a target involve associating temporal measurements. The problem of associating measurements from sensors to the appropriate tracks, especially when missing data, unknown number of targets, and false observations are present, has been treated [4], [5], [6] using approaches for extracting the most probable hypothesis from a set containing multiple hypotheses all compatible with the actual observations. Some recent work on binary proximity sensors in presented in [7].
The extension of the binary problem to multiple target tracking is quite hard due to the fact that by not being able to associate the observations to the targets, the problem moved from a binary issue to a binomial one (instead of having $0-1$ values, we have $0-N_{t}$ values, where $N_{t}$ stands for the number of targets). In [8] we presented a technique based on the sampling of binary matrix and their evaluation through a likelihood function. Then, by applying the Importance Sampling framework, we have been able to achieve an estimation of both position and velocity of multiple targets.

In this article, we change the point of view presented in our previous work [8]. In that precedent paper, we aimed at generating a draw of sensor observations, $S_{t}$, and calculate its likelihood, considered as a product of Bernoulli distribution. That approach, while exact (in term the sense that the likelihood calculated was exact and not approximated) was heavily computer demanding, especially if the number of sensors and/or targets is reasonably important. In the present work, however, we consider the tracking in a binary sensor network as an inverse problem, such that by applying a model $\mathcal{M}$ to the velocity and position vectors, we have an output $S_{t}$, which corresponds to the binary observations by the sensors. Therefore, our aim is to draw the positions and velocities from a well-chosen distribution, apply $\mathcal{M}$, and then evaluate the output by comparison to the observations.
The problem with that approach is that the likelihood of the observations/parameters is hard to calculate when the number of target is bigger than $1$. Therefore, we use a likelihood-free approach, first by defining a simple rejection algorithm based on the Euclidean distance, and then by calculating a more complex pseudo-likelihood and computing a random walk Metropolis-Hastings algorithm.
The paper is organised as follows. First, we describe the general binary target tracking context, and present the state equations. Then, after having quickly introduced the ABC principles, we present the algorithms that we’ll use to estimate the tracks of the targets. We finally present some conclusive simulation results, where we compare the performances of the algorithms in function of a number of varying parameters such as the duration of the tracking, the number of targets and the number of sensors in the area.

The first generalized approximate Bayesian computation algorithm aimed at provide some accurate enough inference with an approximation of the target density [9]. This approach is widely used and popular among different areas such as medicine [10], Several improvements have been proposed since, by improving the approximated likelihood with sequential methods [11, 12, 13], or empirical likelihood [14]. The issue of possible poor acceptance rate is also adressed in [15], which offers a partial least square transformation to choose informative statistics.
The extension of the ABC algorithms provided by the SMC samplers, can be efficient but computer demanding. This limitation, thus, makes these extensions inefficient in our context given our real-time tracking requirements. Interested readers may also refer to [16], which provides an excellent up-to-date survey of ABC methods.
In its most common form (the most very basic ABC algorithm is the approximate rejection algorithm (ABC-Rej) which is detailed in the following section) ABC algorithms draw inference from the following posterior density,

where $\varepsilon$ is a tolerance level, $\rho$ an adequate distance function, and $A_{\varepsilon,\mathsf{y}}$ the set of ”tolerated” observations, meaning,

This algorithm is designed to overcome the issue of having intractable likelihood. It is the simplest one, but it is proved to perform exact inference in the presence of uniform model or measurement error [17]. The details are provided in Alg. 3.2.

The most important feature to have a successful ABC-Rej algorithm is to define a measure $\rho()$, between the model outputs, and to have a reasonable prior distribution. In our situation, we remind that the statistic $\bf c$ is a vector of length $N_{s}$, which values belong to $[0,N_{t}]$. We then simply define $\rho$ as,

the euclidean distance between the vectors ${\bf c}_{1},{\bf c}_{2}$.

This ABC-Rej algorithm provides a reasonable estimate of both position and velocity, but however doesn’t really consider the past knowledge we have of the targets evolutions. Therefore, in addition to the Euclidian distance (Eq. 10), we will also try to build a pseudo likelihood ($f$), to include the information we think relevant to this target tracking.

Then, given that function $f$, we derive a classical MCMC step using the generation of a random uniform variable to select (or not) the new particles.
The function $f$ consist on a summary of all the information (a priori or based on the observations) that we think can help us improve the estimation procedure. Therefore, recalling that the main assumptions were based on a low-manoeuvring target, we will base $f$ on the consecutive bearings and distance between two consecutive measurements.

Usually, the acceptance rates of the ABC-MCMC schemes are relatively low. In order to ensure a reasonable coverage of the target distribution, we therefore need to make sure that the acceptance rate stays at an acceptable level, $23\%$ being the heuristic most common when the parameter space dimension is bigger than $5$ [18], [19].
Improving the acceptance rate is usually a matter of prior distribution and renewal calibration (e.g. variance of the renewal process). As the chosen prior distribution seems rather adequate, we need to focus on the renewal process and on the ABC only features, namely $\rho$ and $\varepsilon$.

The classical ABC-MCMC schemes suffer form two majors drawbacks. First, the samples obtained are dependant, and often highly correlated. While this can be an issue for other features, there is nothing to worry about in our situation. Indeed, the transformation of the parameters (eq. 2) is not bijective, then the use of the ABC-MCMC does not aim at drawing the true posterior distribution of $({\bf x}_{t},{\bf v}_{t})$. Instead, we expect that the acceptance step will select the most likely parameters according to the hypothesis we made on the movement (and represented by $f$ in ABC-RW). The second drawback may only be a concern if the targets are suspected to be highly manoeuvrable. In that situation, the ABC-MCMC scheme offers no warranty to detect the changes of direction. To overcome this, we propose to use an ABC Parallel Temporing algorithm, as presented in [20].
The principle of parallel tempering is to introduce $N$ temperatures and to run in parallel $N$ associated MCMC chains with target distributions being tempered distributions of the target $\pi$. The first chain targets $\pi$, and the tempered chains target flatter versions of $\pi$. Each iteration of the parallel tempering is decomposed into two types of moves. Local moves, to update the chains. And global moves, to swap between the chains.
Transposed to ABC framework, the parallel tempering replaces the increasing temperatures with increasing tolerance values. The local moves are performed using the ABC framework. The global moves (or chains swap), demand a specific treatment. Let $N$ be the size of the chain (also known as the number of particles), and $J$ be the number of parallel chains. Alg. ABC-PT explain the additional step required to compute a parallel tempering algorithm.

The issue with that kind of algorithm is the duration of the estimation process. Indeed, having $J$ chains means that your complexity increases drastically, and that is why this solution is accurate if you have highly turbulent targets.

We assumed so far that the number of targets is known. However, for the aim of surveillance in enemy areas, the exact number of targets to track is unknown, even if some knowledge of the range can be available. This issue has been studied quite widely for regular multi-target tracking, mostly using Particle Filters, by adding some prior information [21], proposing a two-layer algorithm [22, 23], or using a grid-based probabilistic approach. However, none of these methods can be applied in our particular context.
Having an evolving (or unknown) number of targets is an issue that rise a change of the parameter space dimension. This issue has been addressed in the context of MCMC by the use of reversible jump, introduced in [24], and later improved in [25]. The idea is to calculate the likelihood for two different parameter number, and compute and acceptance step just like in the classical rejection algorithm.
This change in the parameter space can also be viewed as a model choice issue, and a review of ABC based model choice is provided in [26]. The techniques presented in this article however are very computer demanding in the sense that a lot of particles need to be drawn before a reasonable estimate is calculated.

To underline the accuracy of our algorithms, we tested them on different simulation scenarios. First, we consider a single target and we compare the performances of the ABC algorithms with the MCMC Random Walk algorithm performed using the known likelihood (2).

We display on figure 2 the evolution of the RMSE for the tracking of a single target, using the MCMC algorithm, the ABC-RW algorithm, and the ABC-PT algorithm. As expected, the MCMC performs better than the two others, the ABC-PT algorithm being able to outperform the ABC-RW algorithm.

Then, we consider the trajectories of $N_{t}$ targets, which evolves according to equation 1, with $F_{t}=I_{N_{t}}$ and ${\bf Q}_{t}=\sigma^{2}I_{N_{t}}$ (with $\sigma^{2}=0.1$). The starting positions of the target are randomly chosen on a circle centred in $0$, and with a radius equal to $40$. The initial velocities of the targets are chosen so that the targets move toward the center of the circle. The duration of the tracking is $30$ seconds. The simulations are computed through the statistical software $\mathsf{R}$.
On figure 3 we aimed at track $5$ targets, remarkably close to each other. The initial position of the targets was known within a range of $10$ meters. The MCMC runs where $5000$-particles long, and we display on this figure the last $250$, for each time period. The trajectories can be finally point-estimated using the Maximum A Posteriori (MAP). The non-turbulent nature of the trajectories ($\sigma_{v}=0.1$) helped ABC-RW algorithm to provide a good estimation of all the targets trajectories.

We display on figures 4 and 5 the particles accepted during the MCMC run, for a specific $t$ (fig. 4), or the whole trajectory (fig. 5). The simulation consist on $3$ targets, with more varying directions than in the previous simulation ($\sigma_{v}=0.3$). The total number of particles in a run is $10000$. The acceptance rate is roughly $0.12$, which is a bit low, but due to the choice of $\varepsilon$.

In figure 4, we also present the histograms estimating the posterior density of the bearings. As we can notice from both figures, our algorithm is able to track the three targets with a high accuracy, which is remarkable since no association rule is defined.

Finally, we evaluated the root mean square error (RMSE) of the estimated position for different number of targets and different number of sensors. Table 1 provides the different RMSE values and their variance as the tracking is being performed for $30$ seconds. The obvious conclusion is that the best algorithm seems to be the one using the Parallel Tempering feature (ABC-PT). However, we recall that the estimation has been performed with $5$ parallel chains, which means $5$ times more computer resources. While it has no importance for offline tracking, real-time tracking is a different issue and the ABC-RW algorithm provides good estimates as well.