In this section, we study two particular MSTA problems, for event detection and two- dimensional localizationtasks in a static setting. We assume that all the tasks in each problem instance are going to be of a single type, and we use this information to re- fine the formulation of each problem. We explore these as examples of non-additive sensor utility models to show that considering such utility schemes is necessary when supporting sensing tasks in heterogeneous sensor networks, such as in an emergency response operation. Below we provide a formal definition of both event detection and 2D-localization MSTA problems in a static setting.
4.3 Two Non-Additive Sensor Utility Models 90
4.3.1
Event Detection
The goal of the detection task is to detect events in a specific location with the highest probability. [121] gives a complex model of sensor assignment, with an objective function based on the probability of detecting certain kinds of events, conditioned on the events occurring and the number of sensors assigned to detect the event in a given location.
We extract the kernel of this problem as follows. Given are collections of sensors and tasks. Each task is to monitor and detect events, if they occur, in a certain location. The utility of a sensor to a task is the detection probability when the event occurs. Let Si → Tjindicate that sensor i is assigned to task j and let pjindicate Tj’s profit. Sensor
detection events are assumed, for simplicity and concreteness, to be independent, under the assumption that false negatives are the result of random failure of the individual sensor components. The objective is then to maximize the sum of cumulative detection probabilities for tasks (weighted by task profits), given the probability eij that a single
sensor Si detects an event for Tj:
X Tj∈T pj(1 − Y Si→Tj (1 − eij)) (4.1)
Equation 4.1 is based upon the model that the responses of sensors are statistically independent of one other. This is based upon the reasonable assumption that the ran- dom fluctuations measured by the sensors are statistically independent given that the sensors are either disparate over space and/or modality. We call the resulting prob- lem the Cumulative Detection Probability maximization problem (MAXCDP). Here the utilities are monotonic increasing but nonlinear, as sensors are assigned. [107] proves that MAXCDP is strongly NP-Hard and that the hardness result remains even for geometric instances, even if sensors and tasks lie on a line.
4.3.2
Two Dimensional Localization
For target localization through triangulation of the bearing measurements, two or more sensors that are not collinear with the target are necessary to ensure full observability of the target’s location. The expected mean squared error when incorporating imperfect bearing measurements is well understood [57, 61]. Specifically, it can be shown that when the bearing measurements are modeled as the true bearings embedded in additive white Gaussian noise (AWGN) of mean zero and variance σ2, then the error covariance
of the (x, y) location of the target is approximately:
R = " n X i=1 1 σ2d2 i cos2θ i − cos θisin θi
− cos θisin θi sin2θi
#−1
where di and θi are the distance and bearing, respectively, from the target event to the
i-th sensor. We choose to model the uncertainty in the calculated target location, U , as a function of the expected mean squared error (MSE), which is U = trace{R}. Alternatively, the uncertainty could be U = det{R} as described in [63]. We prefer the trace because of its physical interpretation as the MSE and because it bounds the determinant. As noted in [61], the error covariance R is related to the inverse of the Fisher Information Matrix. Therefore, clearly this approach relates to the ones covered in Section 4.1 and Chapter 2 as it uses the inverse of maximum-likelihood estimates of target locations as an uncertainty measurement. Such uncertainty estimates are, in fact, the common trait of most of the localization schemes, which as discussed are based on entropy, mean squared error and information-gain approaches.
We consider the case in which only two sensors are used for each localization task, which in most cases provides enough accuracy. More sensors lead to better accuracy but for the purpose of analyzing a non-additive utility problem two sensors are suffi- cient. As before, a sensor may not be assigned simultaneously to more than one task. For the case of sensors 1 and 2 assigned to task j, the uncertainty of the localization
4.3 Two Non-Additive Sensor Utility Models 92
computation is given by:
Uj = σ q D2 1,j+ D22,j | sin(θ1,j− θ2,j)| (4.2)
We refer here to the localization problem as L. In this problem, (disjoint) pairs of sensors are assigned to tasks, in order to minimize the sum of all tasks’ uncertainty values; i.e., to minimizeP
jUj. (Note that σ is simply a scaling constant that without
loss of generality we ignore by setting to 1.) We may also define the quality Qj of
a task’s assignment as Qj = 1/Uj, in which case we may give a dual formulation
L’ of the problem where the goal is to maximize the total assignment quality; i.e., to maximize P
jQj. With the objective thus defined, task j’s quality Qj is maximized
(and its uncertainty Ujis minimized) when the separating angle is 90◦and the distances
are minimal.
In the expression above, D1,j and D2,j are the distances between the target and two
sensors and θ = θ1,j − θ2,j is the smaller angle formed by the three points. Note that
expression 4.2 incorporates two incentives. First, the sensors should ideally be as near to the target as possible; second, the ideal angle formed by the points is 90 degrees. If there happens to be a minimum separating distance between any sensor-target pair (sayp1/2), then there is a minimum possible uncertainty value (in this case, 1). The problem with the weights described is a special case of the NP-hard MAXIMUM
3D MATCHING problem [39]. In a weighted version of that problem, we are given finite element sets X, Y , and Z and weights for the triples in a set T = X × Y × Z, and the task is to choose a maximum-weight subset of T such that no element is used more than once.