Target Tracking with Packet Delays and Losses QoI amid Latencies and Missing Data

Transcription

1 Target Tracking with Packet Delays and Losses QoI amid Latencies and Missing Data Wei Wei, Ting He, Chatschik Bisdikian, Lance Kaplan 3, Dennis Goeckel 4, Don Towsley Department of Computer Science, University of Massachusetts, Amherst, MA 3 IBM T. J. Watson Research Center, 9 Skyline Drive, Hawthorne, NY 53, USA 3 Army Research Laboratory, Adelphi, MD Electrical and Computer Engineering Department, University of Massachusetts, Amherst, MA 3 Abstract In this paper, we investigate how network and measurement components impact the quality of a tracking system. Specifically, we use Bayesian information to quantify the quality of target tracking and investigate how network quality and measurement quality affect the value of Bayesian information. By assuming a Brownian motion mobility model for the target and Gaussian measurement noise for the sensors, we derive a closed-formed formula for the Bayesian information, which indicates that improving measurement quality provides a diminishing gain on tracking quality, while the gain from improving network quality remains constant. We obtain the condition under which a user obtains information gain on the target location from a tracking process. We further use Bayesian information as the metric for the gateway to select which sensor to take measurements with and at what time to control the tracking quality. Last, we generalize the above results to targets that move according to a white noise acceleration model. I. INTRODUCTION Sensor networks have been widely used to enable awareness and intelligent field operations, including both civilian and military applications,. Serving as the data acquisition and distribution layer for many applications, sensors collectively produce information about the phenomena or events of interest, which is then used as a basis of future actions. Due to inherent limitations of sensors, e.g., limited battery, bandwidth, and measurement quality, the reported information is usually, however, a distorted version of ground truth. Therefore it is critical to understand and assess the quality of such information (QoI). Broadly speaking, QoI relates to characterizing (via quality attributes) and assessing (using these attributes) the pertinence and importance of the received information to the task at hand 3. Specializing the previous statement to sensor networks, QoI represents the goodness with which the information available describes the state of the world of interest, and is expressed using quality attributes such as accuracy (e.g., probability of detection of an event, accuracy in tracking an object, probability of misclassification), latency, and provenance (or lineage, describing all entities that have produced or touched this information, and hence may have altered its qualities). The QoI directly impacts the effectiveness of action plans that decision makers make based on the information. Consider a scenario where a generic tracking application is deployed in a field of operations to support military coalition tasks. Groups of sensors belonging to various coalition members are deployed in the field, e.g., along the perimeter of a secured area, to monitor passing targets such as vehicles or humans. Users that want to track such targets interface with sensors via gateways, which are capable of communicating to both users and sensors. Specifically, during a query, the user sends a tracking request to the gateway with a time when the target location is needed. The gateway handles the request by launching tracking commands to sensors, collecting sensor responses, and forwarding these responses back to the user. Based on technologies used, sensor nodes could take measurements of various types, e.g., acoustic sensors measure the sounds generated by the target and video sensors measure the image of the target. Without limiting ourselves to a particular sensing technology, we consider sensor reports as the sensor estimates of target location at the time of measurement (which may be produced collectively by a group of sensors). With the tracking information at hand, the user can recommend action plans (e.g., dispatching troops to intercept a suspicious target) to the proper authorities. To conserve resources such as sensor batteries, it may be desirable to activate only a subset of sensors for each request. On the other hand, using fewer sensors inevitably leads to a degraded tracking accuracy, which lowers the QoI. The question is then how to select sensors and how to schedule measurements to maximize QoI. This is particularly challenging in coalition environments because different members often have different sensing and communication capabilities, including both inherent differences in sensor qualities and deployments and deliberate alterations due to security reasons. Our goal is to study the trends of QoI for the application of target tracking with emphasis on tracking accuracy. In particular, due to network delays and losses, the sensor responses may experience nontrivial delays before reaching the user, or even be lost and not reach the user at all. The questions of interest are: what is the impact of latencies

2 and losses on QoI? Given multiple classes of sensors, each with specific measurement and communication capabilities, how do we select sensors and schedule measurements to minimize such impact? We address the above questions by starting with a simplified model, which allows us to obtain insight without going into tedious mathematical details. Our main contributions are: ) We use Bayesian information of the posterior distribution of target location to quantify the quality of target tracking and investigate how network quality and measurement quality affect the value of Bayesian information. ) We obtain the condition under which a tracking task produces information of the target location. 3) We further use Bayesian information as the metric for the gateway to select the sensor for taking measurements and determine the measurement time to control the tracking quality. 4) We extend the above results to a smoother mobility model with the velocity following a Brownian motion. The problem of target tracking has been extensively studied in the context of sensor networks 45. Our work differs from other works in that we investigate how the network quality (delays and lost packets) impact the quality of the tracking. The remainder of the paper is organized as follows. Section II presents problem setting. Section III derives Bayesian information for the setting. Section IV discusses sensor selection and measurement scheduling. Section V extends the results to a continuous white noise acceleration model. Finally Section VI concludes the paper and presents future work. II. PROBLEM SETTING Consider a sensor network as illustrated in Fig.. Multiple sensor nodes are deployed in a field. A moving target is moving along a straight line according to a Brownian motion. The sensors can measure the target coordinates and the measurement errors follow Gaussian distributions with zero mean. Users that want to track the movement of the target send a tracking request to the gateway along with a prior distribution of the target location, a timestamp of this request, and the time t when the target location is needed. The gateway handles tracking requests from users, sends out tracking commands to the corresponding sensors, collects sensor measurements from the sensors, and forwards the measurement results to users. Users then calculate the posterior distribution of the location based on the measurement results and the prior distribution. We assume the gateway and users are equipped with GPS devices so that they always have accurate times. Furthermore, clocks at the sensors are synchronized with that of the gateway. Next we describe a tracking process in more detail and introduce the notations that are used in the following sections. As shown in Fig., at time, a user has a prior distribution p(θ()) (from prior observations) of the target location gateway command sensor Figure. measurement target request measurement A tracking process. user sensor θ(). The prior distribution is Gaussian N(μ θ(), σθ() ), with mean μ θ() and variance σθ(). The user sends a tracking request to the gateway together with the prior distribution p(θ()). Once the gateway receives the request, it calculates which sensor is the best choice for this tracking task based on the sensor measurement errors, network latencies, and packet loss probabilities, and then sends a measurement command to the sensor to make a measurement at a predetermined time τ. We assume the command arrives at the sensor node by time τ (because the gateway does not have a power constraint as the sensor nodes, it has sufficient power to send the command). The sensor makes a measurement at time τ and sends the result z(τ) back to the gateway with a timestamp τ. The gateway then forwards the measurement result and the timestamp to the user. The user then infers θ(t) at the pre-determined time t > τ. It is clear from the above description that the time when a measurement is used is not the time when it is taken, or the time when the measurement request is initiated. This discrepancy is an inherent property of sensor networks due to packet delays and losses. In this paper, we investigate how packet delays and losses affect the quality of target tracking. For instance, under this condition, we investigate whether the user can obtain information gain on the target location at time t compared to that at time for the tracking task illustrated in Fig. A. Target movement model To ease our discussion, we model the target movement using Brownian motion. For t > t, θ(t ) θ(t ) follows N(, t t ), where θ(t ) and θ(t ) are the target locations at time t and t respectively. As mentioned earlier, we assume at time, θ() N(μ θ(), σθ() ). Then θ(t) θ() N(, t). That is, the conditional probability density function (PDF) of θ(t) given θ() is f(θ(t) θ()) πt e (θ(t) θ()) t.

3 3 sensor Figure. (command, ) (z( ), ) gateway (request,,,t) p( ()) (z( ), ) Time chart of a tracking process. Therefore, the marginal PDF of θ(t) is f(θ(t)) user e (θ(t) θ()) t πt πσθ() (θ() μ θ() ) e σ θ() dθ() e π(σθ() + t) (θ(t) μ θ() ) (σ θ() +t) The above result implies θ(t) N(μ θ(), σ θ() + t). If the measurement made at time τ is received by time t, the user calculates the posterior distribution for θ(t z(τ)) based on the measurement, z(τ), and the PDF of the prior distribution at time, p(θ()). B. Calculating posterior from Gaussian prior and measurement We next calculate the posterior distribution p(θ z) for the random parameter θ given a prior distribution p(θ) and observation z. Let p(z θ) denote the conditional PDF of z given θ, and p(z, θ) p(z θ)p(θ) the joint PDF of z and θ. It then follows from Bayes rule that: p(θ z) The PDF for the prior of θ is p(θ) p(z θ)p(θ). p(z θ)p(θ)dθ e (θ μ θ ) σ θ, () πσθ where μ θ is the mean, and σθ is variance. The PDF for the measurement z is p(z θ) e (z θ) πσz σz, where σz is variance of the measurement error. Moreover, p(z θ)p(θ)dθ (z μ θ ) π(σθ + σ z ) e p(θ z) t (σ θ +σz ) () πσ e (θ μ) σ, (3) where μ zσ θ + μ θσz σθ +, σ σ θ σ z σ z σθ +. (4) σ z The above results indicate that a Gaussian prior with Gaussian measurement error lead to a Gaussian posterior. Furthermore, (4) specifies the mean and the variance of the posterior. The above results are used throughout the paper. C. Bayesian Cramer-Rao Bound and Bayesian Information We next introduce Bayesian information adopting the notations in 6. We first introduce Bayesian information for the case where θ is a vector, and then specify Bayesian information for the scalar case. Let ˆθ(z) denote an estimate of θ, which is a function of the observations z. The estimation error is ˆθ(z) θ and the MSE matrix is Σ E z,θ {ˆθ(z) θˆθ(z) θ T }. (5) The BCRB (Bayesian Cramer-Rao Bound) C provides a lower bound on the MSE matrix Σ. It is the inverse of the Bayesian information matrix (BIM) J. That is Σ C J, where the matrix inequality indicates that Σ C (or equivalently Σ J is a positive semi-definite matrix. Let Δ η φ be the m n matrix of second-order partial derivatives with respect to an m parameter vector φ and n parameter vector η, φ η φ η n Δ η φ..... (6) φ n η φ n η n The BIM for θ is defined as J E z,θ { Δ θ θ ln p(z, θ)}. (7) When θ is a scalar, this reduces to the scalar: J E z,θ { ln p(z, θ)}. (8) θ D. Target tracking with a perfect network We motivate our target tracking with packet delays and losses problem by assuming a perfect network with negligible packet delays and packet losses. At time, a user has a prior distribution N(μ θ(), σθ() ). The user sends a tracking request to the gateway and the gateway sends a tracking command to a sensor. After receiving the tracking command, the sensor takes a measurement at τ. The measurement result z, which follows a Gaussian distribution N(θ(), σz), is sent to the gateway. The gateway forwards the result back to the user. Since the network delay is, the user receives the measurement result z at time t. The posterior distribution at t is then calculated using (3). Note here we have τ t. Prior to the user s request for a measurement, the Bayesian information is /σθ(). After receiving the measurement z,

4 4 the Bayesian information is /σ /σθ() + /σ z > /σθ(). This implies that the measurement always increases the Bayesian information of the target location. In the real world, a network has non-zero network delays and can suffer packet losses. This is especially true for sensor networks due to power constraints. Hence, the time a tracking request is sent, the time a measurement is made and the time an inference is made based on the measurement are all different. One natural question is whether requesting a measurement still leads to information gain on the target location under the above condition. The next section will address this question. III. BAYESIAN INFORMATION AMID DELAYS AND LOSSES In this section, we consider the problem that a user sends out a tracking request at time to the gateway with a prior location distribution, the gateway arranges a sensor to make a measurement z z(τ) at time τ and the result is sent back to the user. The user needs to infer the target location at time t > τ. To shorten the noations, we use μ θ and σθ to denote μ θ() and σθ() respectively in later sections. A. Packet delays only We assume there are no packet losses for now, and extend the model to incorporate packet losses in Section III-B. Let X be a random variable representing packet delay between the sensor and the user with PDF f(x) and CDF F (x). At time t > τ, the user infers the target location θ(t). We next derive the Bayesian information under two cases: when the measurement arrives earlier than t and after t. With probability F (t τ), the measurement packet arrives earlier than t. Hence at time t, the user has an observation z, which is the measurement made by the sensor at time τ. We calculate p(θ(t), z) p(θ(t) z))p(z), where p(z) is the probability that at time t the user has an observation of value z. And p(θ(t) z) is the PDF for the posterior distribution of θ at time t. Note that the event that the user has the observation of value z at time t is equivalent to the following two independent events. First, the measurement made by the sensor at time τ has value z. Second, the packet that contains the measurement arrives at the user before time t, that is, X t τ. Therefore, p(z) F (t τ) p(z θ(τ)) π(σ θ + τ) e (θ(τ) μ θ ) (σ θ +τ) F (t τ) dθ(τ) e (z θ(τ)) σz πσz (θ(τ) μ θ ) π(σ θ + τ) e (σ θ +τ) dθ(τ) (z μ θ ) (σ π(σ z + σθ z +σ + θ +τ) F (t τ). τ)e Note p(z) does not involve θ(t), and has no effect on the Bayesian information. Moreover, where Hence, p(θ(t) z) p(θ(t) θ(τ))p(θ(τ) z)dθ(τ) e (θ(t) θ(τ)) (t τ) π(t τ) π(σ τ ) (θ(t) μτ ) (σ e τ +t τ), e (θ(τ) μτ ) στ dθ(τ) πστ μ τ z(σ θ + τ) + μ θσz σθ + τ +, σ σ τ (σ θ + τ)σ z z σθ + τ +. (9) σ z Δ θ(t) θ(t) log p(z, θ(t)) στ. () (σθ +τ)σ z σ θ +τ+σ z On the other hand, with probability F (t τ), the packet does not arrive before time t. In this case, we say the gateway observes a packet loss, and use s to denote this observation. Then, Hence, p(s, θ(t)) p(s θ(t))p(θ(t)) (θ(t) μ θ ) ( F (t τ)) π(σ θ + t) e Δ θ(t) θ(t) (σ θ +t). log p(s, θ(t)) σθ () + t. In summary, by taking weighted average of () and (), the Bayesian information can be calculated as J(τ, t) F (t τ) σ θ + t + More detail can be found in 7. B. Packet delays and packet losses F (t τ) (σθ +τ)σ z σθ +τ+σ z. () Next we incorporate packet losses into the above formulation. Let p be the packet loss probability between the sensor and the gateway. Again assume the user infers the target location at time t > τ. At time t, the measurement may or may not have arrived. With probability F (t τ)( p), a measurement arrives before time t, and with probability F (t τ)( p), no measurement arrives before time t. Based on this, the Bayesian information is J(τ, t) F (t τ)( p) σ θ + t + F (t τ)( p) (σθ +τ)σ z σθ +τ+σ z (3) We assume F (), which implies that when τ t, we have J(t, t) σθ +t, indicating the inference is based on prior knowledge. On the other hand, when t goes to infinity,

5 5 gain on Bayesian information. This differs from the result on network quality where the improvement on Bayesian information is independent of the current network quality. IV. SENSOR SELECTION AND MEASUREMENT SCHEDULING In this section, we use the Bayesian information obtained from Section III-B to investigate the information gain from tracking, sensor selection, measurement scheduling, and tracking quality control. Figure 3. Bayesian information versus τ, with mean delay of.5,. and.. we have lim t J(τ, t), indicating a long waiting time leads to information decays. Comparing () to (3), we can see that the Bayesian information when only considering delay is no less than that when considering both delay and loss. C. Impact of measurement quality and network quality From (3), one can see that the quality of the network can be represented by q F (t τ)( p), which is the probability that the measurement packet arrives at the user before time t. Note J(τ, t) is an increasing function of q. If p, τ and t are fixed, one can see that all the delay distributions that have the same CDF value at t τ have the same q value. In other words, they have the same network quality. On the other hand, by letting r σ, the expression z for J(τ, t) can be rewritten as J(τ, t) q σ θ + t + q, σθ +τ σθ r+τr+ where r corresponds to the measurement quality of a sensor node. By taking a partial derivative of J(τ, t), we observe that J q σθ + t +, σθ +τ σθ r+τr+ indicating the partial derivative with respect to q is a non-negative constant. This shows J(τ, t) is a monotone increasing, linear function with respect to q. The equal sign holds only when r, i.e., when measurements do not provide any information. In this case, improving network quality does not improve the tracking quality. On the other hand, J r (σθ + τ) q (σθ + t +, (σ θ + τ)(t τ)r) indicating that J r is inversely proportional to (σ θ +t+(σ θ + τ)(t τ)r), which implies that when r is large, further improving measurement quality only provides diminishing A. When does user gain information on target location? The goal for a user to request measurement from the sensor network is to increase the information on the target location. To achieve this goal, we need the Bayesian information at time t to be greater than that at time. That is J(τ, t) F (t τ)( p) σ θ + t + F (t τ)( p) (σθ +τ)σ z σθ +τ+σ z > σ θ It is clear that this goal cannot always be achieved. For example, it is not satisfied for very large t (e.g., when t ). On the other hand, if t is too small, the above requirement is not satisfied either. For example, let t τ, then J(τ, t) σ θ +τ < σ θ have J(τ, t) σ θ +t < σ θ. It is clear that if p, we. This is not surprising since no measurement reaches the user in this case. Next, we use an example to illustrate when a user can gain information on target location. Let σθ σ z, p, and t. Let delay X follow an exponential distribution with mean.5. When τ is between.6 and.96, requirement (4) is satisfied, as shown in Fig. 3. If we allow the delay distribution to have a larger mean., then the range of τ to satisfy requirement (4) is (.63,.9). If we further increase the mean of the delay distribution to., then no value of τ can satisfy requirement (4). B. Optimizing J(τ, t) with respect to τ We next investigate how the gateway can adjust τ, the time for a sensor to take a measurement, to maximize the Bayesian information at time t for the user. Assume F (x) is differentiable, then J(τ, t) τ f(t τ) ( p)( σθ + t F (t τ) (σ z σ θ +σ z τ+σ4 θ +τσ θ +τ ) (σθ + +τ+σ z ) ). ( (σ θ +τ)σ z ) σ θ +τ+σ z f(t τ) (σθ +τ)σ z σθ +τ+σ z In order to obtain the optimal τ, we need to find τ that. When p, the above equation is satisfies J(τ,t) τ (4)

6 6 equivalent to f(t τ) σ θ + t f(t τ) (σθ +τ)σ z σθ +τ+σ z F (t τ) (σ z σ θ +σ z τ+σ4 θ +τσ θ +τ ) (σθ + +τ+σ z ). ( (σ θ +τ)σ z ) σ θ +τ+σ z Note the above equation implies that the optimal value of τ does not depend on the packet loss probability between the sensor and the user. Note J(τ, t) τt ( p)f()( τ σθ + t σθ + τ σz ), and hence when τ t, the Bayesian information has a negative slope. The steepness depends on the value of t and packet loss probability p. Interestingly, when p is large, the slope is less steep. This is because when p is large, a packets is very likely to be lost. Therefore, in this case, allowing less time to transmit the packet causes less damage. C. Varying τ to control the quality of information It is not always necessary to optimize the QoI for the user. For instance, if the user is a coalition member, the system does not need to provide the best QoI for the user. In that case, the gateway can select τ so that the QoI is satisfactory but not optimal. From Fig. 3, we can see that we need to choose τ on the left of the optimal τ to control the QoI. This is because in a real sensor network, clock synchronization is not perfect. Therefore, the time the measurement is taken may not be exactly τ. If we choose τ on the right hand side of the optimal τ, the error of the QoI may be large due to the steep curve on the right hand side. D. Sensor selection and measurement scheduling at gateway The above discussion assumes that the gateway has selected a sensor to make measurements. Next, we discuss the sensor selection and measurement scheduling strategy at the gateway. We assume the gateway knows the delay distributions, and packet loss probabilities between the sensors and the user. Furthermore, the gateway knows the measurement error distributions of the sensor nodes. Moreover, the gateway knows the prior distribution of the target location at time, and that the user needs the target location at time t. Based on all this information, the gateway needs to choose a sensor to make a measurement and determine the measurement time τ. In order to achieve a good Bayesian information at time t, the gateway first divides the time interval, t into multiple subintervals, to calculate the Bayesian information for each sensor at the end of the each subintervals. The gateway then selects the sensor with the largest Bayesian information to take measurement. As for τ, the gateway can use τ b, the τ corresponding to the largest Bayesian information, or the one corresponding to the second best Bayesian information on the left of τ b (based on the insight obtained from Section IV-C). V. EXTENSION TO CONTINUOUS WHITE NOISE ACCELERATION (CWNA) MODEL We next extend the Brownian motion target mobility model to a continuous white noise acceleration (CWNA) model 8 and calculate the Bayesian information matrix for both target position and target velocity. A. Covariance matrix of CWNA model Consider a state vector defined over continuous-time where the first element is target position and the second θ (t) element is target velocity. That is θ(t), where θ (t) θ (t) is the target position at time t and θ (t) is the target velocity at time t. We assume θ(t) is described by θ(t) θ(t) + ω(t), (5) where ω(t) is white noise, satisfying Eω(t) and Eω(t)ω(s) δ(t s). Note that acceleration in this model is white noise, thus velocity is a Wiener process - the integral of white noise. In other words, the velocity changes according to a Brownian motion. One can show that Let B(t) t and EB (t) with mean t θ(t) θ() + t s EB(t) t t t t t t 3 3 t t ω(s)ds. Then t t s ω(s)ds. (6) t s Eω(s)ds, (7) t s t x Eω(s)ω(x)dxds, (t s)(t x) t s δ(s x)dxdt t x (t s) t s dt, t s t t. (8) Therefore, θ(t) follows a multi-variate Gaussian distribution θ () + θ ()t, and covariance matrix C(t) θ () t 3 3 t t t. B. Bayesian information matrix of CWNA model We next calculate the Bayesian information matrix for a CWNA model. Because the general derivation is quite complex to write down, we use an example of specific τ, t, and prior distribution parameters to illustrate the procedure. The derivation of Bayesian information for general τ, t and

7 7 prior distribution parameters can be carried out the same way. Assume θ() follows multivariate Gaussian distribution with mean and covariance matrix A(). We have A() and A (). The prior distribution at time is p(θ (), θ ()) θ () θ () A () θ () θ e () π A() π e θ () θ () θ () θ () e θ() +θ ()θ () θ ()+θ () 4θ (). Next, we calculate the posterior distribution of position and velocity of the target at time t given that a measurement was taken at time τ. With probability F (t τ)( p), the measurement does not arrive at time t. Then at time t, we have observation s, indicating lack of measurement for both position and velocity. In this case, we predict θ(t) according to the mobility model. For the interest of space, we denote θ i (x) as θ ix, where i, and x,,. We have Eθ() T θ + θ θ, and p(θ, θ θ, θ ) π C() e (θ() Eθ())T C ()(θ() E(θ()) e ( /3 (θ θ+θ+θ) + 3 (θ θ)). Therefore, p(θ, θ s) p(θ, θ θ, θ )p(θ, θ )dθ dθ e ( /3 (θ θ+θ+θ) + 3 (θ θ) ) e θ +θθ θ+θ 4θ dθ dθ e 3 5 θ θθ+ 5 θ 6 5 θ+ 8 5 θ. In this case, we have 3 Δ θ(t) θ(t) log p(s, θ(t)) On the other hand, if the measurement arrives before time t, we first predict θ(τ) at time τ, then update it with the measurement data at τ, and finally predict θ(t) for time t based on the update at time τ. We have Eθ() θ () + θ () θ () T, and p(θ, θ θ, θ ) π C() e (θ() Eθ())T C ()(θ() E(θ()) e (θ+ θ θ+ θ) + (θ θ). Therefore, the prediction at time τ is p(θ, θ ) p(θ, θ θ, θ )p(θ, θ )dθ dθ e (θ+ θ θ+ θ) + (θ θ) e θ +θθ θ+θ 4θ dθ dθ e 4 9 θ 6 9 θθ+ 9 θ 36 9 θ 9 θ. Assume that the measurement distributions of position and velocity are independent and both have variance. Let z (z, z ) T be the measurement at time τ. Then the joint measurement distribution is p(z θ, θ ) p(z θ )p(z θ ) e (z θ) e (z θ) π π Hence, the joint posterior distribution of the target position and velocity can be calculated as follows. p(θ, θ z) p(θ, θ θ, θ, z)p(θ, θ z)dθ dθ p(θ, θ θ, θ )p(θ, θ z)dθ dθ e (θ+ θ θ+ θ) + (θ θ) p(θ, θ z)dθ dθ e (θ+ θ θ+ θ) + (θ θ) p(θ, θ )p(z θ, θ )dθ dθ e (θ+ θ θ+ θ) + (θ θ) 9 θ 6 9 θθ+ 9 θ 36 9 θ 9 θ e 4 e (z θ) e (z θ) dθ dθ e (θ+ θ θ+ θ) + (θ θ) 9 θ 6 9 θθ+ 9 θ 36 9 e 4 θ 9 θ e (θ +θ ) dθ dθ e.47θ +.499θ.699θθ+.635θ.367θ. In this case, Δ θ(t) θ(t) log p(z, θ(t))

8 8 to a white noise acceleration model. As future work, we are pursuing in the following directions: ) considering location dependent measurement error models; ) exploring more complex target movement models; 3) develop more sophisticated measurement scheduling algorithms. Figure 4. Bayesian information versus τ, with mean delay of,. and., CWNA model. Hence, the Bayesian information matrix is calculated as J(, ) F ()( p) ( F ()( p)) C. Optimizing J(τ, t) + J(τ, t) with respect to τ Next, we investigate how to choose τ to maximize the Bayesian information tr(j(τ, t)) J(τ, t) + J(τ, t), which corresponds to the sum of the quality of target position and velocity estimates. In Figure 4, we plot tr(j(τ, 5)) with respect to τ for delays with mean,. or., we observe similar phenomenon as in the Brownian motion case: the Bayesian information increases slowly until reaches the peak value and then drops sharply afterwards. VI. CONCLUSIONS AND FUTURE WORK In this paper, we consider a simplified model of a sensor network tracking a target moving according to a Brownian motion. This model allows us to establish fundamental performance trends regarding QoI as a function of the packet loss probability and network delay. Specifically, we use Bayesian information to quantify the target tracking quality and investigate how the network quality and measurement quality affect the value of Bayesian information. By assuming a Brownian motion mobility model for the target and Gaussian measurement noise for the sensors, we derive a closed-formed formula for the Bayesian information, which indicates that improving measurement quality provides a diminishing gain on tracking quality, while the gain from improving network quality remains constant. We obtain the condition under which a user obtains information gain on the target location from a tracking process. We further use Bayesian information as the metric for the gateway to select the sensor for taking measurements and determine the measurement time to control the tracking quality. Last, we generalize the above results to targets moving according VII. ACKNOWLEDGEMENT This work was supported in part by the National Science Foundation under grants EEC and the International Technology Alliance sponsored by the U.S. Army Research Laboratory and the U.K. Ministry of Defence under Agreement Number W9NF The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the National Science Foundation, the US Army Research Laboratory, the U.S. Government, the UK Ministry of Defense, or the UK Government. The US and UK Governments are authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation hereon. REFERENCES I. F. Akyildiz, S. Weilian, Y. Sankarasubramaniam, and E. E. Cayirci, A survey on sensor networks, IEEE Communications Magazine, vol. 4, no. 8, pp. 4,. A. Bharathidasan and V. A. S. Ponduru, Sensor networks: An overview, tech. rep., UC Davis. 3 C. Bisdikian, L. M. Kaplan, M. B. Srivastava, D. J. Thornley, D. Verma, and R. I. Young, Building principles for a quality of information specification for sensor information, in th Int l Conf. on Information Fusion, (Seattle, WA, USA), July 9. 4 M. L. Hernandez, Optimal sensor trajectories in bearings-only tracking, in Seventh International Conference on Information Fusion, (Mountain View, USA), June 4. 5 T. He, P. A. Vicaire, T. Yan, L. Luo, L. Gu, G. Zhou, R. Stoleru, Q. Cao, J. A. Stankovic, and T. F. Abdelzaher, Achieving realtime target tracking using wireless sensor networks, in IEEE RTAS, April 6. 6 K. L. Bell and H. L. V. Trees, Posterior Cramer-Rao bound for tracking target bearing, in 3th Annual Workshop on Adaptive Sensor Array Process, June 5. 7 W. Wei, T. He, C. Bisdikian, D. Goeckel, and D. Towsley, Target tracking with packet delays and losses - QoI amid latencies and missing data, Tech. Rep. UM-CS-9-56, UMASS, 9. 8 Y. Bar-Shalom, T. Kirubarajan, and X.-R. Li, Estimation with Applications to Tracking and Navigation. New York, NY, USA: John Wiley & Sons, Inc.,.