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We investigate sequential event detection problems arising in Wireless Sensor Networks (WSNs). A number of battery–powered sensor nodes of the same sensing modality are deployed in a region of interest(ROI). By an event we mean a random time(and, for spatial events, a random location) after which the random process being observed by the sensor field experiences a change in its probability law. The sensors make measurements at periodic time instants, perform some computations, and then communicate the results of their computations to the fusion centre. The decision making algorithm in the fusion centre employs a procedure that makes a decision on whether the event has occurred or not based on the information it has received until the current decision instant. We seek event detection algorithms in various scenarios, that are optimal in the sense that the mean detection delay (delay between the event occurrence time and the alarm time) is minimum under certain detection error constraints. In the first part of the thesis, we study event detection problems in a small extent network where the sensing coverage of any sensor includes the ROI. In particular, we are interested in the following problems: 1) quickest event detection with optimal control of the number of sensors that make observations(while the others sleep),2) quickest event detection on wireless ad hoc networks, and3) optimal transient change detection. In the second part of the thesis, we study the problem of quickest detection and isolation of an event in a large extent sensor network where the sensing coverage of any sensor is only a small portion of the ROI. One of the major applications envisioned for WSNs is detecting any abnormal activity or intrusions in the ROI. An intrusion is typically a rare event, and hence, much of the energy of sensors gets drained away in the pre–intrusion period. Hence, keeping all the sensors in the awake state is wasteful of resources and reduces the lifetime of the WSN. This motivates us to consider the problem of sleep–wake scheduling of sensors along with quickest event detection. We formulate the Bayesian quickest event detection problem with the objective of minimising the expected total cost due to i)the detection delay and ii) the usage of sensors, subject to the constraint that the probability of false alarm is upper bounded by .We obtain optimal event detection procedures, along with optimal closed loop and open loop control for the sleep–wake scheduling of sensors. In the classical change detection problem, at each sampling instant, a batch of samples(where is the number of sensors deployed in the ROI) is generated at the sensors and reaches the fusion centre instantaneously. However, in practice, the communication between the sensors and the fusion centre is facilitated by a wireless ad hoc network based on a random access mechanism such as in IEEE802.11 or IEEE802.15.4. Because of the medium access control(MAC)protocol of the wireless network employed, different samples of the same batch reach the fusion centre after random delays. The problem is to detect the occurrence of an event as early as possible subject to a false alarm constraint. In this more realistic situation, we consider a design in which the fusion centre comprises a sequencer followed by a decision maker. In earlier work from our research group, a Network Oblivious Decision Making (NODM) was considered. In NODM, the decision maker in the fusion centre is presented with complete batches of observations as if the network was not present and makes a decision only at instants at which these batches are presented. In this thesis, we consider the design in which the decision maker makes a decision at all time instants based on the samples of all the complete batches received thus far, and the samples, if any, that it has received from the next (partial) batch. We show that for optimal decision making the network–state is required by the decision maker. Hence, we call this setting Network Aware Decision Making (NADM). Also, we obtain a mean delay optimal NADM procedure, and show that it is a network–state dependent threshold rule on the a posteriori probability of change. In the classical change detection problem, the change is persistent, i.e., after the change–point, the state of nature remains in the in–change state for ever. However, in applications like intrusion detection, the event which causes the change disappears after a finite time, and the system goes to an out–of–change state. The distribution of observations in the out–of–change state is the same as that in the pre–change state. We call this short–lived change a transient change. We are interested in detecting whether a change has occurred, even after the change has disappeared at the time of detection. We model the transient change and formulate the problem of quickest transient change detection under the constraint that the probability of false alarm is bounded by . We also formulate a change detection problem which maximizes the probability of detection (i.e., probability of stopping in the in–change state) subject to the probability of false alarm being bounded by . We obtain optimal detection rules and show that they are threshold d rules on the a posteriori probability of pre–change, where the threshold depends on the a posteriori probabilities of pre–change, in–change, and out–of–change states. Finally, we consider the problem of detecting an event in a large extent WSN, where the event influences the observations of sensors only in the vicinity of where it occurs. Thus, in addition to the problem of event detection, we are faced with the problem of locating the event, also called the isolation problem. Since the distance of the sensor from the event affects the mean signal level that the sensor node senses, we consider a realistic signal propagation model in which the signal strength decays with distance. Thus, the post–change mean of the distribution of observations across sensors is different, and is unknown as the location of the event is unknown, making the problem highly challenging. Also, for a large extent WSN, a distributed solution is desirable. Thus, we are interested in obtaining distributed detection/isolation procedures which are detection delay optimal subject to false alarm and false isolation constraints. For this problem, we propose the following local decision rules, MAX, HALL, and ALL, which are based on the CUSUM statistic, at each of the sensor nodes. We identify corroborating sets of sensor nodes for event location, and propose a global rule for detection/isolation based on the local decisions of sensors in the corroborating sets. Also, we show the minimax detection delay optimality of the procedures HALL and ALL.