We present a test for the problem of decentralized sequential hypothesis testing, which is asymptotically optimum. By selecting a suitable sampling mechanism at each sensor, communication between sensors and fusion center is asynchronous and limited to 1-bit data. The proposed SPRT-like test turns out to be order-2 asymptotically optimum in the case of continuous time and continuous path signals, while in discrete time this strong asymptotic optimality property is preserved under proper conditions. If these conditions do not hold, then we can show optimality of order-1. Simulations corroborate the excellent performance characteristics of the test of interest.
S EQUENTIAL hypothesis testing, first introduced by Wald [1], is one of the most classical and well-studied problems of sequential analysis with applications in areas such as industrial quality control, signal detection, design of clinical trials, etc [2], [3]. In the last two decades, there has been an intense interest in the decentralized (or distributed) formulation of the problem [4]- [13]. In this setup, the sequentially acquired information for decision making is distributed across a number of sensors and is transmitted to a global decision maker (fusion center), which is responsible for making the final decision.
The main difference in the decentralized version of the problem is that the sensors are required to quantize their observations before transmitting them to the fusion center; in other words, the sensors must send to the fusion center messages that belong to a finite alphabet [4]. This requirement is imposed by the need for data compression, smaller communication bandwidth and robustness of the sensor network, which are crucial issues in application areas such as signal processing, mobile and wireless communication, multisensor data fusion, internet security, robot networks and others [5].
Depending on the local memory that the sensors possess and whether there exists feedback from the fusion center, Veeravalli et. al. [6] proposed five different configurations for the sensor network. In the same work, the authors found the optimal decentralized test -under a Bayesian setting-in the case of full feedback and local memory restricted to past decisions. Moreover, under a Bayesian setting, the case of no feedback and no local memory was treated in [7] while the case of full local memory with no feedback in [8], [9]. However, in the last two cases no exactly optimal decentralized test has been discovered (see [10] for a review).
In this work, we assume that the alphabet consists of two letters for all sensors, i.e. we allow the communication of only 1-bit messages. Moreover, we do not use any feedback and we consider the configuration of partial local memory [11]. Specifically, we assume that at each time instant each sensor has access to the value of a summary statistic -that summarizes its previous observations-and uses this value, together with its current observation, in order to send a quantized signal to the fusion center. Under this configuration, an (order-1) asymptotically optimal scheme was suggested by Mei [11] under a Bayesian setting.
Most schemes in the literature of decentralized detection require synchronous communication of the sensors with the fusion center. However, forcing distant sensors to communicate with the fusion center concurrently can be a very challenging practice. Thus, it is important to develop and analyze schemes where this communication protocol is asynchronous. Examples of asynchronous schemes can be found in [12] and [13].
Taking into account this consideration, we suggest that the sensors communicate with the fusion center asynchronously but also at random times. In particular, we suggest that the times instants at which sensor i communicates with the fusion center be stopping times that depend on the observed information at sensor i. We call this type of sampling adapted.
A special case of adapted sampling is the Lebesgue (or leveltriggered) sampling which induces, naturally, a 1-bit communication between sensors and the fusion center. Lebesgue sampling combined with a Sequential Probability Ratio Test at the fusion center give rise to a detection structure known as Decentralized Sequential Probability Ratio Test (D-SPRT) introduced by Hussain in [12], in a discrete time context. However, Hussain did not provide any theoretical support for this test nor evidence that it is efficient in any sense.
Our main contribution in this work consists in formulating and providing proof of asymptotic optimality of the D-SPRT, under both the discrete and the continuous time setup. Our asymptotic optimality result turns out to be stronger as compared to the scheme proposed in [11], with simulation experiments corroborating our theoretical findings.
The case of continuous time observations, which we analyze in Section IV, is clearly an idealization, since in practice we cannot record the sensor observations continuously. However, studying the problem under such a setup allows us to isolate the loss in efficiency due to discrete sampling of the underlying processes at the sensors. This provides valuable insight that leads to more efficient sampling schemes in the more realistic case of discrete time observations. This paper is organized as follows: Section I contains the Introduction. In Section II, we formulate the sequential hypothesis testing problem for the discrete and continuous time case under a centralized and decentralized setup. Moreover, we introduce the concept of adapted sampling and emphasize on Lebegsue sampling and the D-SPRT. In Section III we recall t
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