Optimal Node Density for Two-Dimensional Sensor Arrays

📝 Abstract

The problem of optimal node density for ad hoc sensor networks deployed for making inferences about two dimensional correlated random fields is considered. Using a symmetric first order conditional autoregressive Gauss-Markov random field model, large deviations results are used to characterize the asymptotic per-node information gained from the array. This result then allows an analysis of the node density that maximizes the information under an energy constraint, yielding insights into the trade-offs among the information, density and energy.

💡 Analysis

The problem of optimal node density for ad hoc sensor networks deployed for making inferences about two dimensional correlated random fields is considered. Using a symmetric first order conditional autoregressive Gauss-Markov random field model, large deviations results are used to characterize the asymptotic per-node information gained from the array. This result then allows an analysis of the node density that maximizes the information under an energy constraint, yielding insights into the trade-offs among the information, density and energy.

📄 Content

arXiv:0805.1262v1 [cs.IT] 9 May 2008 OPTIMAL NODE DENSITY FOR TWO-DIMENSIONAL SENSOR ARRAYS Youngchul Sung†, H. Vincent Poor and Heejung Yu ABSTRACT The problem of optimal node density for ad hoc sensor networks deployed for making inferences about two dimensional correlated random fields is considered. Using a symmetric first order conditional autoregressive Gauss- Markov random field model, large deviations results are used to character- ize the asymptotic per-node information gained from the array. This result then allows an analysis of the node density that maximizes the information under an energy constraint, yielding insights into the trade-offs among the information, density and energy.

1. INTRODUCTION We consider the design of wireless ad hoc sensor networks for making inferences about correlated random fields that can model various physical processes, such as temperature, humidity or the density of a certain gas, in a two-dimensional (2-D) space. In par- ticular, we consider the optimal density problem for sensor net- works deployed for statistical inference such as detection or recon- struction of the underlying field. From the information-theoretic perspective, statistical inference via sensor networks can be viewed as a problem of extracting information about an underlying phys- ical process using networked sensor nodes that consume energy for both sensing and communication. Thus, the optimal density problem can be formulated as follows. Problem 1 Given a sensor network deployed on a fixed coverage area of size 2L × 2L and with total available energy E, find the node density µn that maximizes the total information It obtainable from the network. To address this problem, we model the signal field as a 2-D Gauss- Markov random field (GMRF), and consider the Kullback-Leibler information (KLI) and mutual information (MI) [1] as ways of quantifying inferential performance. (The operational meaning of the KLI is given by its appearance as the error exponent of the miss probability of Neyman-Pearson detection of the signal field in sensor noise, whereas that of the MI is given by its role as a measure of uncertainty reduction.) Our approach to determine the total information obtainable from a sensor network is based on the large deviations principle (LDP). That is, for large networks, the total information is approximately given by the product of the number of sensors and the asymptotic per-node information, or the asymptotic information rate. (The units of these intensive quanti- ties is thus nats/sample.) Although closed-form expressions for the †Y. Sung and H. Yu is with the Dept. of Electrical Engineering, Korea Ad- vanced Institute of Science and Technology (KAIST), Daejeon 305-701, South Ko- rea. Email:ysung@ee.kaist.ac.kr and hjyu@stein.kaist.ac.kr. H. V. Poor is with the Dept. of Electrical Engineering, Princeton University, Princeton, NJ 08544. Email: poor@princeton.edu. The work of Y. Sung was supported in part by Brain Korea 21 Project, the School of Information Technology, KAIST. The work of H. V. Poor was supported in part by the U. S. National Science Foundation under Grants ANI-03- 38807 and CNS-06-25637. asymptotic per-node information are not available for general 2-D signals, for the conditional autoregression (CAR) model closed- form expressions for the asymptotic KLI and MI rates have been determined by the authors in [2]. Based on these expressions for asymptotic information rates and their properties, in the current paper we investigate the problem of optimal node density. It is seen that there exists a density maximizing the total information obtainable under an energy constraint. The optimal density is eas- ily obtained numerically, and the behavior of the total information as a function of the density is explained. 1.1. Related Work The issues of optimal sensor density and optimal sampling have been considered based on LDP in previous work (e.g., [3]). How- ever, most work in this area is based on one-dimensional (1-D) sig- nal or time series models that do not capture the two-dimensionality of actual spatial signals. In contrast, our work is based on the LDP results obtained in [2], where a closed-form expression for the asymptotic KLI rate is obtained in the spectral domain. For a 2-D setting, an error exponent was obtained for the detection of 2-D GMRFs in [4], where the sensors are located randomly and the Markov graph is based on the nearest neighbor dependency enabling a loop-free graph. In that work, however, measurement noise was not considered, unlike the present analysis.
2. SIGNAL MODEL AND BACKGROUND In this section, we briefly introduce our previous work [2] relevant to the sensor density problem. To simplify the problem and gain insight into the 2-D case, we assume that sensors are located on a 2-D lattice In = [−n : 1 : n]2, as shown in Fig. 1, and thus form a 2-D array. We model the underlying physical process as a 2-D GMRF and assume that each sensor has Gaussian measur

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