This paper demonstrates fundamental limits of sensor networks for detection problems where the number of hypotheses is exponentially large. Such problems characterize many important applications including detection and classification of targets in a geographical area using a network of seismic sensors, and detecting complex substances with a chemical sensor array. We refer to such applications as large-scale detection problems. Using the insight that these problems share fundamental similarities with the problem of communicating over a noisy channel, we define the “sensing capacity” and lower bound it for a number of sensor network models. The sensing capacity expression differs significantly from the channel capacity due to the fact that for a fixed sensor configuration, codewords are dependent and nonidentically distributed. The sensing capacity provides a bound on the minimal number of sensors required to detect the state of an environment to within a desired accuracy. The results differ significantly from classical detection theory, and provide an intriguing connection between sensor networks and communications. In addition, we discuss the insight that sensing capacity provides for the problem of sensor selection.
In Section we introduce a sensor network model that accounts for contiguity in a sensor's field of view. Contiguity is an essential aspect of many classes of sensors. For example, cameras observe localized regions and seismic sensors sense vibrations from nearby targets. We demonstrate sensing capacity bounds that account for such sensors by extending results about Markov types @cite_20 , and use convex optimization to compute these bounds. The first result in Section assumes the state of the environment is modeled as a one-dimensional vector. In Section we extend this result to the case where the state of the environment is modeled as a two-dimensional grid. While a one-dimensional vector can model sensor network applications such as border security and traffic monitoring, results about two dimensions significantly increase the type of applications described by our models.
@cite_20
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We bound the number of sensors required to achieve a desired level of sensing accuracy in a discrete sensor network application (e.g. distributed detection). We model the state of nature being sensed as a discrete vector, and the sensor network as an encoder. Our model assumes that each sensor observes only a subset of the state of nature, that sensor observations are localized and dependent, and that sensor network output across different states of nature is neither identical nor independently distributed. Using a random coding argument we prove a lower bound on the 'sensing capacity' of a sensor network, which characterizes the ability of a sensor network to distinguish among all states of nature. We compute this lower bound for sensors of varying range, noise models, and sensing functions. We compare this lower bound to the empirical performance of a belief propagation based sensor network decoder for a simple seismic sensor network scenario. The key contribution of this paper is to introduce the idea of a sharp cut-off function in the number of required sensors, to the sensor network community. In passive monitoring using sensor networks, low energy supplies drastically constrain sensors in terms of calculation and communication abilities. Designing processing algorithms at the sensor level that take into account these constraints is an important problem in this context. Here we study the estimation of correlation functions between sensors using compressed acquisition and one-bit-quantization. The estimation is achieved directly using compressed samples, without considering any reconstruction of the signals. We show that if the signals of interest are far from white noise, estimation of the correlation using @math compressed samples out of @math can be more advantageous than estimation of the correlation using @math consecutive samples. The analysis consists of studying the asymptotic performance of the estimators at a fixed compression rate. We provide the analysis when the compression is realized by a random projection matrix composed of independent and identically distributed entries. The framework includes widely used random projection matrices, such as Gaussian and Bernoulli matrices, and it also includes very sparse matrices. However, it does not include subsampling without replacement, for which a separate analysis is provided. When considering one-bit-quantization as well, the theoretical analysis is not tractable. However, empirical evidence allows the conclusion that in practical situations, compressed and quantized estimators behave sufficiently correctly to be useful in, for example, time-delay estimation and model estimation. When monitoring spatial phenomena, which can often be modeled as Gaussian processes (GPs), choosing sensor locations is a fundamental task. There are several common strategies to address this task, for example, geometry or disk models, placing sensors at the points of highest entropy (variance) in the GP model, and A-, D-, or E-optimal design. In this paper, we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected. We prove that the problem of finding the configuration that maximizes mutual information is NP-complete. To address this issue, we describe a polynomial-time approximation that is within (1-1 e) of the optimum by exploiting the submodularity of mutual information. We also show how submodularity can be used to obtain online bounds, and design branch and bound search procedures. We then extend our algorithm to exploit lazy evaluations and local structure in the GP, yielding significant speedups. We also extend our approach to find placements which are robust against node failures and uncertainties in the model. These extensions are again associated with rigorous theoretical approximation guarantees, exploiting the submodularity of the objective function. We demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real-world data sets. Assume that n sensors with identical range r = f(n) 2n, for some f(n) > 1 for all n, are thrown randomly and independently with the uniform distribution in the unit interval [0,1]. They are required to move to new positions so as to cover the entire unit interval in the sense that every point in the interval is within the range of a sensor. We obtain tradeoffs between the expected sum and maximum of displacements of the sensors and their range required to accomplish this task. In particular, when f(n) = 1 the expected total displacement is shown to be Θ(√n). For sensors with larger ranges we present two algorithms that prove the upper bound for the sum drops sharply as f(n) increases. The first of these holds for f(n) ≥ 6 and shows the total movement of the sensors is O ( √In n (n)) while the second holds for 12 ≤ f(n) ≤ In n - 2 In In n and gives an upper bound of O(ln n f(n)ef(n) 2 ). Note that the second algorithm improves upon the first for f(n) δ In In n - In In Inn. Further we show a lower bound, for any 1 ω f(n) 0. For the case of the expected maximum displacement of a sensor when f(n) = 1 our bounds are (n -1 2) and for any e > 0, O(n-1 2+e). For larger sensor ranges (up to (1 - e)ln n n e > O) the expected maximum displacement is shown to be Θ(ln n n). We also obtain similar sum and maximum displacement and range tradeoffs for area coverage for sensors thrown at random in a unit square. In this case, for the expected maximum displacement our bounds are tight and for the expected sum they are within a factor of √ln n. Finally, we investigate the related problem of the expected total and maximum displacement for perimeter coverage (whereby only the perimeter of the region need be covered) of a unit square. For example, when n sensors of radius > 2 n are thrown randomly and independently with the uniform distribution in the interior of a unit square, we can show the total expected displacement required to cover the perimeter is n 12 + o(n). © 2013 ACM.