Power scheduling of universal decentralized estimation in sensor networks

We consider the optimal power scheduling problem for the decentralized estimation of a noise-corrupted deterministic signal in an inhomogeneous sensor network. Sensor observations are first quantized into discrete messages, then transmitted to a fusion center where a final estimate is generated. Supposing that the sensors use a universal decentralized quantization/estimation scheme and an uncoded quadrature amplitude modulated (QAM) transmission strategy, we determine the optimal quantization and transmit power levels at local sensors so as to minimize the total transmit power, while ensuring a given mean squared error (mse) performance. The proposed power scheduling scheme suggests that the sensors with bad channels or poor observation qualities should decrease their quantization resolutions or simply become inactive in order to save power. For the remaining active sensors, their optimal quantization and transmit power levels are determined jointly by individual channel path losses, local observation noise variance, and the targeted mse performance. Numerical examples show that in inhomogeneous sensing environment, significant energy savings is possible when compared to the uniform quantization strategy.     

Decentralized estimation in an inhomogeneous sensing environment

We consider decentralized estimation of a noise-corrupted deterministic parameter by a bandwidth-constrained sensor network with a fusion center. The sensor noises are assumed to be additive, zero mean, spatially uncorrelated, but otherwise unknown and possibly different across sensors due to varying sensor quality and inhomogeneous sensing environment. The classical best linear unbiased estimator (BLUE) linearly combines the real-valued sensor observations to minimize the mean square error (MSE). Unfortunately, such a scheme cannot be implemented in a practical bandwidth-constrained sensor network due to its requirement to transmit real-valued messages. In this paper, we construct a decentralized estimation scheme (DES) where each sensor compresses its observation to a small number of bits with length proportional to the logarithm of its local signal-to-noise ratio (SNR). The resulting compressed bits from different sensors are then collected and combined by the fusion center to estimate the unknown parameter. The proposed DES is universal in the sense that each sensor compression scheme requires only the knowledge of local SNR, rather than the noise probability distribution functions (pdf), while the final fusion step is also independent of the local noise pdfs. We show that the MSE of the proposed DES is within a constant factor of 25/8 of that achieved by the classical centralized BLUE estimator.     

Linear coherent distributed estimation over unknown channels

We study linear distributed estimation with coherent multiple access channel model and MMSE fusion rule. The flat fading channels are assumed unknown at the fusion center and need to be estimated. We adopt a two-phase approach, which first estimates channels and then estimates the source signal, to minimize the MSE of the estimated signal. We study optimal power allocation under a total network power constraint. We consider the optimal power allocation scheme in which training power and data power for each sensor are optimized, and the equal power allocation scheme in which training power is optimized while data power for each sensor is set equal. In both schemes, the problem is formulated as a constrained optimization problem and analytical closed-form solution is obtained. Analytic results reveal that (i) with estimated channels, the MSE approaches to a finite nonzero value as the number of sensors increases; (ii) the optimal training powers are the same in both schemes; (iii) the MSE performance compared with the case when channels are known shows the penalty caused by channel estimation becomes worse as the number of sensors increases. Simulation results verify our findings.     

Detection Performance Limits for Distributed Sensor Networks in the Presence of Nonideal Channels

Existing studies on the classical distributed detection problem typically assume idealized transmissions between local sensors and a fusion center. This is not guaranteed in the emerging wireless sensor networks with low-cost sensors and stringent power/delay constraints. By focusing on discrete transmission channels, we study the performance limits, in both asymptotic and non-asymptotic regimes, of a distributed detection system as a function of channel characteristics. For asymptotic analysis, we compute the error exponents of the underlying hypothesis testing problem; while for cases with a finite number of sensors, we determine channel conditions under which the distributed detection systems become useless - observing the channel outputs cannot help reduce the error probability at the fusion center. We demonstrate that as the number of sensors or the quantization levels at local sensors increase, the requirements on channel quality can be relaxed     

Large system decentralized detection performance under communication constraints

The problem of decentralized detection in a sensor network subjected to a total average power constraint and all nodes sharing a common bandwidth is investigated. The bandwidth constraint is taken into account by assuming non-orthogonal communication between sensors and the data fusion center via direct-sequence code-division multiple-access (DS-CDMA). In the case of large sensor systems and random spreading, the asymptotic decentralized detection performance is derived assuming independent and identically distributed (iid) sensor observations via random matrix theory. The results show that, even under both power and bandwidth constraints, it is better to combine many not-so-good local decisions rather than relying on one (or a few) very-good local decisions.     

Decision fusion in mobile wireless sensor networks using cooperative multiple symbol differential space time coding

In this article, we propose a cooperative scheme for differential space-time codes (DSTCs) to be applied for mobile wireless sensor networks (WSNs) in order to mitigate multipath fading effect. We assume that sensors make independent local decisions about the existing hypothesis and report their decisions to a fusion center, where the final decision is made. Sensors are divided into groups with two sensors each, where sensors in each pair cooperate to send their decisions as a DSTC. Differential modulation scheme, which does not require knowledge of the instantaneous fading gains, is considered to avoid the channel estimation overhead at the cooperating sensors and the fusion center. Channels between sensors and the fusion center are assumed independent identically distributed (i.i.d) Rayleigh fading channels. Moreover, Jakes-Clarkes’ channel model is considered to model the mobility of sensors and/or the fusion center. Since the complexity of the optimal fusion rule grows up exponentially with the observation interval, suboptimal fusion rules are derived and discussed. Finally, simulation results of the proposed cooperative scheme are provided and the detection capabilities of the derived decision fusion rules are compared.     

Network Lifetime Maximization for Estimation in Multihop Wireless Sensor Networks

We consider the distributed estimation by a network consisting of a fusion center and a set of sensor nodes, where the goal is to maximize the network lifetime, defined as the estimation task cycles accomplished before the network becomes nonfunctional. In energy-limited wireless sensor networks, both local quantization and multihop transmission are essential to save transmission energy and thus prolong the network lifetime. The network lifetime optimization problem includes three components: i) optimizing source coding at each sensor node, ii) optimizing source throughput of each sensor node, and iii) optimizing multihop routing path. Fortunately, source coding optimization can be decoupled from source throughput and multihop routing path optimization, and is solved by introducing a concept of equivalent 1-bit MSE function. Based on the optimal source coding, the source throughput and multihop routing path optimization is formulated as a linear programming (LP) problem, which suggests a new notion of character-based routing. The proposed algorithm is optimal and the simulation results show that a significant gain is achieved by the proposed algorithm compared with heuristic methods.     

Position‐based adaptive quantization for target location estimation in wireless sensor networks using one‐bit data

The problem of target location estimation in a wireless sensor network is considered, where due to the bandwidth and power constraints, each sensor only transmits one‐bit information to its fusion center. To improve the performance of estimation, a position‐based adaptive quantization scheme for target location estimation in wireless sensor networks is proposed to make a good choice of quantizer' thresholds. By the proposed scheme, each sensor node dynamically adjusts its quantization threshold according to a kind of position‐based information sequences and then sends its one‐bit quantized version of the original observation to a fusion center. The signal intensity received at local sensors is modeled as an isotropic signal intensity attenuation model. The position‐based maximum likelihood estimator as well as its corresponding position‐based Cramér–Rao lower bound are derived. Numerical results show that the position‐based maximum likelihood estimator is more accurate than the classical fixed‐quantization maximum likelihood estimator and the position‐based Cramér–Rao lower bound is less than its fixed‐quantization Cramér‐Rao lower bound. Copyright © 2015 John Wiley & Sons, Ltd.     

On the optimality of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels

Distributed detection has been intensively studied in the past. In this correspondence, we consider the design of local decision rules in the presence of nonideal transmission channels between the sensors and the fusion center. Under the conditional independence assumption among multiple sensor observations, we show that the optimal local decisions that minimize the error probability at the fusion center amount to a likelihood-ratio test (LRT) given a particular constraint on the fusion rule. This constraint turns out to be quite general and is easily satisfied for most sensible fusion rules. A design example using a parallel sensor fusion structure with binary-symmetric channels (BSCs) between local sensors and the fusion center is given to illustrate the usefulness of the result in obtaining optimal thresholds for local sensor observations. The study that incorporates the transmission channel in the sensor system design may have potential applications in the emerging field of wireless sensor networks.     

Impact of contextual information integration on pixel fusion

Pixel fusion is used to elaborate a classification method at pixel level. It needs to take into account the as accurate as possible information and take advantage of the statistical learning of the previous measurements acquired by sensors. The classical probabilistic fusion methods lack performance when the previous learning is not representative of the real measurements provided by sensors. The Dempster-Shafer theory is then introduced to face this disadvantage by integrating further information which is the context of the sensor acquisitions. In this paper, we propose a formalism of modeling of the sensor reliability in the context that leads to two methods of integration: the first one amounts to integrate this further information in the fusion rule as degrees of trust and the second models the sensor reliability directly as mass function. These two methods are compared in the case where the sensor reliability depends on an atmospheric disturbance: the water vapor.     

Performance Analysis and Code Design for Minimum Hamming Distance Fusion in Wireless Sensor Networks

Distributed classification fusion using error-correcting codes (DCFECC) has recently been proposed for wireless sensor networks operating in a harsh environment. It has been shown to have a considerably better capability against unexpected sensor faults than the optimal likelihood fusion. In this paper, we analyze the performance of a DCFECC code with minimum Hamming distance fusion. No assumption on identical distribution for local observations, as well as common marginal distribution for the additive noises of the wireless links, is made. In addition, sensors are allowed to employ their own local classification rules. Upper bounds on the probability of error that are valid for any finite number of sensors are derived based on large deviations technique. A necessary and sufficient condition under which the minimum Hamming distance fusion error vanishes as the number of sensors tends to infinity is also established. With the necessary and sufficient condition and the upper error bounds, the relation between the fault-tolerance capability of a DCFECC code and its pair-wise Hamming distances is characterized, and can be used together with any code search criterion in finding the code with the desired fault-tolerance capability. Based on the above results, we further propose a code search criterion of much less complexity than the minimum Hamming distance fusion error criterion adopted earlier by the authors. This makes the code construction with acceptable fault-tolerance capability for a network with over a hundred of sensors practical. Simulation results show that the code determined based on the new criterion of much less complexity performs almost identically to the best code that minimizes the minimum Hamming distance fusion error. Also simulated and discussed are the performance trends of the codes searched based on the new simpler criterion with respect to the network size and the number of hypotheses     

A combined decision fusion and channel coding scheme for distributed fault-tolerant classification in wireless sensor networks

In this paper, we consider the distributed classification problem in wireless sensor networks. Local decisions made by local sensors, possibly in the presence of faults, are transmitted to a fusion center through fading channels. Classification performance could be degraded due to the errors caused by both sensor faults and fading channels. Integrating channel decoding into the distributed fault-tolerant classification fusion algorithm, we obtain a new fusion rule that combines both soft-decision decoding and local decision rules without introducing any redundancy. The soft decoding scheme is utilized to combat channel fading, while the distributed classification fusion structure using error correcting codes provides good sensor fault-tolerance capability. Asymptotic performance of the proposed approach is also investigated. Performance evaluation of the proposed approach with both sensor faults and fading channel impairments is carried out. These results show that the proposed approach outperforms the system employing the MAP fusion rule designed without regard to sensor faults and the multiclass equal gain combining fusion rule     

Robust data fusion for multisensor detection systems

Minimax robust data fusion schemes for multisensor detection systems with discrete-time observations characterized by statistical uncertainty are developed and analyzed. Block, sequential, and serial fusion rules are considered. The performance measures used, and made robust with respect to the uncertainties, include the error probabilities of the hypothesis testing problem in the block fusion case and the error probabilities and expected numbers of samples or sensors in the sequential and serial fusion cases. For different sensor observation statistics, the minimax robust fusion rules are derived for two asymptotic cases of interest: when the number of sensors is large and when the number of times the fusion center collects the local (sensor) decisions is large. Moreover, for the case of identical sensor observation statistics and a large number of sensors, it is shown that there is no loss in optimality, if local tests using likelihood ratios and equal thresholds are used in the sequential fusion rule. In all situations, the robust decision rules at the sensors and the fusion center are shown to make use of likelihood ratios and thresholds that depend on the least-favorable probability distributions of the uncertainty class describing the statistics of sensor observations     

Distributed probabilistic routing for sensor network lifetime optimization

A probabilistic and distributed routing approach for multi-hop sensor network lifetime optimization is presented in this paper. In particular, each sensor self-adjusts their routing probabilities locally to their forwarders based on its neighborhood knowledge, while aiming at optimizing the overall network lifetime (defined as the elapsed time before the first node runs out of energy). The theoretical feasibility and a practical routing algorithm are presented. Specifically, a sufficient distributed condition regarding the neighborhood state for distributed probabilistic routing to achieve the optimal network lifetime is presented theoretically. Based on it, a distributed adaptive probabilistic routing (DAPR) algorithm, which considered both the transmission scheduling and the routing probability evolvement is developed. We prove quantitatively that DAPR could lead the routing probabilities of the distributed sensors to converge to an optimal state which optimizes the network lifetime. Further, when network dynamics happen, such as topology changes, DAPR can adjust the routing probabilities quickly to converge to a new state for optimizing the remained network lifetime. We presented the convergence speed of DAPR. Extensive simulations verified its convergence and near-optimal properties. The results also showed its quick adaptation to both the network topology and data rate dynamics.     

Nonparametric belief propagation for self-localization of sensor networks

Automatic self-localization is a critical need for the effective use of ad hoc sensor networks in military or civilian applications. In general, self-localization involves the combination of absolute location information (e.g., from a global positioning system) with relative calibration information (e.g., distance measurements between sensors) over regions of the network. Furthermore, it is generally desirable to distribute the computational burden across the network and minimize the amount of intersensor communication. We demonstrate that the information used for sensor localization is fundamentally local with regard to the network topology and use this observation to reformulate the problem within a graphical model framework. We then present and demonstrate the utility of nonparametric belief propagation (NBP), a recent generalization of particle filtering, for both estimating sensor locations and representing location uncertainties. NBP has the advantage that it is easily implemented in a distributed fashion, admits a wide variety of statistical models, and can represent multimodal uncertainty. Using simulations of small to moderately sized sensor networks, we show that NBP may be made robust to outlier measurement errors by a simple model augmentation, and that judicious message construction can result in better estimates. Furthermore, we provide an analysis of NBP's communications requirements, showing that typically only a few messages per sensor are required, and that even low bit-rate approximations of these messages can be used with little or no performance impact.     

Distributed Estimation in Energy-Constrained Wireless Sensor Networks

In this paper, we consider distributed estimation of a noise-corrupted deterministic parameter in energy-constrained wireless sensor networks from energy-distortion perspective. Given a total energy budget allowable to be used by all sensors, there exists a tradeoff between the subset of active sensors and the energy used by each active sensor in order to minimize the estimation MSE. To determine the optimal quantization bit rate and transmission energy of each sensor, a concept of equivalent unit-energy MSE function is introduced. Based on this concept, an optimal energy-constrained distributed estimation algorithm for homogeneous sensor networks and a quasi-optimal energy-constrained distributed estimation algorithm for heterogeneous sensor networks are proposed. Moreover, the theoretical energy-distortion performance bound for distributed estimation is addressed and it is shown that the proposed algorithm is quasi-optimal within a factor 2 of the theoretical lower bound. Simulation results also show that the proposed method can achieve a significant reduction in the estimation MSE when compared with other uniform schemes. Finally, the proposed algorithm is easy to implement in a distributed manner and it adapts well to the dynamic sensor environments.     

On rate-constrained distributed estimation in unreliable sensor networks

We study the problem of estimating a physical process at a central processing unit (CPU) based on noisy measurements collected from a distributed, bandwidth-constrained, unreliable, network of sensors, modeled as an erasure network of unreliable "bit-pipes" between each sensor and the CPU. The CPU is guaranteed to receive data from a minimum fraction of the sensors and is tasked with optimally estimating the physical process under a specified distortion criterion. We study the noncollaborative (i.e., fully distributed) sensor network regime, and derive an information-theoretic achievable rate-distortion region for this network based on distributed source-coding insights. Specializing these results to the Gaussian setting and the mean-squared-error (MSE) distortion criterion reveals interesting robust-optimality properties of the solution. We also study the regime of clusters of collaborative sensors, where we address the important question: given a communication rate constraint between the sensor clusters and the CPU, should these clusters transmit their "raw data" or some low-dimensional "local estimates"? For a broad set of distortion criteria and sensor correlation statistics, we derive conditions under which rate-distortion-optimal compression of correlated cluster-observations separates into the tasks of dimension-reducing local estimation followed by optimal distributed compression of the local estimates.     

Impact of Channel Errors on Decentralized Detection Performance of Wireless Sensor Networks: A Study of Binary Modulations, Rayleigh-Fading and Nonfading Channels, and Fusion-Combiners

We provide new results on the performance of wireless sensor networks in which a number of identical sensor nodes transmit their binary decisions, regarding a binary hypothesis, to a fusion center (FC) by means of a modulation scheme. Each link between a sensor and the fusion center is modeled independent and identically distibuted (i.i.d.) either as slow Rayleigh-fading or as nonfading. The FC employs a counting rule (CR) or another combining scheme to make a final decision. Main results obtained are the following: 1) in slow fading, a) the correctness of using an average bit error rate of a link, averaged with respect to the fading distribution, for assessing the performance of a CR and b) with proper choice of threshold, on/off keying (OOK), in addition to energy saving, exhibits asymptotic (large number of sensors) performance comparable to that of FSK; and 2) for a large number of sensors, a) for slow fading and a counting rule, given a minimum sensor-to-fusion link SNR, we determine a minimum sensor decision quality, in order to achieve zero asymptotic errors and b) for Rayleigh-fading and nonfading channels and PSK (FSK) modulation, using a large deviation theory, we derive asymptotic error exponents of counting rule, maximal ratio (square law), and equal gain combiners.     

Asymptotic locally optimal detector for large-scale sensor networks under the Poisson regime

We consider distributed detection with a large number of identical binary sensors deployed over a region where the phenomenon of interest (POI) has spatially varying signal strength. Each sensor makes a binary decision based on its own measurement, and the local decision of each sensor is sent to a fusion center using a random access protocol. The fusion center decides whether the event has occurred under a global size constraint in the Neyman-Pearson formulation. Assuming homogeneous Poisson distributed sensors, we show that the distribution of "alarmed" sensors satisfies the local asymptotic normality (LAN). We then derive an asymptotically locally most powerful (ALMP) detector optimized jointly over the fusion form and the local sensor threshold under the Poisson regime. We establish conditions on the spatial signal shape that ensure the existence of the ALMP detector. We show that the ALMP test statistic is a weighted sum of local decisions, the optimal weights being the shape of the spatial signal; the exact value of the signal strength is not required. We also derive the optimal threshold for each sensor. For the case of independent, identically distributed (iid) sensor observations, we show that the counting-based detector is also ALMP under the Poisson regime. The performance of the proposed detector is evaluated through analytic results and Monte Carlo simulations and compared with that of the counting-based detector. The effect of mismatched signal shapes is also investigated.     

Asymptotic Optimality Theory for Decentralized Sequential Hypothesis Testing in Sensor Networks

The decentralized sequential hypothesis testing problem is studied in sensor networks, where a set of sensors receive independent observations and send summary messages to the fusion center, which makes a final decision. In the scenario where the sensors have full access to their past observations, the first asymptotically Bayes sequential test is developed having the same asymptotic performance as the optimal centralized test that has access to all sensor observations. Next, in the scenario where the sensors do not have full access to their past observations, a simple but asymptotically Bayes sequential tests is developed, in which sensor message functions are what we call tandem quantizer, where each sensor only uses two different sensor quantizers with at most one switch between these two possibilities. Moreover, a new minimax formulation of optimal stationary sensor quantizers is proposed and is studied in detail in the case of additive Gaussian sensor noise. Finally, our results show that in the simplest models, feedback from the fusion center does not improve asymptotic performance in the scenario with full local memory, however, even a one-shot, one-bit feedback can significantly improve performance in the case of limited local memory.