Optimal Fusion Rule Research Articles

Distributed adaptive node-specific signal estimation in a wireless sensor network with noisy links

We consider a distributed signal estimation problem in a wireless sensor network where each node aims to estimate a node-specific desired signal using all sensor signals available in the network. In this setting, the distributed adaptive node-specific signal estimation (DANSE) algorithm is able to learn optimal fusion rules with which the nodes fuse their sensor signals, as the fused signals are then transmitted between the nodes. Under the assumption of transmission without errors, DANSE achieves the performance of centralized estimation. However, noisy communication links introduce errors in these transmitted signals, e.g., due to quantization or communication errors. In this paper we show fusion rules which take additive noise in the transmitted signals into account at almost no increase in computational complexity, resulting in a new algorithm denoted as ‘noisy-DANSE’ (N-DANSE). As the convergence proof for DANSE cannot be straightforwardly generalized to the case with noisy links, we use a different strategy to prove convergence of N-DANSE, which also proves convergence of DANSE without noisy links as a special case. We validate the convergence of N-DANSE and compare its performance with the original DANSE through numerical simulations, which demonstrate the superiority of N-DANSE over the original DANSE in noisy links scenarios.

Adaptive JSPA in distributed colocated MIMO radar network for multiple targets tracking

In this study, the authors present a distributed resource-awareness colocated multiple-input-multiple-output radar network for multiple targets tracking. Specifically, taking posterior Cramer-Rao lower bound as the performance metric, they propose a joint scheduling and power allocation (JSPA) scheme as a sparse optimisation problem. The sparsity of the solution enables the JSPA scheme to jointly optimise the schedule and corresponding transmit power of radars by the non-zero entries with associated amplitudes. The JSPA scheme has the robustness of feasibility even if there is not enough resource to achieve the prescribed performance requirements of targets. In this case, the JSPA scheme will automatically relax the performance bounds for individual targets at the lowest price and decide the number of targets to be tracked according to the priorities. They also propose a global optimal fusion rule that builds the connection between resource allocation and distributed tracking. Simulation results demonstrate the effectiveness of the proposed methods. The JSPA scheme can dynamically adjust the resource utilisation of the tracking of individual targets according to the accuracy demands. The performance of important targets is favoured to be satisfied, whereas a target with low priority may be dropped by the JSPA scheme if necessary.

Optimal Decision Fusion Under Order Effects

Abstract This paper studies an optimal decision fusion problem with a group of human decision makers when an order effect is present. The order effect refers to situations wherein the process of decision making by a human is affected by the order of decisions. In our set-up, all human decision makers, called observers, receive the same data which is generated by a common but unknown hypothesis. Then, each observer independently generates a sequence of decisions which are modeled by employing non-commutative probabilistic models of the data and their relation to the unknown hypothesis. The use of non-commutative probability models is motivated by recent psychological studies which indicate that these non-commutative probability models are more suitable for capturing the order effect in human decision making, compared with the classical probability model. A central decision maker (CDM) receives (possibly a subset of) the observers’ decisions and decides on the true hypothesis. The considered problem becomes an optimal decision fusion problem with observations modeled using a non-commutative (Von Neumann) probability model. The structure of the optimal decision rule at the CDM is studied under two scenarios. In the first scenario, the CDM receives the entire history of the observers’ decisions whereas in the second scenario, the CDM receives only the last decision of each observer. The perfromance of the optimal fusion rule is numerically evaluated and compared with the optimal fusion rule derived when using a classical probability model.

Data Fusion in the Air With Non-Identical Wireless Sensors

In this paper, a multi-hypothesis distributed detection technique with non-identical local detectors is investigated. Here, for a global event, some of the sensors/detectors can observe the whole set of hypotheses, whereas the remaining sensors can either see only some aspects of the global event or infer more than one hypothesis as a single hypothesis. Another possible option is that different sensors provide complementary information. The local decisions are sent over a multiple access radio channel so that the data fusion is formed in the air before reaching the decision fusion center (DFC). An optimal energy fusion rule is formulated by considering the radio channel effects and the reliability of the sensors together, and a closed-form solution is derived. A receive beamforming algorithm, based on a modification of Lozano's algorithm, is proposed to equalize the channel gains from different sensors. Sensors with limited detection capabilities are found to boost the overall system performance when they are used along with fully capable sensors. The additional transmit power used by these sensors is compensated by the designed fusion rule and the antenna array gain. Additionally, the DFC, equipped with a large antenna array, can reduce the overall transmit energy consumption without sacrificing the detection performance.

Optimised fusion rule in cluster-based energy-efficient CSS for cognitive radio networks

ABSTRACTCooperative spectrum sensing (CSS) can solve the problem of the hidden terminal in cognitive radio (CR). Energy consumption and sensing time are high in CSS due to a large number of users. In CSS, the parameters that affect the energy efficiency (EE) of the system are fusion rule, sensing time, transmission power and number of users. In this paper, four fusion rules OR-OR, AND-OR, OR-AND and AND-AND are proposed for cluster-based CSS and optimum fusion rule is determined. It is found that in terms of EE metric, AND-OR rule outperforms the other fusion rules. An iterative algorithm has been proposed which finds the optimum number of CR users in a cluster and total number of clusters that maximises the EE. Quantitatively at SNR = −20 dB, EE is maximum when number of clusters equals to 4 and number of users in the cluster equals to 3 for 12 fixed number of users. The maximum EE is 3.69 Mbits/Hz/joule at sensing time 1.5 ms.

A Soft⁻Hard Combination Decision Fusion Scheme for a Clustered Distributed Detection System with Multiple Sensors.

In the distributed detection system with multiple sensors, there are two ways for local sensors to deliver their local decisions to the fusion center (FC): a one-bit hard decision and a multiple-bit soft decision. Compared with the soft decision, the hard decision has worse detection performance due to the loss of sensing information but has the main advantage of smaller communication costs. To get a tradeoff between communication costs and detection performance, we propose a soft–hard combination decision fusion scheme for the clustered distributed detection system with multiple sensors and non-ideal communication channels. A clustered distributed detection system is configured by a fuzzy logic system and a fuzzy c-means clustering algorithm. In clusters, each local sensor transmits its local multiple-bit soft decision to its corresponding cluster head (CH) under the non-ideal channel, in which a simple and efficient soft decision fusion method is used. Between clusters, the fusion center combines all cluster heads’ one-bit hard decisions into a final global decision by using an optimal fusion rule. We show that the clustered distributed system with the proposed scheme has a good performance that is close to that of the centralized system, but it consumes much less energy than the centralized system at the same time. In addition, the system with the proposed scheme significantly outperforms the conventional distributed detection system that only uses a hard decision fusion. Using simulation results, we also show that the detection performance increases when more bits are delivered in the soft decision in the distributed detection system.

Audit Bit Based Distributed Bayesian Detection in the Presence of Byzantines

This paper considers the problem of distributed Bayesian detection in the presence of Byzantines who seek to degrade detection performance by falsifying data. The paper employs an audit bit based mechanism to mitigate Byzantine attacks that partitions sensors into groups. In this framework, local decisions from the sensors in each group reach the fusion center (FC) via multiple paths, which enable the FC to assess (i.e., to audit) the information that reaches it to improve detection performance. In this paper, we consider the audit bit based distributed detection mechanism in the Bayesian framework and derive the optimal fusion rule in the presence of Byzantines. Using probability of error to characterize detection performance of the FC under the optimal fusion rule, we prove that the proposed mechanism is more robust against Byzantine attacks than previously proposed schemes. We show that using the audit bit based mechanism the FC becomes blind (i.e., no useful information reaches the FC) if all the nodes in the network are Byzantines. Furthermore, the paper also introduces the notion of temporal mitigation, namely of mitigating Byzantine attacks over time using audit bits, as well as characterizes optimal Byzantine attacks over time. Extensive numerical results are provided throughout the paper that provide insights into the audit bit based distributed detection mechanism.

Fusion of Correlated Decisions Using Regular Vine Copulas

In this paper, we propose a regular vine copula based methodology for the fusion of correlated decisions. Regular vine copula is an extremely flexible and powerful graphical model to characterize complex dependence among multiple modalities. It can express a multivariate copula by using a cascade of bivariate copulas, the so-called pair copulas. Assuming that local detectors are single threshold binary quantizers and taking complex dependence among sensor decisions into account, we design an optimal fusion rule using a regular vine copula under the Neyman-Pearson framework. In order to reduce the computational complexity resulting from the complex dependence, we propose an efficient and computationally light regular vine copula based optimal fusion algorithm. Numerical experiments are conducted to demonstrate the effectiveness of our approach.

Dynamic spatial fusion of cloud top phase from PARASOL, CALIPSO, cloudsat satellite data

Cloud phase is one of the important parameters of weather and climate research and core elements of atmospheric cloud parametric inversion. The accuracy of its recognition directly influences the inversion precision of cloud optical thickness, spherical albedo, effective particle radius, ice/liquid water content and other parameters. In this paper, we combine with cloud phase products of spaceborne multi-angle polarimetric radiometer, polarized laser radar, and millimeter wave radar. Next, we put forward a Dynamic Spatial Optimal Fusion (DSOF) algorithm. In this algorithm we construct the spatial optimal fusion rules of Cloud Top Phase (CTP) for multi-source data. Finally, we realize the cloud phase spatial fusion using these rules. We used the typhoon “Lupit” as the research object and verified this method. The standard deviations are 4.67%, 7.18%, 40.14% and 36.51%, respectively, compared the results of fused CTP to that of CALIPSO, CloudSat, POLDER3 and MODIS. The fusion results are most close to the CTP of CALIPSO. The results show that this method can effectively achieve multi-source cloud phase inversion, and provide new technology for the development and data synergy of spaceborne multi-sensor satellite.

Distributed Particle Filtering via Optimal Fusion of Gaussian Mixtures

We propose a distributed particle filtering algorithm based on an optimal fusion rule for local posteriors. We implement the optimal fusion rule in a distributed and iterative fashion via an average consensus algorithm. We approximate local posteriors as Gaussian mixtures and fuse Gaussian mixtures through importance sampling. We prove that under certain conditions the proposed distributed particle filtering algorithm converges in probability to a global posterior locally available at each sensor in the network. Numerical examples are presented to demonstrate the performance advantages of the proposed method in comparison with other distributed particle filtering algorithms.

Low-Complexity Optimal Hard Decision Fusion Under the Neyman–Pearson Criterion

The design of the optimal nonrandomized hard decision fusion rule under the Neyman–Pearson criterion is known to be exponential in complexity. In this letter, we formulate a more generalized version of this problem called the “generalized decision fusion problem (GDFP)” and relate it to the classical $\text{0--1}$ knapsack problem. Consequently, we show that the GDFP has a worst-case polynomial time solution. Numerical results are presented to verify the effectiveness of the proposed solution.

Distributed decision fusion under nonideal communication channels with adaptive topology

Multi-sensor decision fusion has attracted some attention in information fusion field, meanwhile, the distributed target detection has been a well-studied topic in the multi-sensor detection theory. This paper investigates the increase in detection reliability that an adaptive network (with adaptive topologies and nonideal channels and decision fusion rules) can provide, compared with a fixed topology network. We consider a network, consisting of K-local uncertainty sensors and a Fusion Center (FC) tasked with detecting the presence or absence of a target in the Region of Interest (ROI). Sensors transmit binary modulated local decisions over nonideal channels modeled as Gaussian noise or fading channels. Assuming that the signal intensity emitted by a target follows the isotropic attenuation power model, we consider three classes of network topology architectures: (1) serial topology; (2) tree topology, and (3) parallel topology. Under the Neyman–Pearson (NP) criterion, we derive the optimal threshold fusion rule with adaptive topology to minimize the error probability. Extensive simulations are conducted to validate the correctness and effectiveness of the proposed algorithms.

Inherence of Hard Decision Fusion in Soft Decision Fusion and a Generalized Radix-2 Multistage Decision Fusion Strategy

We consider the problem of decision fusion for binary event detection using a sensor network of nodes with non-identical detection and false-alarm probability pairs. We show that a soft decision fusion rule that is used to make a binary decision inherently possesses a hard decision fusion (HDF) part. Revelation of HDF part within the soft decision fusion rule, on the one hand, shows a connection between the two approaches and, on the other hand, enables straightforward computation of network-level detection and false-alarm probabilities. We consider the optimal soft decision fusion rule that minimizes the total probability of error and reveal its inherent HDF part for a network of two nodes. We subsequently use it to develop a radix-2 multistage decision fusion strategy for larger networks since revelation of HDF part for them is quite time-consuming. We consider spectrum sensing by a cognitive radio-enabled wireless sensor network to demonstrate the effectiveness of the proposed strategy. We show that the error performance of the proposed strategy is close to that exhibited by the optimal soft decision fusion rule and is better than many suboptimal hard and soft decision fusion strategies. The overall detection and false-alarm probabilities can be easily computed using the proposed strategy. We also show that the EXOR and EXNOR binary fusion rules are never optimal in minimizing the probability of error in decision making.

Generalized Byzantine Attack and Defense in Cooperative Spectrum Sensing for Cognitive Radio Networks

Cognitive radio (CR) is a revolutionary paradigm to solve the spectrum scarcity problem in wireless networks. In cognitive radio networks, cooperative spectrum sensing is regarded as a promising approach method to significantly improve the performance of spectrum sensing, but it can be threatened by Byzantine attack. The existing defense references have focused on how to mitigate the negative effect of Byzantine attack, but with some strong assumptions, such as the attackers are in minority and/or a trusted node exists for data fusion. This observation motivates us to comprehensively analyze the strategies of Byzantine attack and the fusion center (FC) in the absence of these restrictions. To be specific, we consider a generic Byzantine attack model by analyzing sophisticated malicious behaviors, which goes beyond the existing models for its generalization. Under this generalized attack model, we derive the condition which makes the FC blind from malicious perspective. On this basis, the optimal attack strategy to maximize Bayes risk is analyzed, respectively, in the case of the unknown and known fusion rule. Furthermore, we extend our analysis to the scenario where the FC has the knowledge of the attack strategy by an estimation algorithm and adopts the optimal fusion rule. Thus, we also give the closed form expression, in terms of the optimal attack strategy under different scenarios, sequentially. At last, the extensive numerical results are provided to verify our theoretical analyses and proposed estimation algorithm.

Practical method to predict an upper bound for minimum variance track‐to‐track fusion

This study deals with the problem of track-to-track fusion in a sensor network when the correlation terms between the estimates of the agents are unknown. The proposed method offers an upper bound for the optimal minimum variance fusion rule through construction of the correlation terms according to an optimisation scheme. In general, the upper bound filter provides an estimate that is more conservative than the optimal estimate generated by the minimum variance fusion rule, while at the same time is less conservative than one obtained by the widely used covariance intersection method. From the geometrical viewpoint, the upper bound filter results in the inscribed largest volume ellipsoid within the intersection region defined by the ellipsoids corresponding to the fused estimates while the covariance intersection leads to the external minimum volume ellipsoid over the intersection region. The authors demonstrate the superiority of the proposed method through analysing estimation error and consistency of the fusion filter over Monte-Carlo simulations for a multi-dimensional system.

A message passing approach for decision fusion in adversarial multi-sensor networks

We consider a simple, yet widely studied, set-up in which a Fusion Center (FC) is asked to make a binary decision about a sequence of system states by relying on the possibly corrupted decisions provided by byzantine nodes, i.e. nodes which deliberately alter the result of the local decision to induce an error at the fusion center. When independent states are considered, the optimum fusion rule over a batch of observations has already been derived, however its complexity prevents its use in conjunction with large observation windows.In this paper, we propose a near-optimal algorithm based on message passing that greatly reduces the computational burden of the optimum fusion rule. In addition, the proposed algorithm retains very good performance also in the case of dependent system states. By first focusing on the case of small observation windows, we use numerical simulations to show that the proposed scheme introduces a negligible increase of the decision error probability compared to the optimum fusion rule. We then analyse the performance of the new scheme when the FC makes its decision by relying on long observation windows. We do so by considering both the case of independent and Markovian system states and show that the obtained performance are superior to those obtained with prior suboptimal schemes. As an additional result, we confirm the previous finding that, in some cases, it is preferable for the byzantine nodes to minimise the mutual information between the sequence system states and the reports submitted to the FC, rather than always flipping the local decision.

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.

Improved Decision Fusion Model for Wireless Sensor Networks over Rayleigh Fading Channels

This paper deals with decision fusion in wireless sensor networks (WSNs) over Rayleigh fading channels. The likelihood ratio test (LRT) is considered as the optimal fusion rule when applied at the fusion center (FC). However, applying the LRT at the FC requires both the channel state information (CSI) and the local sensors’ performance indices. Acquiring such information is considered as an overhead in energy and bandwidth constrained systems such as WSNs. To avoid these drawbacks, we propose a modification to the traditional three-layer system model of a WSN where the LRT is applied as a local decision making method at the sensors level. Applying the LRT at the sensors level does not require the CSI or the local sensors’ performance indices. It only requires the signal-to-noise ratio (SNR). Moreover, a new fusion rule based on selection combining (SC) is suggested. This fusion method has the lowest complexity when compared to other diversity combining based fusion rules such as the equal gain combiner (EGC) and the maximum ratio combiner (MRC). Simulation results show that the performance of the proposed model outperforms the traditional model. In addition, applying the EGC at the FC in the proposed model provides comparable performance to the traditional model that applies the LRT at the FC.

Impact of access contention on cooperative sensing optimisation in cognitive radio networks

The tradeoff between decreasing interference to primary user (PU) and increasing secondary user's (SU's) throughput is of great importance for cooperative sensing in cognitive radio networks. Non-ideal spectrum sensing in PHY and multiple SUs' access contention in MAC jointly impact SUs' transmission and the tradeoff. In this study, the authors investigate the joint impact from a cross-layer perspective. First, they quantify the reliability of cooperative sensing and compute SUs' transmission probability under the conditions of non-ideal sensing and access contention. Closed-form expressions of the interference probability to PU and SUs' throughput are derived. Specially, two widely-used contention-based MAC protocols, i.e. slotted Aloha and distributed coordination function, are studied. Then, they formulate the sensing-throughput tradeoff problem by using interference probability to PU, rather than the detection probability, as the constraint. Finally, a 2-dimension search algorithm is proposed to obtain the optimal solution, including the optimal fusion rule, sensing duration and detection threshold. Simulation results validate the outperformance of the cross-layer scheme. They also demonstrate how the optimal solution varies with some key parameters, i.e. PU's signal-to-noise ratio and the number of contending SUs.

Unified Analysis of Cooperative Spectrum Sensing Over Composite and Generalized Fading Channels

In this paper, we investigate the performance of cooperative spectrum sensing (CSS) with multiple antenna nodes over composite and generalized fading channels. We model the probability density function (PDF) of the signal-to-noise ratio (SNR) using the mixture gamma (MG) distribution. We then derive a generalized closed-form expression for the probability of energy detection, which can be used efficiently for generalized multipath as well as composite (multipath and shadowing) fading channels. The composite effect of fading and shadowing scenarios in CSS is mitigated by applying an optimal fusion rule that minimizes the total error rate (TER), where the optimal number of nodes is derived under the Bayesian criterion, assuming erroneous feedback channels. For imperfect feedback channels, we demonstrate the existence of a TER floor as the number of antennas of the CR nodes increases. Accordingly, we derive the optimal rule for the number of antennas that minimizes the TER. Numerical and Monte-Carlo simulations are presented to corroborate the analytical results and to provide illustrative performance comparisons between different composite fading channels.