2.07.5 Distributed detection

The distributed detection problem is in general more difficult to handle than the estimation problem. There is an extensive literature on distributed detection problem, but there is still a number of open problems. According to decision theory, an ideal centralized detector having error-free access to all the measurements collected by a set of nodes, should form the likelihood ratio and compare it with a suitable threshold [5]. Denoting with and the two alternative hypotheses, i.e., absence or presence of the event of interest, and with the joint probability density function of the whole set of observed data, under the hypothesis , many decision tests can be cast as threshold strategies where the likelihood ratio (LR) is compared with a threshold , which depends on the decision criterion. This is true, for example, for two important formulations leading to the Bayes approach and to the Neyman-Pearson criterion, the only difference between the two’s being the values assumed by the threshold . The detection rule decides for if the threshold is exceeded or for , otherwise. In formulas,

(7.112)

Ideally, with no communication constraints, every node should then send its observation vector , with to the fusion center, which should then use all the received vectors to implement the LR test, as in (7.112). In reality, there are intrinsic limitations due to, namely: (a) the finite number of bits with which every sensor has to encode the measurements before transmission; (b) the maximum latency with which the decision has to be taken; (c) the finite capacity of the channel between sensors and fusion center. The challenging problem is then how to devise an optimum decentralized detection strategy taking into account the limitations imposed by the communication over realistic channels. The global problem, in the most general setting, is still an open problem, but there are many works in the literature addressing some specific cases. The interested reader may check the book [50] or the excellent tutorial reviews given in [51–53]. The situation becomes more complicated when we take explicitly into account the capacity bound imposed by the communication channel and we look for the number of bits to be used to quantize the local observations before transmitting to the fusion center. This problem was addressed in [54,55], where it was shown that binary quantization is optimal for the problem of detecting deterministic signals in Gaussian noise and for detecting signals in Gaussian noise using a square-law detector. The interesting indication, in these contexts, is that the gain offered by having more sensor nodes outperforms the benefits of getting more detailed (nonbinary) information from each sensor. A general framework to cast the problem of decentralized detection is the one where the topology describing the exchange of information among sensing nodes is not simply a tree, with all nodes sending data to a fusion center, but it is a graph. Each node is assumed to transmit a finite-alphabet symbol to its neighbors and the problem is how to find out the encoding (quantization) rule on each node. A class of problems admitting a message passing algorithm with provable convergence properties was proposed in [10]. The solution is a sort of distributed fusion protocol, taking explicitly into account the limits on the communication resources. An interesting and well motivated observation model is a correlated random field, as in many applications the observations concern physical quantities, like temperature or pressure, for example, which being subject to diffusion processes, are going to be spatially and temporally correlated. One of the first works addressing the detection of a known signal embedded in a correlated Gaussian noise was [56]. Using large deviations theory, the authors of [57] study the impact of node density, assuming that observations become increasingly correlated as sensors are in closer proximity of each other. More recently, the detection of a Gauss-Markov Random field (GMRF) with nearest-neighbor dependency was studied in [8]. Scaling laws for the energy consumption of optimal and sub-optimal fusion policies were then presented in [9]. The problem of energy-efficient routing of sensor observations from a Markov random field was analyzed in [58].

A classification of the various detection algorithms depends on the adopted criterion. A first important classification is the following:

In the first case, the observation is distributed across the nodes, but the decision is centralized. This case has received most of the attention. The interested reader may check, for example, the book [50] or the tutorial reviews given in [51–53]. In the second case, also the decision is decentralized. This case has been considered only relatively recently. Some references are, for example, [59–65].

An alternative classification is between

In the first case, the network collects a given amount of data along the time and space domains and then it takes a decision. In the second case, the number of observations, either in time or in terms of number of involved sensors, is not decided a priori, but it is updated at every new measurement. The network stops collecting information only when some performance criterion is satisfied (typically, false alarm and detection probability) [66–68].

One of the major difficulties in distributed detection comes from establishing the optimal decision thresholds at local and global level. The main problem is how to optimize the local decisions, taking into account that the final decisions will be only the result of the interaction among the nodes. Taking a local decision can be interpreted as a form of source coding. The simple (binary) hypothesis testing can be seen in fact as a form of binary coding. Whenever the observations are conditionally independent, given each hypothesis, the likelihood ratio test at the sensor nodes is indeed optimal [69]. However, finding the optimal quantization levels is a difficult task. Even when the observations are i.i.d., assuming identical decision rules is very common and apparently well justified. Nevertheless there are counterexamples showing that nonidentical decision rules are optimal [69]. Identical decision rules in the i.i.d. case turns out to be optimal only asymptotically, as the number of nodes tends to infinity [70].

A simple example may be useful to grasp some of the difficulties associated with distributed detection. For this purpose, we briefly recall the seminal work of Tenney and Sandell [71]. Let us consider two sensors, each measuring a real quantity , with . Based on its observation , sensor i decides whether the phenomenon of interest is present or not. In the first case, it sets the decision variable , otherwise, it sets . The question is how to implement the decision strategy, according to some optimality criterion. The approach proposed in [71] is a Bayesian approach, where the goal of each sensor is to minimize the Bayes risk, which can be made explicit by introducing the cost coefficients and the observation probability model. Let us denote by the cost of detector 1 deciding on , detector 2 deciding on , when the true hypothesis is . Denoting by the prior probability of event and by the joint pdf of having , observing the pair and deciding for the pair , the average risk can be written as

(7.113)

In this case, each node observes only its own variable and takes a decision independently of the other node. Hence, we can set

(7.114)

Expanding the right hand side by explicitly summing over index i, we get

(7.115)

Considering that and ignoring all terms which do not contain , we get

(7.116)

The average risk is minimized if is chosen as follows

(7.117)

This expression shows that the optimal local decision rule is a deterministic rule. After a few algebraic manipulations, (7.117) can be rewritten, equivalently, as [50]

(7.118)

where is the LR at node 1. Eq. (7.118) has the structure of a LRT. However, note that the threshold on the right hand side of (7.118) depends on the observation , through the term , which incorporates the statistical dependency between the observations and . Hence, Eq. (7.118) is not a proper LRT.

The situation simplifies if the observations are conditionally independent, i.e., . In such a case, the threshold can be simplified into

(7.119)

Since , (7.119) can be rewritten as

(7.120)

Hence, the threshold to be used at node 1 is a function of , i.e., on the decision taken by node 2. At the same time, the threshold to be used by node 2 will depend on the decision rule followed by node 1. This means that, even if the observations are conditionally independent and the decisions are taken autonomously by the two nodes, the decisions are still coupled through the thresholds. This simple example shows how the detection problem can be rather complicated, even under a very simple setting.

In the special case where , , and , i.e., there is no penalty if the decisions are correct, the penalty is 1, when there is one error, and the penalty is 2 when there are two errors, the threshold simplifies into

(7.121)

Hence, in this special case, the two thresholds are independent of each other and the two detectors become independent of each other.

After having pointed out through a simple example some of the problems related to distributed detection, it is now time to consider in more detail the cases where the nodes send their (possibly encoded) data to the FC or they take local decisions first and send them to the FC. In both situations, there are two extreme cases: (a) there is only one FC; (b) every node is a potential FC, as it is able to take a global decision.

2.07.6 Beyond consensus: distributed projection algorithms

In many applications, the field to be reconstructed by a sensor network is typically a smooth function of the spatial coordinates. This happens for example, in the reconstruction of the spatial distribution of the power radiated by a set of transmitters. The problem is that local measurements may be corrupted by local noise or fading effects. An important application of this scenario is given by cognitive networks. In such a case, a secondary node would need to know the channel occupation across space, to find out unoccupied channels, within the area of interest. This requires some sort of spectrum sensing, but in a localized area. The problem of sensing is that wireless propagation is typically affected by fading or shadowing effects, so that a sensor in a shadowed location might indicate that a channel is unoccupied, while this is not true. To avoid this kind of error, which would lead to undue channel occupation from opportunistic users, it is useful to resort to cooperative sensing. In such a case, nearby nodes exchange local measurements to counteract the effect of shadowing.

The problem with local averaging operations is that they should reduce the effect of fading, but without destroying valuable spatial variations. In the following, we recall a distributed algorithm proposed in [72] to recover a spatial map of a field, using local weighted averages where the weights are chosen so as to improve upon local noise or fading effects, but without destroying the spatial variation of the useful signal.

Let us consider a network composed of N sensors located at positions , and denote the measurement collected by the ith sensor by , where represents the useful field while is the observation error. Let us also denote by , a set of linearly independent spatial functions defining a basis for the useful signal. The useful signal can then be represented through the basis expansion model

(7.137)

In vector notation, introducing the N-size column vector and similarly for the vector , we may write

(7.138)

where is the matrix whose mth column is , is an r-size vector of coefficients and is the useful signal. The spatial smoothness of the useful signal field may be captured by choosing the functions to be the low frequency components of the Fourier basis or low-order 2 D polynomials. For instance, if the space under monitoring is a square of side L, we may choose the set

(7.139)

In practice, the dimension r of the useful signal subspace is typically much smaller than the dimension N of the observation space, i.e., of the number of sensors. We can exploit this property to devise a distributed denoising algorithm.

If we use a Minimum Mean Square Error (MMSE) strategy, the goal is to find the useful signal vector that minimizes the mean square error

(7.140)

The solution is well known and is given by [44]:

(7.141)

Our goal is actually to recover the vector , rather than . In such a case, the estimate of is

(7.142)

The operation performed in (7.142) corresponds to projecting the observation vector onto the subspace spanned by the columns of . Assuming, without any loss of generality (w.l.o.g.), the columns of to be orthonormal, the projector simplifies into

(7.143)

The centralized solution to this problem is then very simple: The fusion center collects all the measurements , compute and then recovers from (7.143).

The previous approach is well known. The interesting point is that the MMSE solution can be achieved with a totally decentralized approach, where every sensor interacts only with its neighbors, with no need to send any data to a fusion center. The proposed approach is based on a very simple iterative procedure, where each node initializes a state variable with the local measurement, let us say , and then it updates its own state by taking a linear combination of its neighbors’ states, similarly with what happens with consensus algorithms, but with coefficients computed in order to solve the new problem.

More specifically, denoting by , the N-size vector containing the states of all the nodes, at iteration k, and by the vector containing the initial measurements collected by all the nodes, the vector evolves according to the following linear state equation:

(7.144)

where is typically a sparse (not necessarily symmetric) matrix. The network topology is reflected into the sparsity of . In particular, the number of nonzero entries of, let us say, the ith row is equal to the number of neighbors of node i. In a WSN, the neighbors of a node are the nodes falling within the coverage area of that node, i.e., within a circle centered on the location of the node, with radius dictated by the transmit power of the node and by the power attenuation law. Our goal is to find the nonnull coefficients of that allow the convergence of to the vector given in (7.143). In general, not every network topology guarantees the existence of a solution of this problem. In the following, we will show that a solution exists only if each node has a number of neighbors greater than the dimension r of the useful signal subspace.

Let us denote by the orthogonal projector onto the r-dimensional subspace of spanned by the columns of , where denotes the range space operator and is a full-column rank matrix, assumed, w.l.o.g., to be semi-unitary. System (7.144) converges to the desired orthogonal projection of the initial value vector onto , for any given , if and only if

(7.145)

i.e.,

(7.146)

Resorting to basic algebraic properties of discrete-time systems, it is possible to derive immediately some basic properties of . In particular, denoting with OUD the Open Unit Disk, i.e., the set , a matrix is semistable if its spectrum satisfies and, if , then 1 is semisimple, i.e., its algebraic and geometric multiplicities coincide. If is semistable, then [73, p.447]

(7.147)

where denotes group generalized inverse [73, p.228]. Furthermore, setting, without loss of generality, the matrix in the form , (7.147) can be rewritten as

(7.148)

But is the projector onto the null-space of . Hence, we can state the following:

Alternatively, the previous conditions can be rewritten equivalently in the following form

The previous conditions do not make any explicit reference to the sparsity of matrix . However, when we consider a sparse matrix, reflecting the network topology, additional conditions are necessary to make sure that the previous conditions are satisfied. In other words, not every network topology is able to guarantee the asymptotic projection onto a prescribed signal subspace. One basic question is then what network topology is able to guarantee the convergence to a prescribed projector. We provide now the conditions on the sparsity of , or equivalently , guaranteeing the desired convergence.

From condition (i) of Proposition 1, given the matrix , must satisfy the equation . Let us assume that every row of has K nonzero entries and let us indicate with the set of the column indices corresponding to the nonzero entries of the jth row of . Hence, every row of must satisfy the following equation

(7.149)

To guarantee the existence of a nontrivial solution to (7.149), the matrix on the left hand side must have a kernel of dimension at least one. This requires K to be strictly greater than r, the dimension of the signal subspace. Since the number of nonzero entries of, let us say the jth, row of is equal to the number of neighbors of node j plus one (the coefficient multiplying the state of node i itself), this implies that the minimum number K of neighbors of each node must be at least equal to the dimension r of the signal subspace. Of course this condition is necessary but not sufficient. It is also necessary to check that the sparse matrix built with rows satisfying (7.149), with , had rank . This depends on the location of the nodes and on the specific choice of the orthogonal basis.

An example can be useful to illustrate the benefits achievable with the proposed technique. We consider the case where the observation is corrupted by a multiplicative, spatially uncorrelated, noise, which models, for example a fading effect. Let us denote with the measurement carried out from node i, located in the point of coordinates , where denotes the useful field, whereas represents fading. We consider, for instance, a useful signal composed by transmitters and we assume a polynomial power attenuation, so that the useful signal measured at the point of coordinates is

(7.150)

where is the power emitted by source i, located at , and specifies the power spatial spread. Furthermore, fading is modeled as a spatially uncorrelated multiplicative noise. The sensor network is composed of 2500 nodes uniformly distributed over a 2D grid. All the transmitters use the same power, i.e., in (7.150), and the noise has zero mean and variance .

In this case, it is useful to apply a homomorphic filtering to the measured field. In particular, we take the of the measurement, thus getting . To smooth out the undesired effect of fading, we assume a signal model composed by the superposition of 2D sinusoids, so that the columns of the matrix in (7.138) are composed of signals of the form , and , with . We set the initial value of the state of each node equal to and we run the distributed projection algorithm described above. After convergence, we simply take the of the result.

Figure 7.18 shows an example of application. In particular, the spatial behavior of the useful signal power is shown in the top left plot, while the observation corrupted by fading is reported in the top right figure. It is useful to consider that, in the example at hand, the useful signal would require a Fourier series expansion with an infinite number of terms to null the modeling error. Conversely, in our example, we used two different orders, and . The corresponding reconstructions are shown in the bottom figures. From Figure 7.18 it is evident the capability of the proposed distributed approach to provide a significant attenuation of the fading phenomenon, without destroying valuable signal variations.

2.07.7 Minimum energy consensus

Although distributed algorithms to achieve consensus have received a lot of attention because of their capability of reaching optimal decisions without the need of a fusion center, the price paid for this simplicity is that consensus algorithms are inherently iterative. As a consequence the iterated exchange of data among the nodes might cause an excessive energy consumption. Hence, to make consensus algorithms really appealing in practical applications, it is necessary to minimize the energy consumption necessary to reach consensus. The network topology plays a fundamental role in determining the convergence rate [74]. As the network connectivity increases, so does the convergence rate. However, a highly connected network entails a high power consumption to guarantee reliable direct links between the nodes. On the other hand, if the network is minimally connected, with only neighbor nodes connected to each other, a low power is spent to maintain the few short range links, but, at the same time, a large convergence time is required. Since what really matters in a WSN is the overall energy spent to achieve consensus, in [76,77] it was considered the problem of finding the optimal network topology that minimizes the overall energy consumption, taking into account convergence time and transmit powers jointly. More specifically, in [77] it is proposed a method for optimizing the network topology and the power allocation across every link in order to minimize the energy necessary to achieve consensus. Two different types of networks are considered: a) deterministic topologies, where node positions are arbitrary, but known; b) random geometries, where the unknown node locations are modeled as random variables. We will now review the methodology used in both cases.

2.07.8 Matching communication network topology to statistical dependency graph

In Section 2.07.2.2, we saw that the topology of a sensor network observing a random field should depend on the structure of the graph describing the observed field. In this section, we recall a method proposed in [82,83] to design the topology of a wireless sensor network observing a Markov random field in order to match the structure of the dependency graph of the observed field, under constraints on the power used to ensure the sensor network connectivity. As in [82,83], our main task is to recover the sparsity of the dependency graph and to replicate it at the sensor network level, under the constraint of limiting the transmit power necessary to establish the link among the nodes. Also in this case, searching for an optimal topology is a combinatorial problem. To avoid the computational burden of solving the combinatorial problem, we propose an ad hoc relaxation technique that allows us to achieve the solution through efficient algorithms based on difference of convex problems.

Let us assume to have a network composed of N nodes, each one observing a spatial sample of a Gaussian Markov Random Field. We denote by the vector of the observations collected by the N nodes and we assume that has zero mean and covariance matrix . The statistical dependency among the random variables is well captured by the structure of the Markov graph whose vertices correspond to the random variables and whose links denote statistical dependencies among the variables. As discussed in 2.07.2.2 the main feature of a Markov graph is that it is sparse and there is no link between two nodes if and only if their observations are statistically independent. Moreover, if the random vector is also Gaussian, with covariance matrix , the sparsity of the Markov graph is completely specified by the sparsity of the precision matrix, which is the inverse of the covariance matrix, i.e., . On the other hand, the topology of the WSN can also be described by a graph, having adjacency matrix such that only if there is a physical link between nodes i and j. We use a simple propagation model such that there is a link between node i and j if the power received by node j exceeds a minimum power . The received power depends on the power used by node i to transmit to node j and on the distance between nodes i and j through the equation

(7.164)

where is the path loss exponent.

As proposed in [82,83], our goal is to design the topology of the WSN, and hence its adjacency matrix , in order to match as well as possible the topology of the dependency graph, compatibly with the power expenditure necessary to establish each link in the network. Without any power constraint, we would choose to be equal to , so as to reproduce the same sparsity of the dependency graph. Adding the power constraints, we will end up, in general, with a matrix different from . We measure the difference between the two matrices and using the so called Burg divergence, defined as

(7.165)

Even though the Burg divergence does not respect all the prerequisites to be a distance, it holds true that , if and only if , otherwise, the divergence is strictly positive. If the matrices and are definite positive, the expression in (7.165) coincides with the Kullback-Leibler divergence between the probability density function (pdf) of two Gaussian random vectors having zero mean and precision matrices and or, equivalently, covariance matrices and .

2.07.9 Conclusions and further developments

In this article we have provided a general framework to show how an efficient design of a wireless sensor network requires a joint combination of in-network processing and communication. In particular, we have shown that inferring the structure of the graph describing the statistical dependencies among the observed data can provide important information on how to build the sensor network topology and how to design the flow of information through the network. We have illustrated several possible network architectures where the global decisions, either estimation or hypothesis testing, are taken by a central node or in a totally decentralized way. In particular, various forms of consensus have been shown to be instrumental to achieve globally optimal performance through local interactions only. Consensus algorithms have then been generalized to more sophisticated signal processing techniques able to provide a cartography of the observed field. In a decentralized framework, the network topology plays an important role in terms of convergence time as well as structure of the final consensus value. Considering that most sensor networks exchange information through a wireless channel, we have addressed the problem of finding the network topology that minimizes the energy consumption required to reach consensus. Finally, we have showed how to match the network topology to the Markov graph describing the observed variables, under constraints imposed by the power consumption necessary to establish direct links among the sensor nodes.

Even though the field of distributed detection and estimation has accumulated an enormous amount of research works, there are still many open problems, both in the theoretical as well as in the application sides. In the following we make a short list of possible topics of future interest.

Appendix A

In this appendix we briefly review some important notations and basic concepts of graph theory that have been adopted in the previous sections (for a more detailed introduction to this field see [13]).

References

1. Akyildiz IF, Su W, Sankarasubramaniam Y, Cayirci E. Wireless sensor networks: a survey. Comput Networks 2002;393–422.

2. Giridhar A, Kumar PR. Toward a theory of in-network computation in wireless sensor networks. IEEE Commun Mag. 2006;98–107.

3. Gupta P, Kumar P. Critical power for asymptotic connectivity in wireless networks. In: McEneaney W, Yin G, Zhang Q, eds. Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H Fleming. Boston, MA: Birkhauser; 1998.

4. Giridhar A, Kumar PR. Computing and communicating functions over sensor networks. IEEE J Sel Areas Commun. 2005;755–764.

5. Kay SM. Fundamentals of Statistical Signal Processing, Detection Theory. vol II Englewood Cliffs, NJ: Prentice-Hall PTR; 1998.

6. Whittaker J. Graphical Models in Applied Multivariate Statistics. John Wiley & Sons 1990.

7. Lauritzen SL. Graphical Models. Oxford University Press 1996.

8. Anandkumar A, Tong L, Swami A. Detection of GaussMarkov random fields with nearest-neighbor dependency. IEEE Trans Inform Theory. 2009;55:816–827.

9. Anandkumar A, Yukich JE, Tong L, Swami A. Energy scaling laws for distributed inference in random fusion networks. IEEE J Sel Areas Commun. 2009;1203–1217.

10. Kreidl OP, Willsky AS. An efficient message-passing algorithm for optimizing decentralized detection networks. IEEE Trans Autom Contr. 2010;563–578.

11. Gastpar M. Information-theoretic bounds on sensor network performance. In: Swami A, Zhao Q, Hong Y-W, Tong L, eds. Wireless Sensor Networks—Signal Processing and Communications Perspectives. John Wiley & Sons 2007; Chapter 2.

12. Cover TM, Thomas JA. Elements of Information Theory. Hoboken, NJ: John Wiley & Sons; 2006.

13. Godsil C, Royle G. Algebraic Graph Theory. New York: Springer; 2001.

14. Ren W, Beard RW, Atkins EM. Information consensus in multivehicle cooperative control: collective group behavior through local interaction. IEEE Control Syst Mag. 2007;27(2):71–82.

15. Olfati-Saber R, Fax JA, Murray RM. Consensus and cooperation in networked multi-agent systems. Proc IEEE. 2007;95:215–233.

16. Barbarossa S, Scutari G. Decentralized maximum-likelihood estimation for sensor networks composed of nonlinearly coupled dynamical systems. IEEE Trans Signal Process. 2007;55(7):3456–3470.

17. Scutari G, Barbarossa S, Pescosolido L. Distributed decision through self-synchronizing sensor networks in the presence of propagation delays and asymmetric channels. IEEE Trans Signal Process. 2008;56(4):1667–1684.

18. Jadbabaie A, Morse S. Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Trans Auto Control. 2003;48:988–1001.

19. Xiao L, Boyd S. Fast linear iterations for distributed averaging. Syst Control Lett. 2004;53:65–78.

20. Nedić A, Ozdaglar Asuman. Cooperative distributed multi-agent optimization. In: Palomar DP, Eldar YC, eds. Convex Optimization in Signal Processing and Communications. Cambridge University Press 2010.

21. Sardellitti S, Giona M, Barbarossa S. Fast distributed average consensus algorithms based on advection-diffusion processes. IEEE Trans Signal Process. 2010;58(2):826–842.

22. Fiedler M. Algebraic connectivity of graphs. J Czech Math. 1973;23:298–305.

23. Horn R, Johnson CR. Matrix Analysis. Cambridge University Press 1985.

24. Hatano Y, Das AK, Mesbahi M. Agreement in presence of noise: pseudogradients on random geometric networks. In: Proceedings of the IEEE Conference on Decision and Control (CDC), Seville, Spain. 2005;6382–6387.

25. Tahbaz Salehi A, Jadbabaie A. On consensus over random networks. In: Proceedings of the 44th Allerton Conference, UIUC, Illinois, USA. 2006;27–29.

26. Tahbaz Salehi A, Jadbabaie A. Consensus over ergodic stationary graph processes. IEEE Trans Automat Contr. 2010;55.

27. Aysal TC, Coates M, Rabbat M. Distributed average consensus using probabilistic quantization. In: Proceedings of the IEEE/SP Workshop on Statistical Signal Processing Workshop (SSP), Maddison, Wisconsin, USA. 2007;640–644.

28. Huang M, Manton J. Stochastic approximation for consensus seeking: mean square and almost sure convergence. In: Proceedings of the IEEE Conference on Decision and Control (CDC), New Orleans, LA, USA. 2007;306–311.

29. Kar S, Moura JMF. Distributed consensus algorithms in sensor networks with imperfect communication: link failures and channel noise. IEEE Trans Signal Process. 2009;57(5):355–369.

30. Lipshitz SP, Wannamaker RA, Vanderkooy J. Quantization and dither: a theoretical survey. J Audio Eng Soc. 1992;40:355–375.

31. Wannamaker R, Lipshitz S, Vanderkooy J, Wright J. A theory of nonsubtractive dither. IEEE Trans Signal Process. 2000;48(2):499–516.

32. Schuchman L. Dither signals and their effect on quantization noise. IEEE Trans Commun Technol COMM-12 1964;162–165.

33. Polyak B. Introduction to Optimization. New York: Optimization Software Inc.; 1987.

34. Barbarossa S, Scutari G. Bio-inspired sensor network design: distributed decision through self-synchronization. IEEE Signal Process Mag. 2007;24(3):26–35.

35. Schizas ID, Ribeiro A, Giannakis GB. Consensus in Ad Hoc WSNs with noisy links: Part I—Distributed estimation of deterministic signals. IEEE Trans Signal Process. 2008;56:350–364.

36. Schizas ID, Giannakis GB, Roumeliotis SI, Ribeiro A. Consensus in Ad Hoc WSNs With noisy links: Part II—Distributed estimation and smoothing of random signals. IEEE Trans Signal Process. 2008;56(2):1650–1666.

37. Cattivelli F, Lopes CG, Sayed AH. Diffusion recursive least-squares for distributed estimation over adaptive networks. IEEE Trans Signal Process. 2008;56(5):1865–1877.

38. Braca P, Marano S, Matta V. Enforcing consensus while monitoring the environment in wireless sensor networks. IEEE Trans Signal Process. 2008;56(7):3375–3380.

39. Dimakis AG, Kar S, Moura JMF, Rabbat MG, Scaglione A. Gossip algorithms for distributed signal processing. Proc IEEE. 2010;98(11):1847–1864.

40. Bertrand A, Moonen M. Consensus-based distributed total least squares estimation in Ad Hoc wireless sensor networks. IEEE Trans Signal Process. 2011;59(5):2320–2330.

41. Li L, Scaglione A, Manton JH. Distributed principal subspace estimation in wireless sensor networks. IEEE J Sel Top Signal Process. 2011;5(4):725–738.

42. Kar S, Moura JMF, Ramanan K. Distributed parameter estimation in sensor networks: nonlinear observation models and imperfect communication. IEEE Trans Inform Theory. 2012;58:3575–3605.

43. Mateos G, Schizas ID, Giannakis GB. Distributed recursive least-squares for consensus-based in-network adaptive estimation. IEEE Trans Signal Process. 2009;57(11):4583–4588.

44. Kay SM. Fundamentals of Statistical Signal Processing, Estimation Theory. vol I Englewood Cliffs, NJ: Prentice-Hall PTR; 1993.

45. Bertsekas DP, Tsitsiklis JN. Parallel and Distributed Computation: Numerical Methods. Belmont, MA: Athena Scientific; 1989.

46. Boyd S, Parikh N, Chu E. Distributed Optimization and Statistical Learning Via the Alternating Direction Method of Multipliers. Now Publishers May 2011.

47. Tibshirani R. Regression Shrinkage and Selection via the Lasso. J Roy Stat Soc. 1996;58:267–288.

48. Mateos G, Bazerque JA, Giannakis GB. Distributed sparse linear regression. IEEE Trans Signal Process. 2010;58:5262–5276.

49. Xiao JJ, Cui S, Luo ZQ, Goldsmith AJ. Power scheduling of universal decentralized estimation in sensors networks. IEEE Trans Signal Process. 2006;54(2):413–422.

50. Varshney PK. Distributed Detection and Data Fusion. New York: Springer-Verlag; 1997.

51. Viswanathan R, Varshney PK. Distributed detection with multiple sensors: Part I—Fundamentals. Proc IEEE. 1997;85:54–63.

52. Blum RS, Kassam SA, Poor HV. Distributed detection with multiple sensors: Part II—Advanced topics. Proc IEEE. 1997;85:64–79.

53. Chamberland JF, Veeravalli V. Wireless sensors in distributed detection applications. IEEE Signal Process Mag. 2007;16–25.

54. Chamberland JF, Veeravalli V. Decentralized detection in sensor networks. IEEE Trans Signal Process. 2003;407–416.

55. Chamberland JF, Veeravalli V. Asymptotic results for decentralized detection in power constrained wireless sensor networks. IEEE J Sel Areas Commun. 2004;1007–1015.

56. Willett P, Swaszek PF, Blum RS. The good, bad, and ugly: distributed detection of a known signal in dependent gaussian noise. IEEE Trans Signal Process. 2000;3266–3278.

57. Chamberland JF, Veeravalli V. How dense should a sensor network be for detection with correlated observations? IEEE Trans Inform Theory 2006;5099–5106.

58. Anandkumar A, Tong L, Swami A. Energy efficient routing for statistical inference ofMarkov random fields. In: Proceedings of CISS 2007, Baltimore. 2007;643–648.

59. Alanyali M, Saligrama V, Savas O, Aeron S. Distributed Bayesian hypothesis testing in sensor networks. In: 2004;5369–5374. Proceedings of the American Control Conference, Boston, MA. vol. 6.

60. Saligrama V, Alanyali M, Savas O. Distributed detection in sensor networks with packet losses and finite capacity links. IEEE Trans Signal Process. 2006;54(11):4118–4132.

61. Aldosari SA, Moura JMF. Topology of sensor networks in distributed detection. In: Proceedings of the IEEE International Conference on Acoustic Speech Signal Processing (ICASSP), Toulouse, France. 2006;V1061–V1064.

62. S. Kar, J.M.F. Moura, Consensus based detection in sensor networks: topology optimization under practical constraints, Workshop Information Theory in Sensor Networks, Santa Fe, NM, 2007.

63. Bajovic D, Jakovetic D, Xavier J, Sinopoli B, Moura JMF. Distributed detection via gaussian running consensus: large deviations asymptotic analysis. IEEE Trans Signal Process. 2011;59:4381–4396.

64. Cattivelli F, Sayed AH. Distributed detection over adaptive networks using diffusion adaptation. IEEE Trans Signal Process. 2011;59(5):1917–1932.

65. Xiao L, Boyd S, Lall S. A space-time diffusion scheme for peer-to-peer least-squares estimation. In: Proceedings of International Conference on Information Processing in Sensor Networks, Nashville, TN. 2006;168–176.

66. Marano S, Matta V, Willett P, Tong L. Cross-layer design of sequential detectors in sensor networks. IEEE Trans Signal Process. 2006;54:4105–4117.

67. Mei Y. Asymptotic optimality theory for decentralized sequential hypothesis testing in sensor networks. IEEE Trans Inform Theory. 2008;54:2072–2089.

68. Sardellitti S, Barbarossa S, Pezzolo L. Distributed double threshold spatial detection algorithms in wireless sensor networks. In: Proceedings of the IEEE 10th Workshop on Signal Processing Advances in Wireless Communication (SPAWC ’09), Perugia, Italy. 2009;51–55.

69. Tsitsiklis JN. Decentralized detection. Adv Stat Signal Process. 1993;2:297–344.

70. Tsitsiklis JN. Decentralized detection by a large number of sensors. Math Control Signal Syst. 1988;1(2):167–182.

71. Tenney RR, Sandell Jr NR. Detection with distributed sensors. IEEE Trans Aerosp Electron Syst. 1981;17:98–101.

72. Barbarossa S, Scutari G, Battisti T. Distributed signal subspace projection algorithms with maximum convergence rate for sensor networks with topological constraints. In: ICASSP 2009. 2009.

73. Bernstein DS. Matrix Mathematics. Princeton University Press 2005.

74. Boyd S, Ghosh A, Prabhakar B, Shah D. Randomized gossip algorithms. IEEE Trans Inform Theory. 2006;52:2508–2530.

75. S. Barbarossa, G. Scutari, T. Battisti, Cooperative sensing for cognitive radio using decentralized projection algorithms, in: IEEE 10th Workshop on Signal Processing in Wireless Communication, SPAWC’ 09, pp. 116–120.

76. Sardellitti S, Barbarossa S, Swami A. Average consensus with minimum energy consumption: optimal topology and power allocation. In: Proceedings of the European Signal Processing Conference (EUSIPCO 2010), Aalborg, Denmark. 2010;189–193.

77. Sardellitti S, Barbarossa S, Swami A. Optimal topology control and power allocation for minimum energy consumption in consensus networks. IEEE Trans Signal Process. 2012;60(1):383–399.

78. Jeflea A. A Parametric study for solving nonlinear fractional problems. An St Univ Ovidius Constanta. 2003;11:87–92.

79. Dinkelbach W. On nonlinear fractional programming. Manag Sci. 1967;13:492–498.

80. Hatano Y, Mesbahi M. Agreement over random networks. IEEE Trans Automat Contr. 2005;50:1867–1872.

81. Aysal TC, Barner KE. Convergence of consensus models with stochastic disturbances. IEEE Trans Inform Theory. 2010;56:4101–4113.

82. Sardellitti S, Barbarossa S. Energy preserving matching of sensor network topology to the statistical graphical model of the observed field. In: 17th International Conference on Digital Signal Processing (DSP), Corfu, Greece. 2011.

83. S. Barbarossa, S. Sardellitti, Energy Preserving Matching of Sensor Network Topology to Dependency Graph of the Observed Field, IEEE Trans. Signal Process. (in press).

84. Donoho D. Compressive sensing. IEEE Trans Inform Theory. 2006;52:1289–1306.

85. Gasso G, Rakotomamonjy A, Canu S. Recovering sparse signals with a certain family of non-convex penalties and DC programming. IEEE Trans Signal Process. 2009;57:4686–4698.

86. Tao PD, An LTH. A D.C optimization algorithms for solving the trust-region subproblem. SIAM J Optim. 1998;8:476–505.

87. Zdunek R, Cichocki A. Fast nonnegative matrix factorization algorithms using projected gradient approaches for large-scale problems. Comput Intell Neurosci. 2008.

88. Brualdi RA, Ryser HJ. Combinatorial Matrix Theory. Cambridge, UK: Cambridge University Press; 1991.

89. Mezard M, Parisi G, Zee A. Spectra of Euclidean random matrices. Nuclear Phys B. 1999;559:689–701.

90. Bordenave C. Eigenvalues of Euclidean Random Matrices, Random Structures and Algorithms. vol. 33 NY, USA: John Wiley & Sons; December 2008; pp. 515–532.

91. Rai S. The Spectrum of a random geometric graph is concentrated. J Theor Probab. 2007;20:119–132.