Random Access Sensor Networks: Field Reconstruction From Incomplete Data

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1 Random Access Sensor Networks: Field Reconstruction From Incomplete Data Fatemeh Fazel Northeastern University Boston, MA Maryam Fazel University of Washington Seattle, WA Milica Stojanovic Northeastern University Boston, MA Abstract We address efficient data gathering from a network of distributed sensors deployed in a challenging field environment with limited power and bandwidth. Utilizing the low-rank property of the sensing field, we leverage results from the matrix completion theory to integrate the sensing procedure with a simple and robust communication scheme based on random channel access. Results show that the space-time map of the sensing field can be recovered efficiently, using only a small subset of sensor measurements, collected over a fading random access channel. I. INTRODUCTION Wireless sensor networks greatly facilitate long-term monitoring of the natural environment [1]. Such networks comprise a large array of battery-powered sensors, a central collection unit referred to as the Fusion Center (FC) and a communication scheme for the sensors to transmit their data to the FC. Once the network is deployed, access to the sensors is limited, hence re-charging the batteries is not easy. As a result, longterm deployment entails energy-aware sensing and efficient communication. Due to the size of the network, and the energy, bandwidth and time constraints of the data acquisition process, missing or partial information presents a ubiquitous problem in many applications such as geosciences [2], industrial and environmental monitoring, and monitoring surface deformations related to oil fields [3]. Thus, field recovery inevitably has to rely on incomplete data. Based on the principles of compressed sensing and random channel access, in [4], [] we proposed an integrated architecture for sensing and communication, referred to as Random Access Compressed Sensing (RACS). This scheme targets the recovery of slowly varying fields from sub-sampled data, gathered while the process is approximately unchanged. RACS utilizes the fact that most natural signals have a sparse representation in the frequency domain. In the current work, we relax this assumption and take into consideration the temporal variations of the field over much longer periods of time. Employing results from the lowrank matrix recovery problem [6] and the matrix completion theory [7] [9], we address the design of a random access network for long term monitoring of temporally varying fields. Research funded in part by ONR grant N , NSF grant , and NSF CAREER grant ECCS The matrix completion theory addresses the recovery of a low-rank matrix from a subset of its entries. Matrix completion is applied to field monitoring in [], where the authors propose an online algorithm for subspace tracking. The process of gathering the data, which is the focus of our work, was not however in the scope of this paper. In the context of cognitive radios, collaborative spectrum sensing using matrix completion is studied in [11]. In [12], the authors discuss sampling strategies over a grid network to improve energy efficiency. The communication aspect, however, is not addressed, and the time variations of the process are not considered. The sensing and communication aspects of the wireless sensor networks are commonly treated independently. Our contribution is in unifying the sensing scheme, based on lowrank matrix recovery, with a simple and robust communication scheme using random access. The proposed integrated sensing and communication scheme relies only on a few assumptions about the statistical properties of the sensing field, and is thus applicable to a variety of fields. The rest of the paper is organized as follows: In Section II, we integrate the sensing scheme, based on matrix completion, with the communication scheme, using random access. In Section III, a probabilistic model for the system is provided upon which the network design guidelines of Section IV are then based. In Section V, we study the inherent trade-offs in choosing the system parameters. Finally, concluding remarks are provided in Section VI. II. RECONSTRUCTION FROM INCOMPLETE DATA We consider a sensor network with N nodes, where each node measures the field at a given sensing rate λ 1. We will rely on the matrix completion theory to determine the required sensing rate λ 1, which is the objective of the design procedure. A. Field Model The network measures a process u(x, y, t) where x and y denote positions in space and t denotes time. We assume that node i is located at position (x i, y i ), where i {1,..., N}. The observations of the field are available at discrete intervals of time separated by t. Assuming a stationary process, the discretization interval t depends on the temporal autocorrelation of the process R(τ), which quantifies the average correlation between two samples of the process separated

2 by time τ. In other words, R(τ) = 1 N N i=1 R i(τ), where R i (τ) = E{u i (t + τ)u i (t)} denotes the autocorrelation of the process observed by sensor i. The value of t is chosen as the time lag during which the samples of the signal are sufficiently correlated, i.e., t is chosen such that R( t) = qr() (1) where q is the desired level of the correlation. 1 The process thus remains almost unchanged during t, and packets received from distinct nodes during this time interval are regarded as pertaining to the same snapshot of the process. Precise synchronization among nodes, which is difficult to achieve in a large-scale network, is thus not needed. The appropriate value for t strikes a balance between the required sensing rate and the redundancy of sampling. Consequently, as we will show in Section V, it provides a trade-off between the consumed energy and the quality of reconstruction. With a slight abuse of notation, let u i (m) denote the signal measured by sensor node i at any time t [(m 1) t, m t]. Now the sensing field is described by the space-time matrix of sensor measurements as u 1 (1) u 1 (2)... u 1 (M) U =... (2) u N (1) u N (2)... u N (M) The matrix U is of dimension N M, where for large-scale and long-term monitoring applications, both M and N are large. We assume that U is a low rank matrix, i.e., its rank r is much smaller than N and M, as is the case with many natural signals. To illustrate this fact, we generate an example process which will serve as our test case. We assume a large field over which k = sources are randomly placed, each generating an exponentially decaying signal (e.g., heat, sound, etc). A total of N = 2 sensors are distributed over the field to monitor the process. At time t, the observed process at coordinate (x i, y i ) is given by u i (t) = k A j e j p (xi a j ) 2 +(y i b j ) 2 (3) j=1 where (a j, b j ) are the coordinates, A j is the strength, and p j is the decay coefficient of the j-th source, respectively. The map of the process at initial time t = is shown in Fig. 1. The process then evolves over time as the sources move along random trajectories. In order to sample the continuous-time process, and to form the data matrix U, we first determine the appropriate t. Fig. 2 shows the normalized temporal autocorrelation function of the process. For a desired value of q, say q =.98, we note that t = 1 sec. The process is thus unchanged during t = 1 sec. The resulting data matrix U is 1 The function R(τ) may not be completely known a-priori, since it depends on the process being sensed; however, we can assume that it is approximately known from historical data. In Section V, we discuss the dependence of system performance on the choice of t. y x Fig. 1. The map of the process U(x, y, t) at time t =, where the location of the targets is shown by circles. As time progresses, the sources move randomly in the space. The readings of the sensors u i (m) are collected to form the data matrix U (Eq. (2)). normalized autocorrelation time lag τ [sec] Fig. 2. Normalized temporal autocorrelation function of the process, R(τ), averaged over all the sensors. The autocorrelation decreases to.98% of its maximum value at t = 1 sec. formed by joining M consecutive snapshots of the process at the chosen t intervals. The normalized singular values of U are plotted in Fig. 3, for several values of M. It is clear that U has only a small number of significant eigenvalues, i.e., it is low-rank. B. Communication Scheme Each sensor node generates measurement packets randomly at an average rate of λ 1 packets/sec and transmits its packets immediately on a random access channel. Without loss of generality, we assume that packet generation at each node follows an independent Poisson process. Each packet contains the quantized and binary encoded measurement information as well as the location tag for the generating sensor. Once a packet is received, the FC assigns a time stamp to it according to the packet s reception time

3 normalized singular value of the data matrix M=3 M= M= 7 1 index of singular value Fig. 3. Singular values of the data matrix U, normalized with respect to the maximum singular value, plotted for several values of M. We notice that the matrix has a few significant eigenvalues while the rest are negligible. In a random access network, packet collisions are inevitable. Thus, some packets are lost, while others may be received in error due to the communication noise. The FC discards the erroneous packets and collects the rest over an interval M t. It then arranges these packets into an N M matrix X, whose entries correspond to the successfully received packets. Since the sensors transmit at random moments in time, and because packet losses due to collisions and communication noise are random as well, X can be regarded as containing a uniformly random subset of the entries of U. Specifically, the entries of X are x i (m) = u i (m) if the i-th sensor transmitted during the m-th interval, and if the transmission was successful, otherwise x i (m) =. We leverage results from the matrix completion problem which asserts that under some conditions, an unknown lowrank matrix can be recovered from observing a sufficient number of its entries by solving a convex optimization problem, as long as the locations of the observed entries are picked uniformly at random. The number of entries required for recovery is on the order of O(rn(log n) 2 ) where n = max{n, M} [7]. Thus, once the FC collects a sufficient number of packets, it can recover the data matrix U using matrix completion algorithms. C. Recovery The sensing field is reconstructed from the useful packets collected during an observation interval of duration M t, by minimizing the nuclear norm of the matrix U, subject to the constraint u i (m) = x i (m) for the observed entries. This is a convex optimization problem, and a number of numerical algorithms can be used to solve it. Some examples are low-rank matrix fitting algorithm LMaFit [13], fixed point continuation with approximate singular value decomposition (FPCA) [14] and accelerated proximal gradient APGL [1]. In this paper, we use FPCA for recovery, which solves the nuclear norm regularized least squares problem. III. SUCCESSFUL PACKET RECEPTION Given a Poisson distribution of packet arrival times, the probability of n packets arriving concurrently at the FC is P (n) = (2Nλ 1T p ) n e 2Nλ1Tp (4) n! where T p is the packet duration. In a fading channel, successful packet reception is defined as the event in which the received signal-to-interference-and-noise ratio γ stays above a threshold b, where typically b = 2 6 [16]. In other words, the probability of successful reception is given by { } X p s = Prob > b () I n + P N where X represents the power of the desired packet, I n represents the total interference power (caused by the n interfering packets), and P N is the noise power. Assuming a Rayleigh model for the fading statistics, the probability of successful reception is given by [17] p s = e b/γ e 2Nλ 1T p b 1+b (6) where γ is the average received SNR. Within the interval t there may be more than one packet transmitted by a single node. If the FC has successfully received a packet from a given node in this interval, then the extra packet(s) are redundant since the process does not change in t. Hence, not all of the received packets at the FC are useful. Note that if a given packet is lost, a repetition may still be received successfully. For a particular node, let N 1 ( t) denote the number of packets generated during t, that are successfully received. If this number is greater than or equal to 1, the FC will keep one such successfully delivered packet and discard any other packets received from that node. Hence, the number of useful packets generated at each node during t is given by { N, N1 ( t) = ( t) = (7) 1, N 1 ( t) 1 The probability of receiving a useful packet from a node is p g = Prob{N ( t) = 1}, which can be expressed as p g = t Tp l=1 (λ 1 t) l e λ1 t [1 (1 p s ) l ] (8) l! where the first term is the probability that the node generates l packets during t, and the second term is the probability that one or more of the generated packets are received successfully. Note that the maximum number of packets that are generated by a single node during t is t T p 1. The expression (8) then reduces to p g = 1 e psλ1 t (9) From the expression (7), the effective average number of packets received from a given node during t is E{N ( t)} = p g. The average effective arrival rate of useful

4 packets at the FC is thus λ = Np g = N t t (1 e p sλ 1 t ) () The total number of packets that are used in the reconstruction process, K(λ, M t), is the number of received packets left after discarding the unsuccessful and repetitive packets. Thus, the arrival of useful packets follows a Poisson process with an effective average arrival rate λ given by Eq. (). i.e., the probability of receiving k useful packets in the interval t is P K (k) = (λ M t) k e λ M t. (11) k! IV. NETWORK DESIGN The target sampling density η s is defined as the ratio of the required number of observed entries to the total number of entries and is given by η s = Crn(log n) 2 /NM (12) where C is a constant. The actual sampling density, which is a random variable, is obtained by dividing the number of useful measurements collected in M t by the size of the data matrix, i.e., η = K(λ, M t)/nm. (13) A well-designed system thus must satisfy η η s (14) However, since η is random, a probabilistic approach to system design has to be used. We thus define the probability of sufficient sensing as the probability that the target sampling density η s is exceeded at the FC during the observation interval. We then specify the performance requirement as the minimum probability of sufficient sensing, P s, such that Prob {η η s } P s. (1) For large values of the Poisson parameter M tλ, the normal distribution closely approximates the Poisson distribution [18]. Thus, for a given η s and a desired P s, the corresponding λ s can be approximately determined from the condition ( ) MNη s M tλ s Q P s (16) M tλ s where Q( ) is the complementary cumulative distribution function of the standard normal distribution. The resulting λ s represents the minimum average rate of useful packets that meets the sufficient sensing requirement. In other words, λ λ s (17) The design objective is to determine the per-node sensing rate λ 1 that ensures sufficient sensing. The condition (17) implies that λ 1s λ 1 λ 1c (18) where λ 1s and λ 1c are the solutions to λ = λ s, as shown in λ [packet/sec] λ =λ s λ 1s λ 1m λ 1c λ [packet/sec] 1 Fig. 4. The average arrival rate of useful packets, λ, plotted versus the per-node sensing rate λ 1. The solutions to λ = λ s are shown in the figure. Fig. 4. We are only interested in those values of λ 1 for which the system is stable,i.e., for which increasing λ 1 results in an increased number of useful packets. The desired value of the per-node sensing rate thus lies in the stable region λ 1s λ 1 λ 1m (19) where λ 1m is the point at which λ reaches its maximum value. V. DESIGN CONSIDERATIONS We now study the trade-offs in choosing the system parameters. We define two performance metrics: a) the energy consumption; b) the recovery error. The energy consumed by the network to recover one second of the process is given by Γ = Nλ 1 P T T p () where P T is the average transmission power per packet per node. Clearly, a smaller sensing rate λ 1 translates into lower energy consumption. The recovery error is defined as ϵ = U Û F U F (21) where U is the data matrix given by Eq. (2), Û is the recovered data matrix at the FC, and F is the Frobenius norm. The system parameters that need to be determined are M and t. To illustrate the trade-offs, we use the test case of Fig. 1. Specifically, we intend to track this process as it evolves over time, with system parameters given by b = 2, P s =.9, γ = 2 db and T p = 2 msec. For the chosen range of values of M shown in Fig. 3, the rank r of the example data matrix can be approximated as fixed. Fig. shows that there is an optimum value of M, at which λ 1 is smallest, and hence, the energy consumption is the least. This is explained by noting that, assuming a fixed

5 λ 1s [packt/sec].7.6 recovery error M t [sec] Fig.. The per-node sensing rate, λ 1s, plotted versus M, for N = 2. The rank r is assumed to remain fixed at r =. Fig. 7. The recovery error plotted versus t, for the example field (M = N = 2, P s =.9, B = 38. kbps, b = 2, γ = 2 db)..7.6 M= N/2 M=N M=2N λ 1s [packet/sec] y t [sec] x Fig. 6. The per-node sensing rate, λ 1s, plotted versus t, for several values of M, when N = 2. rank r, η s achieves its minimum at M = N. Thus, from an energy point of view, an appropriate choice for the number of columns of the data matrix is M N. For a fixed M and N, Fig. 6 shows that λ 1s decreases with t. Thus, in order to minimize the energy consumption, t has to be chosen as the largest value possible. However, the choice of t also affects the recovery error. The effect of t on the quality of reconstruction depends on the correlation properties of the sensing field, i.e., the rate of change of the field. For the example field, Fig. 7 shows the recovery error plotted versus t. The choice of t thus involves a trade-off between the energy consumption and the desired accuracy of recovery. To visually illustrate the quality of the recovered field, Figs. 8 and 9 show the recovered snapshots of the field, with t = 1 sec and t = 2 sec, where the corresponding recovery errors are 3% and 9%, respectively. In general, both r and t are characteristics of the process that is being sensed, and as Fig. 8. The recovered map with t = 1 sec. We have employed a sensing rate λ 1s =.7 packet/sec, resulting in a sampling density η =.7. The data matrix U is then recovered with a recovery error on the order of 3%. Clearly, increasing the sampling density will reduce the recovery error further. such may not be exactly known and furthermore, may vary with time. The recovery error ϵ is thus influenced by the inaccuracies in estimating r (or equivalently, η s ) and t. VI. CONCLUSIONS Field recovery from incomplete data is a ubiquitous problem in monitoring of environmental and industrial phenomena. Using the matrix completion theory, we proposed a method for data acquisition in a wireless sensor network, deployed for long-term monitoring of large fields. By integrating sensing and communication, an efficient strategy is proposed using which the space-time map of the process is completely recovered from only a small set of measurement packets received over a fading and noisy random access channel. Network design depends on certain properties of the sensing field, such as the sampling interval and the rank of the data matrix, which may not be known a-priori and hence must be

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