On the trade-off between energy efficiency and estimation error in compressive sensing☆
Introduction
Compressive sensing (CS) (a.k.a. compressed sensing) refers to the process of reconstructing a signal that is supposed to be sparse or compressible [2]. It has found wide applications in communications and networking. For example, in cognitive radio networks, spectrum sensing is a critical component in dynamic spectrum access and enforcement of spectrum usage and sharing [3], [4], [5], [6], [7]. Given the wide range of activities in space, time, and frequency, it would be extremely challenging and costly to have a full range and dense sampling. In such situations, compressive sensing becomes a powerful tool for efficient spectrum sensing. Generally in a wireless sensor network (WSN), spatially distributed sensors are used to monitor physical or environmental conditions [8], [9]. The sparse signals in wireless sensor networks are obtained by collecting readings from sensor nodes by a server (or, a data processing center) through wireless transmissions. The server will then use CS to process the sparse data to achieve certain design goals (e.g., recovering the missing data).
CS has received considerable interest from the wireless community recently. For example, several CS papers have focused on network optimization and scheduling, where two important factors, i.e., network performance and power consumption, are considered in the design of CS schemes. The objectives of the schemes presented in these papers are to prolong the life time of wireless sensor nodes, while meeting certain network performance requirements. Energy can be conserved by turning off some sensor nodes while keeping the rest active. Network performance can be measured in terms of surveillance quality such as the number of active nodes [10], [11], network coverage [12], the minimum degree of connection [13], [14], and surveillance delay [15], [16]. The proposed schemes usually aim to achieve a trade-off between network performance and power consumption.
Other CS papers have laid emphasis on signal compression and reconstruction. These papers seek a trade-off between compression efficiency and reconstruction quality. For instance, the conventional CS approach is based on orthonormal basis. An important example of this approach is wavelet transform. The original sensor data is compressed and delivered [17]; therefore fewer network resource will be needed for transmitting the compressed data [18], [19], [20].
In this paper, we investigate effective CS schemes for addressing the fundamental trade-off between energy efficiency and estimation error. As discussed, a compressed signal can be obtained by collecting readings from a subset of the deployed sensor nodes, so that the rest of the nodes can be turned off to save energy. At the data processing center, the intact signal is reconstructed via signal estimation. Therefore, there is a fundamental trade-off between how many active nodes to choose (thus how much energy to spend) and the corresponding accuracy of prediction. Recently, only a few papers have investigated the problem of joint network optimization and signal processing. In [21], sensor nodes are divided into subgroups and all the readings can be recovered from one of the subgroups using isotonic regression. The objective is to maximize the number of subgroups while keeping the estimation error below a tolerable threshold. In [22], nodes are grouped into pairs. In each pair, the node with lower battery capacity goes to sleep, while the node with higher battery capacity estimates the measurement of the sleeping node using linear regression. In [23], [24], the authors investigate the problem of reconstructing a distributed signal through the collection of a small number of sensor readings, where CS is applied in conjunction with principle component analysis. The data of the entire network is recovered with Bayesian estimation.
In particular, we aim to develop effective CS schemes in WSNs and seek to balance the trade-off between energy saving and recovery accuracy, which are in terms of the number of active sensor nodes and the corresponding estimation error. We examine three classes of CS approaches in related work discussed above. We first implement the Bayesian estimation approach presented in [23], and propose an enhancement to this approach by scaling the variance of the sensor readings. We next revisit the isotonic regression approach presented in [21], and propose an improved method based on nearly isotonic regression. Finally, we briefly introduce two polynomial regression approaches, i.e., linear regression and quadratic regression, which are used as benchmarks in the performance evaluation. Bayesian estimation selects active nodes randomly, while isotonic regression chooses active nodes based on estimation errors, which results in better reconstruction quality. Unlike LR and QR, isotonic regression considers the distribution of readings for more accurate estimation. The proposed enhancements are evaluated with trace-driven simulations. We find considerable gaps between the original approaches and the proposed enhancements in the simulation results. We also find that the near isotonic regression method achieves the best performance among all the CS schemes examined in this paper.
The remainder of this paper is organized as follows. In Section 2, we describe the system model and assumptions. We introduce Masiero’s method and its enhancement in Section 3. In Section 4.1, we discuss Koushanfar’s method and the nearly isotonic regression based enhancement. In Section 5, we present two polynomial regression schemes. We evaluate the proposed enhancements with trace-driven simulations in Section 6. Section 7 concludes this paper.
Section snippets
System model and assumptions
The network model considered in this paper consists of autonomous sensor nodes that are deployed in an area to monitor physical conditions (e.g., spectrum availability) and a data processing center (or, a server), as illustrated in Fig. 1. We assume that the sensor nodes collect information data following a synchronized slot structure. Once a node obtains its data in a time slot, it transmits the data to the data processing center through its wireless interface in the same time slot.
We assume
Bayesian estimation approaches
In Bayesian Estimation (BE), the parameter set θ is obtained by maximizing the posterior probability density function (pdf)where is the data set and the plausible model. Since does not include θ, maximizing the posterior probability is equivalent to maximizing the product of and .
In [24], Masiero et al. propose an approach in which Principal Component Analysis (PCA) and CS are jointly considered. In the following, we first
Isotonic regression approaches
With Bayesian Estimation, we are able to manage L, the number of active nodes, and the percentage of power saved is computed as (N − L)/N. However, there is no consideration of the distance between the true measurements xk and the recovered measurements . In many applications, the quality of reconstruction is no less important than energy saving. In this section, we introduce Isotonic Regression (IR) that is able to address this problem.
Isotonic regression aims to find a weighted least-square
Polynomial regression approaches
If we treat the readings from sX as predictor variables and those from sY as response variables, it is natural to use polynomial regression to estimate the readings of sY from those of sX. In this section, we examine how to use Linear Regression (LR) and Quadratic Regression (QR) in CS.
Performance evaluation
In this section, we evaluate the performance of all the methods with trace-driven simulations. The simulation code was written in MATLAB. We use the traces from the sensors deployed in an outdoor environment [28]. There are N = 16 sensors that measure the illumination in a field. The range of the measurements is from 0 to 255 and they are collected from time k = 1 to 150. The scatter plot of the measured data is presented in Fig. 2.
We define Mean Square Error (MSE) as:
Conclusion
In this paper, we investigated three classes of CS approaches with focus on balancing the trade-off between energy efficiency and estimation error in a WSN environment. We first introduced Bayesian Estimation based approaches and propose an enhanced scheme by scaling the variance of projected random variables. We then adopted NIR to improve the performance of CIR. LR and QR approaches are also examined for comparison purpose. We compared the three classes of CS approaches with trace-driven
Acknowledgments
This work is supported in part by the US National Science Foundation (NSF) under Grants CNS-0953513, and through the NSF Broadband Wireless Access and Applications (BWAC) Center site at Auburn University.
Donglin Hu received the M.S. degree from Tsinghua University, Beijing, China, in 2007 and the B.S. degree from Nanjing University of Posts and Telecommunications, Nanjing, China in 2004, respectively, all in electrical engineering. He received the M.S. degree in Probability and Statistics from Auburn University, Auburn, AL, in 2011, and the Ph.D. degree in Electrical and Computer Engineering from Auburn University in 2012. Currently he is a postdoctoral research fellow in the Department of
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Donglin Hu received the M.S. degree from Tsinghua University, Beijing, China, in 2007 and the B.S. degree from Nanjing University of Posts and Telecommunications, Nanjing, China in 2004, respectively, all in electrical engineering. He received the M.S. degree in Probability and Statistics from Auburn University, Auburn, AL, in 2011, and the Ph.D. degree in Electrical and Computer Engineering from Auburn University in 2012. Currently he is a postdoctoral research fellow in the Department of Electrical and Computer Engineering at Auburn University. His research interests include cognitive radio networks, femtocell networks, network modeling, cross-layer design, performance analysis, and algorithm optimization for wireless networks and multimedia communications.
Shiwen Mao received Ph.D. in electrical and computer engineering from Polytechnic University, Brooklyn, NY, USA (now Polytechnic Institute of New York University) in 2004. He was a research staff member with IBM China Research Lab from 1997 to 1998. He was a Postdoctoral Research Fellow/Research Scientist in the Bradley Department of Electrical and Computer Engineering at Virginia Polytechnic Institute and State University (Virginia Tech), Blacksburg, VA, USA from 2003 to 2006. Currently, he is the McWane Associate Professor in the Department of Electrical and Computer Engineering, Auburn University, Auburn, AL, USA. His research interests include cross-layer optimization of wireless networks and multimedia communications, with current focus on cognitive radios, femtocell networks, 60 GHz mmWave networks, free space optical networks, and smart grid. He is on the Editorial Board of IEEE Transactions on Wireless Communications, IEEE Communications Surveys and Tutorials, Elsevier Ad Hoc Networks Journal, Wiley International Journal of Communication Systems, and ICST Transactions on Mobile Communications and Applications. He is the Director of E-Letter of the Multimedia Communications Technical Committee (MMTC), IEEE Communications Society for 2012–2014. He is a coauthor of TCP/IP Essentials: A Lab-Based Approach (Cambridge University Press, 2004). He was awarded the McWane Endowed Professorship in the Samuel Ginn College of Engineering for the Department of Electrical and Computer Engineering, Auburn University in August 2012. He received the US National Science Foundation Faculty Early Career Development Award (CAREER) in 2010. He is a co-recipient of The 2004 IEEE Communications Society Leonard G. Abraham Prize in the Field of Communications Systems and The Best Paper Runner-up Award at The Fifth International Conference on Heterogeneous Networking for Quality, Reliability, Security and Robustness (QShine) in 2008. He was named 2012 Exemplary Editor of IEEE Communications Surveys and Tutorials. He also received Auburn Alumni Council Research Awards for Excellence–Junior Award in 2011 and two Auburn Author Awards in 2011. Dr. Mao holds one US patent.
Nedret Billor received her Ph.D. from University of Sheffield, UK. She is a professor in the Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA. Her research interests include Outlier detection in regression and multivariate data, Robust multivariate and functional data analysis. She has authored numerous publications in journals which have garnered over many citations. She is a fellow of the Royal Statistical Society, elected member of the International Statistical Institute.
Prathima Agrawal is the Samuel Ginn distinguished professor of electrical and computer engineering and the director of the Wireless Engineering Research and Education Center, Auburn University, Auburn, Alabama. Prior to her present positions, she worked in Telcordia Technologies (formerly Bellcore), Morristown, New Jersey, and at AT&T/Lucent Bell Laboratories, Murray Hill, New Jersey. She created and served as head of the Networked Computing Research Department in Murray Hill. She is widely published and holds 51 US patents. She is a Fellow of IEEE. She received the BE and ME degrees in electrical communication engineering from the Indian Institute of Science, Bangalore, India, and the PhD degree in electrical engineering from the University of Southern California in 1977.