Decentralized Hypothesis Testingin Sensor Networks
ALLA TARIGHATI
Doctoral Thesis in Electrical EngineeringStockholm, Sweden 2016
TRITAEE 2016:177ISSN 16535146ISBN 9789177291855
Signal Processing DepartmentKTH, School of Electrical Engineering
SE100 44 StockholmSWEDEN
Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlaggestill offentlig granskning for avlaggande av teknologie doktorsexamen i elektro ochsystemteknik onsdagen den 30 november 2016 klockan 10.00 i sal F3, Lindstedtsvagen 26, Stockholm.
2016 Alla Tarighati, unless otherwise stated.
Tryck: Universitetsservice US AB
To my beloved family!
Abstract
Wireless sensor networks (WSNs) play an important role in the future ofInternet of Things IoT systems, in which an entire physical infrastructurewill be coupled with communication and information technologies. Smartgrids, smart homes, and intelligent transportation systems are examples ofinfrastructure that will be connected with sensors for intelligent monitoring and management. Thus, sensing, information gathering, and efficientprocessing at the sensors are essential.
An important problem in wireless sensor networks is that of decentralized detection. In a decentralized detection network, spatially separatedsensors make observations on the same phenomenon and send informationabout the state of the phenomenon towards a central processor. The centralprocessor (or the fusion center, FC) makes a decision about the state of thephenomenon, base on the aggregate received messages from the sensors. Inthe context of decentralized detection, the object is often to make the bestdecision at the FC. Since this decision is made based on the received messages from the sensors, it is of interest to optimally design decision rules atthe remote sensors.
This dissertation deals mainly with the problem of designing decisionrules at the remote sensors and at the FC, while the network is subjectto some limitation on the communication between nodes (sensors and theFC). The contributions of this dissertation can be divided into three (overlapping) parts. First, we consider the case where the network is subjectto communication rate constraint on the links connecting different nodes.Concretely, we propose an algorithm for the design of decision rules at thesensors and the FC in an arbitrary network in a personbyperson (PBP)methodology. We first introduce a network of two sensors, labeled as therestricted model. We then prove that the design of sensors decision rules,in the PBP methodology, is in an arbitrary network equivalent to designingthe sensors decision rules in the corresponding restricted model. We alsopropose an efficient algorithm for the design of the sensors decision rules inthe restricted model.
Second, we consider the case where remote sensors share a commonmultiple access channel (MAC) to send their messages towards the FC, andwhere the MAC channel is subject to a sum rate constraint. In this situation,
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the sensors compete for communication rate to send their messages. Wefind sufficient conditions under which allocating equal rate to the sensors,so called rate balancing, is an optimal strategy. We study the structure ofthe optimal rate allocation in terms of the Chernoff information and theBhattacharyya distance.
Third, we consider a decentralized detection network where not onlyare the links between nodes subject to some communication constraints,but the sensors are also subject to some energy constraints. In particular,we study the network under the assumption that the sensors are energyharvesting devices that acquire all the energy they need to transmit theirmessages from their surrounding environment. We formulate a decentralizeddetection problem with system costs due to the random behavior of theenergy available at the sensors in terms of the Bhattacharyya distance.
Keywords: Wireless sensor networks, decentralized detection, personbyperson optimization, multiple access channels, rate allocation, Chernoffinformation, Bhttacharyya distance, energy harvesting.
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Sammanfattning
Tradlosa sensornatverk spelar en viktig roll i framtiden av IoT (eng. Internet of Things) system, dar en hel fysisk infrastruktur kommer att koppla med kommunikations och informationsteknik. Smarta nat, smartahem och intelligenta transportsystem ar exempel pa infrastruktur som skaanslutas med sensorer for intelligent overvakning och hantering. Saledes aravkanning, informationsinsamling och effektiv informationsbearbetning avsensordata viktigt.
Ett centralt problem i tradlosa sensornatverk ar decentraliserad detektion. I ett decentraliserad detektionsnatverk, gor spatialt separeradesensorer observationer av samma fenomen och skickar information omtillstandet for fenomenet mot en central processor. Den centrala processorn (eller fusionscentrum, FC), tar ett beslut om tillstandet hos fenomenet,med anvandning av de sammanlagda mottagna meddelandena fran sensorerna. Inom ramen for decentraliserad detektion ar syftet framst att goradet basta beslutet i FC. Eftersom detta beslut fattas i enlighet med de mottagna meddelanden fran sensorerna, ar det av intresse att optimalt utformabeslutsregler for fjarrsensorerna.
Denna avhandling behandlar framst med problemet att utforma beslutsregler pa fjarrsensorer och FC, medan natverket ar foremal for nagonbegransning av kommunikationen mellan noderna (sensorerna och FC). Detvetenskapliga bidraget i denna avhandling kan delas in i tre (overlappande)delar. Forst, ar fallet dar natverket ar foremal for kommunikationshastighetbegransningar pa lankarna som forbinder olika noder. Konkret foreslar vi enalgoritm for utformningen av beslutsregler vid sensorerna och FC i ett godtyckligt natverk i en personforperson (PBP) metodik. Vi introducerar ettforenklat natverk av tva sensorer, en sa kallat restricted model. Vi bevisaratt utformningen av sensorernas beslutsregler, under en PBP metodik, i ettgodtyckligt nat motsvarar utforma sensorernas beslutsregler i motsvaranderestricted model. Sedan sa foreslar vi en effektiv algoritm for konstruktionav sensorernas beslutsregler i det forenklade natverket.
Sedan betraktar vi scenariot dar fjarrsensorer har en gemensam multiple access channel (MAC) for att skicka sina meddelanden till FC, ochdar MACkanalen ar foremal for en begransning av summan av de individuella bithastigheterna. I den situationen, konkurrerar sensorerna om
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kommunikationshastigheten for att skicka sina meddelanden. Vi definierartillrackliga villkor under vilka det ar en optimal strategi att fordela likahastighet till sensorerna. Vi studerar strukturen for en optimal fordelning avhastigheterna i termer av Chernoffinformation och Bhattacharyyaavstand.
Sists sa betraktar vi ett decentraliserat detektionsnatverk dar inte baralankarna mellan noderna ar foremal for kommunikationsbegransningar, utandar aven sensorerna ar ocksa foremal energibegransningar. Mer precist sastuderar vi natverket under antagandet att sensorerna ar energiinsamblandeoch forvarvar alla den energi de behover for att sanda sina meddelanden fransin omgivning. Vi formulerar det decentraliserade detektionsproblemet medsystemkostnader som modellerar det slumpmassiga beteendet av energin ide individuella sensorerna i termer av Bhattacharyyaavstandet.
Nyckelord: Tradlosa sensornatverk, decentraliserad detektion, personforperson optimering, flera atkomstkanaler, fordelningshastighet, Chernoffinformation, Bhattacharyyaavstand, energiinsamblande.
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List of Papers
The thesis is based on the following papers:
[A] A. Tarighati, and J. Jalden, Bayesian design of decentralized hypothesis testing under communication constraints,in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Process. (ICASSP), May 2014, pp. 76247628.
[B] A. Tarighati, and J. Jalden, Bayesian design of tandem networks for distributed detection with multibit sensor decisions,IEEE Trans. Signal Process., vol. 63, no. 7, pp. 18211831, April2015.
[C] A. Tarighati, and J. Jalden, A general method for the design oftree networks under communication constraints, in Proc. 17thInt. Conf. Information Fusion (FUSION), July 2014, pp. 17.
[D] A. Tarighati, and J. Jalden, Rate allocation for decentralized detection in wireless sensor networks, in Proc. 16th IEEEInt. Workshop Signal Process. Advances in Wireless Commun. (SPAWC), June 2015, pp. 341345.
[E] A. Tarighati, and J. Jalden, Optimality of rate balancing inwireless sensor networks, IEEE Trans. Signal Process., vol. 64,no. 14, pp. 37353749, July 2016.
[F] A. Tarighati, J. Gross, and J. Jalden, Decentralized detection inenergy harvesting wireless sensor networks, in Proc. EuropeanSignal Process. Conf. (EUSIPCO), Aug. 2016.
[G] A. Tarighati, J. Gross, and J. Jalden, Decentralized hypothesis testing in energy harvesting wireless sensor networks,IEEE Trans. Signal Process., to be submitted, available:arXiv:1605.03316.
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Acknowledgements
This dissertation could not have been finished without the help and support from many professors, colleagues, friends and my family. It is mygreat pleasure to acknowledge people who have given me guidance, helpand encouragement.
First and foremost, I would like to thank my main advisor Assoc. Prof. Joakim Jalden for the continuous support of my Ph.D study andrelated research, for his patience, motivation, and immense knowledge. Hisguidance helped me throughout my research and writing of this dissertation.Without his breadth of knowledge in the area of signal processing, finishingmy dissertation would have remained a distant dream.
I would like to thank my coadvisor Prof. Mats Bengtsson for giving methe opportunity to pursue my Ph.D studies at the department of SignalProcessing, KTH. I would like to thank you also for your essential help inLatex.
I would like to express my sincere appreciation to DeWiNe researchmembers, specially Assoc. Prof. Tobias Oechtering who engaged in valuablediscussion with me during the first years of my research on distributed detection, and also acted as the internal advanced reviewer of this dissertation.I would like to thank my coauthor Assoc. Prof. James Gross for helping meto transform one of my research interests into an article, and sharing thosejoyful moments watching football games together.
I am always indebted to my undergraduate advisor Assoc. Prof. Kamran Kazemi. His sincere dedication to his students was both admirable andinspirational. My interest in research began when I started working withhim. I feel honored to have had a chance to work with him.
It has been my pleasure to work with colleagues in the departments ofsignal processing and communications theory, KTH. I would like to thankMajid Nasiri, Majid Gerami, M. Reza Gholami, Hamed Farhadi, Amirpasha Shirazinia, Nafiseh Shariati, Shahab Nazari, Farshad Naghibi, ServehShalmashi, Ehsan Olfat, Hossein Shokri, Arash Owrang, and Nima NajariMoghadam for making my lunch/fika breaks such an amazing times. I wouldlike to thank Marie Maros, Arun Venkitaraman, Klas Magnusson, RasmusBrandt, Pol Del Aguila Pla, Ehsan Olfat, Arash Owrang, Martin Sundin,and Nima Najari Moghadam (monthlyfikagroupmembers) for having great
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times together playing and laughing.I also would like to acknowledge Rasmus Brandt, Klas Magnusson, Ma
trin Sundin, and Tove Schwartz for their patience in helping me with mySwedish language. I would like to thank Tove Schwartz for taking careof all the administrative works. Thank you for brightening our mood byarranging fikas, museum tours, going iceskating, etc.
It was a privilege to share my office with Nima Najari Moghadam. Iwould like to thank him for being a great office mate during these years. Ialso thank my friends (too many to list here but you know who you are!)for providing support and friendship that I needed.
I wish to acknowledge those who have helped me with proofreading thisdissertation: Majid Gerami, Nima Najari Moghadam, and Farshad Naghibi.
I would like to express my gratitude to Prof. Pramod K. Varshney for thetime and effort spent on acting as opponent and to the grading committee:Dr. Sorour Falahati, Prof. Subhrakanti Dey, and Prof. Pierluigi Salvo Rossi.
A special word of thanks also goes to my family for their continuoussupport and encouragement. For my parents Zinat and Mohammad, whoraised me with love and supported me in all my pursuits, and gave mefreedom in my life. I would like to thank my brother Ali and my sistersFisa and Parisa, for their neverending supports and encouragement. Iknow I always have you to count on when times are rough. I would liketo express my gratitude to my grandmother Mamaniran. As my first everteacher, Mamaniran has played an important role in the development ofmy identity and shaping the individual that I am today.
Finally, I am grateful for the love, encouragement, and tolerance of mylovely wife Tahereh, who has made all the difference in my life. Withoutyour patience and sacrifice, I could not have completed this dissertation.Words are not enough to express all my gratitude to you.
Alla TarighatiStockholm, November 2016
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Contents
Abstract i
Sammanfattning iii
List of Papers v
Acknowledgements vii
Contents ix
Acronyms xiii
I Summary 11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Decentralized detection in wireless sensor networks . 11.2 Decentralized hypothesis testing problems . . . . . . 21.3 Stateoftheart problems in distributed detection . . 51.4 Performance measures . . . . . . . . . . . . . . . . . 101.5 Energy constrained wireless sensor networks . . . . . 16
2 Our contributions . . . . . . . . . . . . . . . . . . . . . . . . 182.1 Problem of designing sensors decision functions . . . 182.2 Problem of rate allocation in wireless sensor networks 282.3 Problem of decentralized detection in energy harvest
ing sensor networks . . . . . . . . . . . . . . . . . . . 353 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
II Included papers 51
A Bayesian Design of Decentralized Hypothesis Testing Under Communication Constraints A1
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1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . A3
3 Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . A5
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . A7
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . A10
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A11
B Bayesian Design of Tandem Networks for Distributed Detection With Multibit Sensor Decisions B1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . B1
2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . B5
3 DM Design Through a Restricted Model . . . . . . . . . . . B7
3.1 Formation of the Restricted Model . . . . . . . . . . B7
3.2 Design of DMs in the Restricted Model . . . . . . . B11
3.3 Complexity of the proposed method . . . . . . . . . B15
4 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . B16
4.1 Binary hypothesis testing . . . . . . . . . . . . . . . B18
4.2 M ary hypothesis testing . . . . . . . . . . . . . . . B22
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . B23
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B25
C A General Method for the Design of Tree Networks UnderCommunication Constraints C1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . C1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . C3
3 Restricted Model . . . . . . . . . . . . . . . . . . . . . . . . C6
4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . C11
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . C14
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C15
D Rate Allocation for Decentralized Detection in WirelessSensor Networks D1
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . D1
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . D3
3 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . D5
4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . D7
4.1 Benitz and Bucklews Method . . . . . . . . . . . . . D8
4.2 Numerical Method . . . . . . . . . . . . . . . . . . . D9
4.3 The Error Probability Performance of Sensor NetworksD11
5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . D11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D12
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E Optimality of Rate Balancing in Wireless Sensor Networks E11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . E12 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . E33 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . E114 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . E17
4.1 Laplacian Observations . . . . . . . . . . . . . . . . E174.2 Gaussian Observations . . . . . . . . . . . . . . . . . E18
5 Simulation Results and Discussions . . . . . . . . . . . . . . E216 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . E277 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . E29
7.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . E297.2 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . E307.3 Proof of Lemma 4 . . . . . . . . . . . . . . . . . . . E31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E35
F Distributed Detection in Energy Harvesting Wireless Sensor Networks F11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . F12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . F33 Analyzing an Energy Harvesting Sensor . . . . . . . . . . . F64 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . F95 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . F11References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F12
G Decentralized Hypothesis Testing in Energy HarvestingWireless Sensor Networks G11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . G12 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . G3
2.1 System Model . . . . . . . . . . . . . . . . . . . . . . G32.2 Energy Harvesting Sensors . . . . . . . . . . . . . . G7
3 Design of Energy Harvesting Sensors . . . . . . . . . . . . . G104 Error Probability Performance of Networks . . . . . . . . . G185 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . G216 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . G21
6.1 Proof of (20) . . . . . . . . . . . . . . . . . . . . . . G216.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . G22
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G24
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Acronyms
WSN Wireless sensor network
IoT Internet of Things
i.i.d. Independent and identically distributed
MAP Maximum aposteriori
SNR Signaltonoise ratio
PBP Personbyperson
FC Fusion center
DM Decision maker
LRQ Likelihoodratio quantizer
ROC Receiver operating characteristic
OOK Onoff keying
MAC Multiple access channel
BAC Binary asymmetric channel
LLR Loglikelihood ratio
LRT Likelihoodratio test
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Part I
Summary
1 Introduction
1.1 Decentralized detection in wireless sensor networks
Wireless sensor networks have gained worldwide attention over the pastdecade [13]. Small sensors with limited processing, computing resourcesand communication capabilities are used in wireless sensor networks. Thesesensors are able to sense, measure, and gather information from their surrounding environment. The sensors are also able, based on local decisionprocesses, to transmit the sensed data [46].
Statistical inferences about the environment such as detection, parameter estimation and tracking, are some of the emerging applications of wireless sensor networks. In all these applications we are faced with a decisionmaking problem where information is distributed across the sensors in thenetwork. In all these group decisionmaking problems we need to select aparticular course of action based on our observations regarding a certainphenomenon. If the course of action is from a set of possible options, ourproblem is a decentralized detection problem [711].
In a decentralized detection system, spatially separated sensors observea common phenomenon. If the sensors are able to communicate all theirdata to a central processor for data processing, the sensors only act as datacollectors and no processing at the sensors is needed. In this situation,the data processing is centralized in nature and optimal algorithms can beimplemented. On the other hand, if there are communication constraintson the sensors, some preliminary data processing at each sensor needs to becarried out and a compressed or quantized version of the received data isthen given as the sensor output. According to the network arrangement, theoutput of each sensor is sent either to another sensor or to a fusion center(FC) where the received information is appropriately combined to yield thefinal inference [1214].
Decentralized detection problems are found in many practical situations.As an example, consider a radar detection system, where spatially separatedradars are observing an area. According to their observations, they sendsome data towards a central processor or FC. Based on the received datafrom the radars, the FC makes a decision regarding the presence or absenceof a target in that area. As another example, consider a digital communication system in which one of several possible waveforms is transmittedand the goal is to determine the transmitted symbol according to the noisyobservations at the sensors.
In the context of decentralized hypothesis testing, each sensor is anintelligent unit, and is therefore often referred to as a decision maker (DM).In what follows, we shall discuss decentralized hypothesis testing problemsin different topologies.
S1 Sn SN
Phenomenon H H
Fusion Center
x1 xn xN
u1 un uN
H
Figure 1: Decentralized hypothesis testing scheme in a parallelnetwork.
1.2 Decentralized hypothesis testing problems
In a decentralized detection problem, we may be faced with a binary problemin which we are looking for a yes or no type of answer. For example, in theradar detection system described previously (see Section 1.1), our goal isto determine if the target is present (yes) or not (no). These problems areknown as binary hypothesis testing problems and the two hypotheses areusually denoted by H0 and H1. We may also be faced with a more generalsituation with multiple hypotheses. An example of this problem is thedigital communication system also described previously (see Section 1.1),where the set of possible waveforms consists of M > 2 different symbols.In the M ary hypothesis testing problems, we denote the hypotheses by1
H0, H1, . . . , HM1.
The design of decentralized hypothesis testing algorithms depends on theunderlying sensor network topology (or configuration), i.e., the arrangementof the sensors in a network [4]. The most common topology is the paralleltopology, which is extensively considered in the literature (see [9,1520] andreferences therein) and is shown in Figure 1. In this configuration, a sensorSn, n = 1, . . . , N , makes a private observation, denoted by xn, of the common phenomenon H and sends a message un towards the FC. The FC aftercombining all the messages u1, . . . , uN received from the sensors S1, . . . , SN
1The definition of a hypothesis set for M ary problems for notational simplicity insome publications is as {H1, . . . ,HM}.
1. INTRODUCTION 3
S1 Sn SN
Phenomenon H H
x1 xn xN
MAC
u1 un uN
Fusion Center
H
Figure 2: Decentralized hypothesis testing scheme in a network,where the sensors send their data through a MAC channel to theFC.
makes a global decision H in favor of a hypothesis from hypothesis set H.In this configuration there is no communication between the sensors andeach sensortoFC channel is a oneway channel from the sensor.
Although a large body of research in decentralized detection has beendevoted to the case where the sensors transmit their messages towards theFC through parallel access channels, in wireless sensor networks the wirelessmedium is typically shared among the sensors. In other words, the sensortoFC channels are modeled as common multiple access channels (MAC)[2126]. This configuration is shown in Figure 2.
In both configurations shown in Figures 1 and 2, the sensors make private observations on the underlying phenomenon and transmit their output,known as local decisions, directly to the FC through wireless channels. Inanother popular structure, known as the tandem (or serial) topology, sensors are connected in series [2734]. In this configuration, each sensor isconnected to two neighboring sensors: its predecessor and its successor.Two exceptions to this rule are the first sensor, which does not have anypredecessor, and the last sensor which does not have any successor. Anetwork of sensors arranged in tandem is also shown in Figure 3. In thisconfiguration, each sensor Sn, n = 2, . . . , N 1 makes an observation xn
4 Summary
S1 Sn SN
Phenomenon H H
u1 un1 un uN1 uN
x1 xn xN
Figure 3: Decentralized hypothesis testing scheme in a tandemnetwork.
and receives the output un1 of its predecessor as its inputs, and makes adecision un to send to its successor Sn+1. Sensor S1 makes a decision u1only based on its private observation x1, and the decision of the last sensor,SN , is the final decision of the network.
Distributed detection in the tandem topology is closely connected todecision making in social networks [3540]. In social networks, each agentchooses an action/decision by optimizing its local utility function and subsequent agents then choose their actions/decisions using their private observations together with the actions/decisions of previous agents [38, 41].In social networks, the agents typically do not reveal their raw private observations on the underlying state of nature, while they do reveal theiractions/decisions (e.g., votes and recommendations). This can be viewed asa low resolution (or in the context of decentralized detection as quantized)version of their observations and their interaction with other agents in thenetwork.
In contrast to decentralized detection networks where the agents decisions are made in such a way that a global performance measure is optimized, in social networks the agents decisions are made in such a waythat their local performance measure is optimized. In other words, in socialnetworks agents selfishly try to optimize their outputs [42].
In addition to the extreme cases of parallel and tandem networks, thereare other configurations such as tree topologies which are combinations ofthese cases [4348]. An example of such a configuration is shown in Figure 4.In this situation, some of the nodes make observations while other nodesact as relays which combine the input messages and forward their outputto another node. All the observed data eventually arrive at the FC, wherethe final decision about the present hypothesis is made.
In the following section, state of the art problems for each of the aforementioned configurations will be discussed. However, before discussing theproblems, let us define the parameters that will be repeatedly used through
1. INTRODUCTION 5
FC
H
Figure 4: Decentralized hypothesis testing scheme in a tree network. Each sensor is shown by a circle and the observations arenot shown.
out this thesis. Let Xn and Un, for some 1 n N , be random variablescorresponding to observation xn and output un of sensor Sn, respectively.The phenomenon H is also modeled as a random variable drawn froma set H , {H0, H1, . . . , HM1} with corresponding apriori probabilities0, 1, . . . , M1 for a general M ary hypothesis testing problem. We alsodefine fXH (x1, . . . , xN h), for h H, as the joint conditional distribution ofthe observations at sensors S1 to SN , where by definition X , X1, . . . , XN .When the observations at the sensors, conditioned on the hypothesis, areindependent the joint conditional distribution decouples as
fXH (x1, . . . , xN h) = fX1H (x1h) . . . fXN H (xN h) .
We also define Xn as the observation space of Xn, and Urn as the messagespace of output un.
1.3 Stateoftheart problems in distributed detection
For the parallel topology shown in Figure 1, two main problems need to beconsidered: the design of decision rules at the sensors, and the design ofdecision rules at the FC. To optimize a performance metric of the network,the decision rules at the FC and the local decision rules at the sensors shouldbe designed jointly. Assuming perfect knowledge of the system parameters,the optimum design of the FC rule is conceptually a straightforward task[4, 9]. However, even for smallsized networks, the design of decision rulesat the sensor nodes is a formidable task [49].
6 Summary
Given a realization xn, sensor Sn in a parallel topology makes a decisionun according to its own private observation xn using a decision functionn(), i.e.,
n(xn) = un , (1)
and the FC makes the final decision h in favor of a hypothesis from the setH by combining all the received messages from the sensors using a decisionfunction FC(), i.e.,
FC(u1, . . . , uN) = h . (2)
Note that, in this situation the sensor makes a decision only based on itsown observation and not the observations at the other sensors, since thereis no communication between the sensors. In what follows, we shall firstdiscuss the structure of optimal sensors decision rules and then formulatestateoftheart problems in distributed detection networks.
If there is no communication constraint on the channel from the sensorto the FC, the sensor gives its own observation as its output, i.e., un = xn.However, in practical situations the sensortoFC channel is subject to somecommunication constraints. Let us assume that this channel is an errorfreebut rateconstrained channel of rate rn, for a positive integer rn. In otherwords, this channel is capable of reliably carrying rn bits to the FC. In thissituation, each sensor Sn is a quantizer and its decision rule is a mappingfrom the observation space Xn to the message space Urn , {1, . . . , 2rn},i.e., n : Xn Urn . We are then led to the problem of finding quantizationstrategies at the sensors with respect to optimizing a performance measure.This problem was extensively considered in the quantization literature [5055] and decentralized detection literature [9,5658]. It is well known that forseveral specific observation distribution models, when observations at thesensors are independent given the hypothesis, likelihoodratio quantizers(LRQ) are optimal. Therefore the problem of finding optimal decision rulesat the sensors reduces to that of quantization thresholds [9, 15, 16, 5961].
In the simple case of binary hypothesis testing where the sensors arearranged in parallel and each sensor sends a single bit towards the FC, theoptimal decision rule at each sensor, as discussed above, is a likelihoodratiotest (LRT) as
fXnH (xnH1)fXnH (xnH0)
un=1
un=0
n, (3)
where n is a scalar constant. If the likelihoodratio is monotone in observation xn, (3) admits the following form
xnun=1
un=0
Tn, (4)
for a constant Tn which depends on n and an observation model at thesensor. In other words, in this case, without loss of generality we can assume
1. INTRODUCTION 7
that a quantizer is directly applied to the observations, rather than to thelikelihoodratio (see [58, 62] for more detailed discussion). Furthermore, wecan without loss of generality assume that observations at the sensors arefrom the set of real numbers, i.e., Xn R; even for nonreal observationsthe likelihoodratios at the sensors are from the real set.
Returning now to the parallel configuration, let us consider the detection problem in which our aim is to design the decision rules of the sensornodes and the FC by optimizing a performance measure, subject to the rateconstraints of the sensortoFC channels. This problem can be formulatedas:
Optimize: some performance measure,
variables: 1, . . . , N , FC,
subject to: r1, . . . , rN .
(5)
Note that so far we have not discussed the choice of the performance metricand we postpone this issue to Section 1.4.
This problem was considered quite extensively in the literature, specially for the binary hypothesis test and when the sensortoFC channelsare onebit channels, and for conditionally independent observations at thesensors [9, 16]. Even under these simplified assumptions, the problem ofdesigning decision rules at the sensors and the FC is a demanding task andin most cases, finding globally optimal decision rules at the sensors is mathematically intractable. To overcome this difficulty, we can use a personbyperson (PBP) optimization method for the design of decision rules atthe sensors and the FC. In the personbyperson optimization approach, weoptimize the decision rule of each sensor while assuming fixed decision rulesat all other sensors and the FC. This approach is only guaranteed to reach alocally optimal solution and not necessarily a globally optimal solution [4].
This specific problem was considered by Longo et al. in [17] for general correlated observations at the peripheral nodes. The peripheral nodes(scalar quantizers) were to be cooperatively designed according to a systemwide measure of performance. They argued that the natural criterion ofoptimizationin their case the power of a NeymanPearson testmade thedesign procedure intractable. They therefore proposed to instead optimizea measure of similarity between the conditional distributions of the jointindex space, i.e., the full set of quantizer outputs. Longo et al. obtainedtheir final result using a cyclic personbyperson optimization algorithm formaximizing the Bhattacharyya distance [63] between the conditional probability distributions under the two hypotheses. We shall discuss this choiceof performance measure in detail in Section 1.4.
The design problem is even more challenging when the sensors share aMAC channel to send their outputs towards the FC, as shown in Figure 2.In this situation the detection problem involves the design of decision rules
8 Summary
at the sensor nodes and the FC and, at the same time, allocation of ratesto the sensors in such a way that some communication constraints imposedby the MAC channel are satisfied. The MAC channel model, as in [21], canbe considered as a set of errorfree channels that are subject to a sum rateconstraint. We can mathematically formulate this problem as:
Optimize: some performance measure,
variables: 1, . . . , N , FC, r1, . . . , rN
subject to:
N
n=1
rn R ,(6)
for a positive integer R, where R is sum rate capacity of the MAC channel.For the case of a binary hypothesis test where the observations at the
N remote sensors are independent and identically distributed (i.i.d.) conditioned on the true hypothesis H , Chamberland and Veeravalli [21] studiedthe structure of an optimal sensor configuration. Their study was in termsof the optimal number of sensors (N) and the optimal sensors rate allocation. They found sufficient conditions under which having N = R one bit(binary) sensors is optimal. Although the condition of having only rate onesensors leads to a very simple network design, it is in many cases simplynot practically feasible to have an arbitrary number of sensors. The maximum number of sensors deployed in practice is often limited by hardware orspatial constraints. Therefore in [62,64] we have considered a more generalcase which extends the results in [21]. Concretely, we have considered thecase where N is fixed or limited apriori, and our aim is to optimally selectthe set of rates in the network. We shall discuss these results in Section 2.2.
The optimal design of decision rules for the sensors arranged in tandem,according to Figure 3, has also generated a significant amount of interest inthe past [27,29,33,34,65,66]. Sensor Sn, n = 2, . . . , N in a tandem networkmakes a decision according to its own observation xn and the decision un1of its predecessor Sn1 using a decision function n : Xn Urn1 Urn ,i.e.,
n(xn, un1) = un , (7)
where sensor SN serves as the FC and therefore for its output message setwe have UrN = H. Sensor S1 only uses its direct observation x1 to make adecision u1 using a decision function 1 : X1 Ur1 , i.e.,
1(x1) = u1 . (8)
As in the parallel topology, if there is no communication constraint on thechannels between the sensors, each sensor only gives its inputs as outputand the problem reduces to a centralized problem and the optimal fusiondecision N can be found accordingly. Let us consider the case where the
1. INTRODUCTION 9
channel between sensors Sn and Sn+1 is an errorfree but rateconstrainedchannel of rate rn. Without loss of generality, we can assume that the outputmessage un, n = 1, . . . , N 1 is from the set Urn = {1, 2, . . . , 2rn}, while theoutput message of SN (the FC) is from the set H = {H0, . . . ,HM1} for anM ary hypothesis testing problem. In this situation the detection problemis to design the decision rules of the sensor nodes and the FC by optimizinga performance measure, subject to the rate constraints of sensortosensorchannels, or equivalently:
Optimize: some performance measure,
variables: 1, . . . , N ,
subject to: r1, . . . , rN1 .
(9)
The optimal design of the sensors in a tandem network was previouslystudied in [27,29,34] under the assumption that the observations at the sensors were, conditioned on the hypothesis, independent. This scenario hasalso been generalized in [33] to the case of conditionally dependent observations. While in [27,29,34] a binary hypothesis testing and binary messagesbetween the sensors were considered, [33] relaxed these assumptions andconsidered general M ary hypothesis testing, however with only M valuedmessages (i.e., Urn = M) for M 2 and n = 1, . . . , N .
Considering the optimal performance limits of tandem networks, it wasshown in [16,28] that for distributed networks with two sensors the optimalparallel network is outperformed by the optimal tandem network. However,for any network of more than two sensors, parallel networks perform betterthan serial networks, and for any given distributed detection problem withi.i.d. observations there exist a number of sensors at which the parallelnetwork becomes better [16]. Moreover, the error probability in the caseof a parallel topology with any logical decision functions, goes to zero veryquickly as the number of sensors increases. This does however not holdin general for the tandem topology. In other words, as was shown in [65],the rate of error probability decay of the tandem network is always subexponential in the total number of sensors, while the error probability decayof a parallel network is exponential in the total number of sensors [67].
There are also some significant studies regarding the asymptotic performance of parallel and tandem networks, for instance [28, 65, 6769].In [28,68] the problem of M ary hypothesis testing in tandem networks wasconsidered when the sensors were allowed to send M valued messages. Theyhave shown that in this situation a necessary and sufficient condition for theprobability of error to asymptotically go to zero is that the loglikelihood ratio of the observation at each sensor, between any two arbitrary hypotheses,is unbounded in magnitude. We can then conclude that in the general casewith potentially bounded loglikelihood ratios, having strictly more messages than the number of hypotheses is needed to have zerolimiting error
10 Summary
probability. In the case of a binary hypothesis testing (i.e., M = 2) and forthe case of bounded loglikelihood ratios, Cover has proposed an algorithmin [68] with a fourvalued message (Urn = 4) which results in zerolimitingprobability of error under each hypothesis. Koplowitz has generalized thisidea in [69], where he has shown that, even if the loglikelihood ratios arebounded, (M +1)valued messages are sufficient for achieving zerolimitingprobability of error in M ary hypothesis testing.
Considering a tandem network of fixed size N , Papastavrou and Athans[28] proposed a simple but suboptimal scheme for the design of sensors. Intheir method, each sensor is optimized for locally minimal error probabilityat its output, instead of for globally optimal performance. For this particular scheme, they have also shown that a necessary and sufficient conditionto achieve zerolimiting probability of error is that the loglikelihood ratioof the observation of each sensor be unbounded from both above and below.While optimizing the performance (e.g. minimizing the error probability)locally at the output of each sensor makes the algorithm relatively simple,it has a side effect that the messages are then constrained to be M valued(i.e., Urn = M) for the M ary hypothesis testing problem. This is due tothe fact that a onetoone relation between the sensor output messages andthe hypotheses is needed in the definition of the local error probabilities.Therefore, even though it is known that increasing the number of communication messages can improve the performance of a parallel network [70], theproblem of designing the sensors in a tandem network for arbitraryvaluedmessages remains largely open [27]. AlIbrahim and AlHakeem [66] alsohave exemplified this point by allowing the first sensor in a tandem networkof two sensors to communicate twobit messages instead of onebit messages. They have observed a significant improvement in the performance ofthe network for binary hypothesis testing. One way to view this result isas follows: multibit (soft) decisions are able to transmit more informationto the FC for the final decision than a binary (hard) decision would. Thedifficulty is in figuring out how to best capture and quantize this additionalinformation and this is one problem that we shall address in Section 2.1 ofthis thesis.
In the following section we will consider different choices of performancemeasure and discuss their properties.
1.4 Performance measures
In the Bayesian formulation, a decision at the FC is made in favor of ahypothesis based on given prior information, namely apriori probability ofhypotheses, and hypothesisconditional probabilities. The optimal decision
1. INTRODUCTION 11
rule is the Bayes test that minimizes the Bayes risk which is equal to
R =M1
i=0
M1
j=0
CijjP(HiHj
), (10)
where Cij represents the cost of deciding on the presence of Hi while Hj is
true and P(HiHj
)represents the probability of deciding on the presence
of Hi while Hj is true [9].When no cost is assigned for a correct decision and unit cost is assigned
for a wrong decision, i.e.,
Cij =
{1 i 6= j ,0 i = j ,
Bayes risk simplifies to Bayes error probability and Bayes rule reduces tothe maximum aposteriori (MAP) decision rule [71]. According to the MAP
rule, the FC based on its input z decides on hypothesis Hi if [72]
iPZH
(zHi
)= max
{0PZH (zH0) , . . . , M1PZH (zHM1)
}, (11)
for a general M ary hypothesis testing problem, where PZH (zHj) is thehypothesisconditional probability of input z Z at the FC, and where iis the apriori probability of hypothesis Hi. The expected error probabilitywhen using the MAP rule at the FC is [73]
PE = 1
zZ
max{0PZH (zH0) , . . . , M1PZH (zHM1)
}. (12)
In practice, Bayes error probability is quite tricky to calculate and in thefollowing we will discuss some of these difficulties in the design of sensornetworks. We will start with the parallel network and find hypothesisconditional probabilities and then discuss the calculation of hypothesisconditional probabilities for the tandem network.
For a network of N sensors arranged in parallel or MAC, as in Figures 1and 2 respectively, the hypothesisconditional probability is
PZH (zHj) = PUH (u1, . . . , uN Hj) , (13)
where U , U1, . . . , UN , and where the complete input set at the FC isZ = U , Ur1 . . . UrN . When the observations Xn at the sensors,conditioned on the hypothesis, are independent, the hypothesisconditionalprobability decouples as
PZH (zHj) = PU1H (u1Hj) . . . PUN H (uN Hj) . (14)
12 Summary
This is due to the fact that each sensor output Un is only a function of itscorresponding input Xn. Since functions of independent random variablesare also independent, the independence of sensors inputs results in theindependence of sensors outputs.
Now consider a sensor Sn in the parallel network. Given a sensor decisionrule n and the true hypothesis Hj H, the probability mass functionassociated with the message Un = n(Xn) Urn is calculated as
PUnH (unHj) = Pr(n(Xn) = unHj
)
=
x1n (un)
fXnH (xHj) dx,(15)
where 1n (un) is the set of observations x X that satisfy n(x) = un.This can be generalized to the joint probability mass function associatedwith the message set U = U1, . . . , UN at the FC as
PUH (u1, . . . , uN Hj) =
x111 (u1)
. . .
xN1N
(uN )
fXH (x1, . . . , xN Hj) dx1 . . . dxN .(16)
As (13) shows, in order to design the decision rules at the sensors, such thatthe MAP error probability in (12) is minimized, we need to find the jointprobability mass functions which are found from the joint conditional distribution of the observations at the sensors. This makes the direct use of (12)as a performance measure for the design of sensor rules in a parallel networklimited. However, using some mathematical reformulations in [74], we haveshown that using the same complexity order as the existing methods (forexample the proposed method in [17]) we can use (12) in a parallel networkto design the sensors decision rules, in a personbyperson framework.
For a network of N sensors arranged in tandem, as in Figure 3, thehypothesisconditional probability is
PZH (zHj) = PXN ,UN1H (xN , uN1Hj) , (17)
where for consistency we assume a discrete observation space Xn at (thesensors and therefore) the FC in a tandem network. Note that, althoughwe restrict our attention to discrete observation spaces, Xn can be used toapproximate observations in a continuous space using finegrained binning.Longo et al. [17] used this idea by representing each bin, or interval, inthe continuous observation space by an index xn from the discrete set Xn.The complete input set for the tandem network at the FC is then Z =XN UrN1.
Using the same argument as in (14), for conditionally independent observations at the sensors, the hypothesisconditional probability at the FC
1. INTRODUCTION 13
decouples as
PZH (zHj) = PXN H (xN Hj)PUN1H (uN1Hj) . (18)
Note that for conditionally independent observations at the sensors, (18)generalizes to a continuous observation space. To do this, we can replaceprobability mass function PXN H (xN Hj) with probability density functionfXN H (xN Hj).
Now consider sensor Sn for 2 n N in the tandem network. Givenits sensor rule n and the true hypothesis Hj H, the probability massfunction associated with its message Un = n(Xn, Un1) Urn is calculatedas
PUnH (unHj) = Pr(n(Xn, Un1) = unHj
)
=
(xn,un1)1n (un)
PXn,Un1H (xn, un1Hj) ,(19)
where 1n (un) is the set of tuples (xn, un1) that satisfy n(xn, un1) = un.For sensor S1, the probability mass function associated with its messageU1 = 1(X1) Ur1 is calculated as
PU1H (u1Hj) =
x111 (u1)
PX1H (x1Hj) , (20)
with the corresponding definition for 11 (u1). Equations (19) and (20)show that every FC output uN depends on all the observations x1, . . . , xNthrough the sensor functions 1, . . . , N in a complicated recursive way. Thiscan also be seen from (7) and (8) for any FC decision h H as follows:
h = uN = N (xN , uN1)
= N (xN , N1 (xN1, uN2))
...
= N (xN , N1 (xN1, N2 (xN2, . . . , 2 (x2, 1 (x1)) . . .))) .
(21)
These mathematical formulations show that finding the decision rules atthe sensors which directly minimize the MAP error probability in a tandemnetwork is even more challenging than in a parallel network for arbitrarynetwork size. However, for smallsized networks (i.e., N = 2) and conditionally independent observations at the sensors, the MAP rule provides acomputationally tractable tool for the design of such networks, and in [75],we have used this idea for the design of an arbitrarysized tandem network
14 Summary
in a personbyperson framework. We have further generalized this ideain [76] for the design of sensor decision rules in an arbitrary tree topology.
Although the personbyperson methodology provides a computationally tractable tool for the design of sensor decision rules in a wireless sensornetwork, it does not necessarily lead to a globally optimal design and theproblem of understanding the characteristics of an optimal network remainsopen. One way to tackle this problem in a binary hypothesis testing is tostudy the structure of optimal networks in terms of their error exponent. Inwhat follows we will be discussing the Chernoff information and the Bhattacharyya distance as two performance metrics for the design of decisionrules of sensors and the FC of a network [52, 63, 77].
Consider a binary hypothesis testing problem where sensors are arrangedin parallel or MAC. The expected error probability when using the MAPrule at the FC is [see (12)]
PE = 1
u
max{0PU H (uH0) , 1PU H (uH1)
}
=
u
p (u)(1max
{PHU (H0u) , PHU (H1u)
})
=
u
p (u)(min
{1 PHU (H0u) , 1 PHU (H1u)
})
=
u
min{0PUH (uH0) , 1PU H (uH1)
}.
(22)
Since the Bayes error probability in (22) minimizes the average probabilityof error, bounding it tightly is crucial in hypothesis testing [78]. To upperbound the Bayes error in (22), let us replace the min function by the smoothpower function: namely for a, b > 0, we have
min{a, b} ab1, (0, 1).
Therefore we obtain the following Chernoff bound for the error probability[79]
PE 0 11
u
(PU H (uH0)
) (PU H (uH1)
)1
u
(PU H (uH0)
) (PU H (uH1)
)1,
(23)
which follows from 0, 1 1 and holds for any (0, 1). Since (23) is truefor any (0, 1) we can take the minimum of it to achieve the Chernoffinformation bound as follows
PE eCr(), (24)
1. INTRODUCTION 15
where Cr() is the Chernoff information at the input of the FC for the givenrate allocation r , r1, . . . , rN and sensor rules , 1, . . . , N ,
Cr() , log min(0,1)
u
(PU H (uH0)
) (PU H (uH1)
)1. (25)
The inequality in (24) suggests maximizing the Chernoff information inplace of minimizing the error probability as a performance measure in thedesign of sensor networks, with the cost of solving an optimization problem. The Chernoff information was used as the performance measure inhypothesis testing problems, see [21, 8082]. Lee and Sung [80] consideredthe performance of mismatched likelihood ratio detectors for binary hypothesis testing problems. Using large deviation theory, they have derivedthe maximum Bayesian error exponent for a mismatched detector and haveshown that the maximum Bayesian error exponent is given by the Chernoffinformation. In [81], Fabeck and Mathar designed the decision rule at thesensors by locally optimizing the Chernoff information. In [82], Chamberland and Veeravalli studied the performance of power constrained wirelesssensor networks in a parallel topology when the channels between the sensors and the FC are subject to additive noise, while in [21], they studied thestructure of an optimal network of sensors arranged in a MAC configurationwhich is subject to a sum rate constraint.
While the Chernoff information provides a tight upper bound for theprobability of error at the FC, for many applications and for hypothesisconditional distributions finding the optimal parameter can be mathematically intractable. In these situations a relatively looser upper bound can beused. By setting = 0.5 in (25) we obtain the Bhattacharyya distance [63]as
Br() , log
u
PUH (uH0)PU H (uH1) . (26)
It immediately follows that
Br() Cr()
as the Bhattacharyya distance can be obtained from (25) with = 0.5 inplace of optimization over as noted above. The Bhattacharyya distancealso provides an upper bound on the Bayesian error probability of the MAPdetector according to [83]
PE 01e
Br() (27)
which can be found from (23). As mentioned above, the main reason for using the Bhattacharyya distance in place of the Chernoff information is thatit will increase the tractability of the problem by dropping the optimization over . Furthermore, considering the Bhattacharyya distance instead
16 Summary
of the Chernoff information (in terms of mathematical tractability) has thefollowing benefit: the Bhattacharyya distance of a network of sensors, arranged in parallel or MAC and with independent observations, is equal tothe sum of the Bhattacharyya distances of individual sensors. Thus, a network which maximizes the Bhattacharyya distance at the FC is a networkwith individually optimized sensors.
It should however be noted that the Bhattacharyya distance has beenfrequently used in the past as a performance measure in the design of distributed detection systems [17, 52]. It will be used in this thesis for theproblem of designing decision rules at the sensors and allocating rates tothe sensors arranged in MAC configuration [62]. The Bhattacharyya distance will also be used as the performance measure for the design of sensorsin a network of energy harvesting sensors [84,85], as we shall discuss in thenext section.
1.5 Energy constrained wireless sensor networks
In wireless sensor networks, a large number of sensors with small batteriesand limited lifetimes are often used. Their limited lifetimes are a majorlimitation of using them [86]. In other words, the sensors work as long astheir batteries last and this implies that the network itself also has a limitedlifetime. To increase the lifetime of batterypowered sensor networks manysolutions have been proposed [8792], e.g., by choosing the best modulationstrategy [89], or by exploiting powersaving modes (sleep/listen) periodically[91]. However, in all of these methods the aim is to find an energy usagestrategy to maximize the lifetime of a network, and therefore the lifetimeremains bounded and finite. An alternative way of dealing with this problemis to use energy harvesting devices at the sensor nodes. An energy harvestingdevice is capable of acquiring energy from nature or from manmade sources[9395].
Energy harvesting technologies provide a promising future for wirelesssensor networks, such as self sustainability and an effectively perpetual network lifetime which is not limited by the sensor batterys lifetime [9597].While acquiring energy from the environment makes it possible to deploywireless sensor networks in situations which are impossible using conventional batterypowered sensors, it poses new challenges related to the management of the harvested energy. These new challenges are due to therandomness of the amount of energy available at a sensor, since the sourceof energy might not be available at all times that we may want to use thesensor nodes [96, 98106].
Figure 5 shows a network of energy harvesting sensors, where sensorsare arranged in parallel. Each sensor Sn, n = 1, . . . , N during time intervalt makes an observation xn,t and sends a message un,t towards the FC. Inthis configuration the output message of the sensor depends not only on its
1. INTRODUCTION 17
Fusion Center
S1 SNSn
Phenomenon Ht H
u1,t un,t uN,t
Ht
x1,t xn,t xN,te1,t
w1,t
en,t
wn,t
eN,t
wN,t
Figure 5: Decentralized hypothesis testing scheme in a parallelnetwork of energy harvesting sensors.
observation, but also on its battery charge, i.e.,
n(xn,t, bn,t) = un,t ,
where bn,t denotes the battery charge of sensor Sn at transmission time t. Acentral problem in this configuration is how to design the sensors decisionrules in such a way that a performance measure is optimized. However,this problem is more challenging than the conventional problem of designing sensors decision rules in a parallel network, due to the randomness ofthe battery charges. Note that in this situation, energy (denoted by en,t)arrives to the sensor according to a random process and in general the FCis not aware of the battery charges. To the best of our knowledge, the problem of decentralized hypothesis testing using energy harvesting sensors hasnot been considered before and in [84,85] we have considered this problem.Concretely, in these works, we have formulated a decentralized detectionproblem with system costs due to the random behavior of the energy available at the sensors and we have shown how the problem formulation changes(compared to the unconstrained case) when we consider the energy featuresin the problem of designing decision rules at the sensors in the network.
18 Summary
2 Our contributions
In this section we discuss our contributions to the problem of decentralizedhypothesis testing in wireless sensor networks. In the first part, we shallconsider the problem of designing decision rules at the sensors and theFC in different configurations. We have studied this problem in [7476]labeled as Papers AC. In the second part, we shall consider the problem ofrate allocation in wireless sensor networks. We have studied this problemin [62, 64] labeled as Papers DE. Finally, we shall consider the problemof decentralized detection in energy harvesting sensor networks. We havestudied this problem in [84, 85] labeled as Papers FG. For each paper,we will briefly discuss the system model and assumptions and present thecentral results.
2.1 Problem of designing sensors decision functions
Paper A: Bayesian design of decentralized hypothesis testing under communication constraints [74]In this paper we have considered a binary hypothesis testing problem in theparallel topology as in Figure 1, where the peripheral nodes observe a common phenomenon and send their possibly correlated observations towardsthe FC via errorfree but rateconstrained channels. The objective in thispaper is to design the sensor nodes decision rules n, n = 1, . . . , N and theFC decision rule FC such that the Bayesian error probability PE at the FCis minimized.
This problem was studied by Longo et al. [17], who considered eachsensor to be a scalar quantizer that satisfies the rateconstraints. They proposed a cyclic personbyperson optimization algorithm to design the quantizers. Longo et al. argued that the natural criterion of optimizationthepower of a NeymanPearson testmade the design procedure intractable.They therefore proposed to instead maximize the Bhattacharyya distance(26) between the conditional probability distributions under the two hypotheses.
The main contribution of our work was to show that, contrary to previous claims, it is in fact possible, within a personbyperson framework, tominimize the probability of error directly, while having the same complexityorder as in the previous work by Longo et al. To this end, we have proposedcombining the idea of finegrained observationbinning used in [17] with amethod for updating the hypothesisdependent mass functions of the jointindex space (13).
The benefits of the proposed method are that: it takes into accountthe apriori probability of the hypotheses in the design procedure, and itis applicable to general M ary hypothesis testing problems and not onlybinary hypothesis tests. Furthermore, the proposed method shows better
2. OUR CONTRIBUTIONS 19
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0
Errorprobability
bC
bC
bC
bC
bC
bC
bCbC
bC
bC bC bC
bC
bC
Longo et al.s methodProposed method
bC
bC
Figure 6: Error probability performance of the designed networksusing different methods for different apriori probabilities.
performance in terms of both the receiver operating characteristic (ROC)and the error probability. To exemplify these results, we have considered anetwork of N = 2 sensors with the following observation model:
H0 :
[x1x2
]=
[n1n2
],
H1 :
[x1x2
]=
[aa
]+
[n1n2
],
(28)
where n =
[n1n2
]is a zeromean Gaussian vector of covariance matrix
=
(2 r2
r2 2
),
where r = 0.9 is the spatial correlation coefficient. As in [17], we haveassumed that per channel signaltonoise ratio (E/2) is 5 dB and channelrates are the same, i.e., r1 = r2. We divided the interval [4,+4] (containing0.9997 of the total probability mass for each DM) into 256 bins, and designedthe sensors for various values of apriori probabilities and channel rates
20 Summary
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PF
PD
Unconstrained case (r = )Longo et al.s method for r = 2Proposed method for r = 2
Figure 7: The complete ROC curve achievable using the proposeddesign method, Longo et al.s method, and the optimal centralizeddesign.
(in bits/sample). Figure 6 shows the error probability performance of thesensor network designed using the proposed method and using Longo et al.smethod, and Figure 7 illustrates the achievable ROC curves for both designmethods. Note that the close relation between the error probability and theROC makes it possible to compare the results of the proposed method withthose of Longo et al.s in terms of the ROC curve as well. These figuresimply that although the proposed design method is based on the Bayesianerror probability, it also outperforms the previous method in terms of theROC curve.
To see how the two methods work, let us consider the decision regionsfound using these methods in Figure 8, when 0 = 0.5 and with the sameparameters as before, i.e., r = 0.9, r1 = r2 = 2 and the same observationmodel as in (28) and for a = 0.5. In this figure, the decision boundaries foreach sensor are shown by solid blue lines and the final decision by the FC(which uses the optimal MAP rule) in each region is shown by (red colored)zeros and ones, where zero and one correspond to H0 and H1, respectively.We further show sample values p(x1, x2) by cyan and their conditional meanvalues by solid black circles. Also the corresponding decision regions usinglikelihood test [107] (for centralized design) are shown by dashed lines inboth subfigures. As we can see, the decision regions in the proposed methodare such that more likely observations are classified correctly, while less likely
2. OUR CONTRIBUTIONS 21
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observations may not be classified correctly. In other words, the proposedmethod performs better in the detection of more probable observations,while it performs worse in the detection of less likely observations, whichresults in a better overall performance for the system.
Consider again the decision regions in Figure 8. We can see that the
22 Summary
designed sensors using Longo et al.s method are monotone quantizers ofrate 2: each axis is divided into 22 nonoverlapping intervals, each intervalcorresponds to a sensor output message from the set {0, 1, 2, 3}, while thedesigned sensors using the proposed method are not monotone quantizers.Though it is known that monotone quantizers are often optimal decisionfunctions at the sensors with conditionally independent observations, theyare not proved to be in general optimal decision functions when the sensorsmake correlated observations. This can also be seen from our simulationresults in Figure 8.
Paper B: Bayesian design of tandem networks for distributed detection with multibit sensor decisions [75]In this paper we have considered the problem of decentralized hypothesistesting where several peripheral nodes are arranged in tandem as in Figure 3. We have assumed that the observations at the sensors, conditionedon the true hypothesis, are independent. Furthermore, the channels between every two successive sensors are errorfree but rateconstrained. Theobjective in this paper is to design the sensor nodes decision rules such thatthe error probability at the FC (the last sensor) is minimized.
This problem was previously considered in [27, 68] for a binary hypothesis test, where the sensors were able to send onebit messages. Thoughthese studies then were extended to M ary hypothesis testing problems,the output of each sensor was constrained to be from an M valued messageset. In this paper we have proposed a cyclic numerical design algorithm forthe design of sensors in a personbyperson methodology, where the number of communicated messages was not necessarily equal to the number ofhypotheses.
Our main contribution is to introduce a numerical methodology for designing sensors decision rules in a tandem network, where each sensor iscapable of sending arbitraryvalued messages. We have designed the sensorsin such a way that the final error probability of the network is minimized.We have proposed a modified personbyperson optimization in which eachsensors decision rule is jointly designed with the FC (the last sensor). Inother words, we have designed each sensor under the assumption that theFC always employs the optimal MAP rule to its inputs.
First we have shown that the design of each sensor Sn, n = 1, . . . , N 1in this framework is equivalent to the design of sensor S in a network labeledas a restricted network, as shown in Figure 9, where SN in both networksuses the MAP rule. This means that regardless of the network size, eachsensor in a tandem network can be designed using the restricted model witha fixed computational burden.
2. OUR CONTRIBUTIONS 23
S P (uN1u,H) SN
Phenomenon H
u uN1 uN
y H yN
Figure 9: Restricted model used for the design of sensors in tandemnetworks.
In other words, in the proposed personbyperson methodology (wherea sensor is designed together with the FC while all other sensors are keptfixed), the design of the sensor is equivalent to the design of sensor S inthe restricted model. We have then found parameters of the restrictedmodel according to the structure of the network: P (uN1u,H) was foundaccording to the other sensors (fixed) decision rules, y was found accordingto the sensor observation and the input from its predecessor, except for thefirst sensor in the network, while yN had the same distribution as the FCobservation.
Next, we have proposed a computationally efficient algorithm for thedesign of sensor S in the restricted model. We have further found theoverall complexity of the proposed algorithm for the design of sensors in atandem network.
In order to illustrate the benefits of the proposed method, we have compared the performance of designed networks using our proposed methodwith those of Cover [68] and Swaszek [27] for binary hypothesis testing.Figure 10 shows the performance of networks with different numbers ofsensors designed using different methods. We assumed each real valued observation consists of an antipodal signal a in unitvariance additive whiteGaussian noise, i.e.,
H0 : xi = a+ ni ,H1 : xi = +a+ ni .
We also defined the per channel SNR for the binary hypothesis test as E ,a2, and assumed that the hypotheses are equally likely (0 = 1 = 0.5).
As noted before, the proposed method can be applied to general M aryhypothesis testing problems. To see this, we considered a ternary hypothesistesting problem in which each real valued observation consists of a knownsignal sm,m = 0, 1, 2 in unit variance additive white Gaussian noise. Wehave assumed an equal distance signal set in the interval [a, a], i.e., the
24 Summary
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Swaszek optimum rateoneCover twostate unboundedLRProposed method rateoneProposed method ratetwoProposed method ratethreeProposed method ratefourOptimum linear detector
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Figure 10: Error probability performance of tandem networks withdifferent channel rates for a binary hypothesis testing problem asa function of number of sensors. Error probability of an unconstrained tandem network and existing methods for rateone channels are also shown.
test signal set was {a, 0, a}. The observation model at each sensor was
Hm : xi = sm + ni, m = 0, 1, 2.
Similar to the binary hypothesis test, we assumed equally likely hypothesesand defined the SNR as E , a2. Figure 11 shows the performance ofnetworks with different numbers of sensors designed using the proposedmethod for different channel rates. As we can see from Figure 10 andFigure 11, networks with multibit (soft) message sensors outperform thoseof binary (hard) message sensors and by increasing the channel rates (lengthof transmitted messages) the performance of the networks improves.
2. OUR CONTRIBUTIONS 25
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Figure 11: Error probability performance of tandem networks withdifferent channel rates for different numbers of sensors with anunconstrained tandem network, for a ternary hypothesis testingproblem.
Paper C: A general method for the design of tree networks undercommunication constraints [76]This paper mainly generalizes the results in [75]. In this paper, we haveconsidered a distributed detection problem where several nodes are arrangedin an arbitrary tree topology. Communication channels between the sensorsare again rateconstrained but errorfree and observations at the sensors areconditionally independent.
We have proposed a cyclic design procedure using the expected minimumerror probability by adopting a personbypersonmethodology for the designof sensors decision rules in the network. Concretely, we design the decisionrule of each sensor jointly together with the FC while all the other sensorsin the network are kept fixed. In order to obtain a tractable solution duringthe design of a sensor, say Sn, all other sensors were modeled using a Markovchain. We have further shown that the design of sensor Sn jointly with theFC is analogous to the design of a special case of a network with only twonodes, which is again referred to as the restricted model. The restrictedmodel for the design of sensors in a tree network is shown in Figure 12.
26 Summary
k P (wz,H) FC
P (yH) P (vH)PhenomenonH
z w
y v
Figure 12: Restricted model for the design of nodes in tree topology.
Next, we have shown how, according to the structure of the problem andMarkovian properties of sensors in a network, parameters of the restrictedmodel can be found in a computationally efficient way.
As an example consider the tree network shown in Figure 13. Let usassume only sensors S1, . . . , S4 make observations (labeled as leaves) andsensors S5, S6 are relays that summarize their received messages from theirneighboring nodes and send their messages towards the FC. We also assumethat the observation model at each leaf consists of an antipodal signal ain additive unitvariance white Gaussian noise as
H0 : xi = a+ ni ,H1 : xi = +a+ ni .
We define the per channel SNR for a binary hypothesis test again as E ,a2, and we assume that the hypotheses are equally likely (0 = 1 = 0.5).We further assume leaftorelay and relaytoFC channels have the samerate r. Using the proposed algorithm, we have designed the tree networkfor different rates r. In Figure 14, error probability performance of the
FC
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Figure 13: An example of a tree network.
2. OUR CONTRIBUTIONS 27
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Figure 14: Error probability performance of a designed tree network shown in Figure 13 for different channel rates.
designed tree network for different rates is shown. As expected, increasingchannel rates results in better error probability performance.
28 Summary
2.2 Problem of rate allocation in wireless sensor net
works
Paper D: Rate allocation for decentralized detection in wirelesssensor networks [64]In this paper we have considered the problem of rate allocation to sensorsarranged as in Figure 2, in which the sensors make noisy observations ofa binary phenomenon and send a quantized information towards a FC tomake the final decision. The sensors send their messages through a MACchannel which is modeled as a sumrate constrained channel, with capacityR.
Using the Chernoff information (25), Chamberland and Veeravalli [21]studied the structure of an optimal network model in terms of the optimalnumber of sensors and their rate allocation. They proved that, if for agiven observation model there exists a (onebit) quantization rule whichleads to the transfer of at least half of the Chernoff information containedin each raw observation, an optimal strategy is to employ N = R sensorseach quantizing its observation to a onebit message. Although their resultgreatly simplifies the design of sensors in a MAC network, it suffers from theconstraining assumption of having an unlimited number of active sensors.In fact, due to cost and space constraints, it may not be feasible to haveN = R sensors in a network.
In this paper we have addressed the problem of finding an optimal rateallocation r when the total number of active sensors N is fixed apriori.We assumed R = mN , where m is a positive integer. Using the Chernoffinformation at the FC as a performance measure, we have found conditionsunder which uniform rate allocation to the sensors in the network is anoptimal rate allocation. These conditions are captured in Theorem 1 of thepaper and is restated as follows:
Theorem: Given a sensor design method, if for a single sensor Sn the resulting Chernoff information Crn is a discrete concave function of rate rn,a uniform rate allocation across sensors is optimal.
As stated above, the optimality of the uniform rate allocation in thenetwork relies on the concavity of the Chernoff information of the sensordesign method. In order to show the benefits of the results, we have exploredthis point numerically for a couple of sensor design methods. First, we haveconsidered Benitz and Bucklews [54] method for the design of a sensorrule (or quantization rule) in detection with i.i.d. observations. When theobservation model at any sensor consists of an antipodal signal a in anadditive unitvariance white Gaussian noise, or
H0 : xi = a+ ni ,H1 : xi = +a+ ni ,
2. OUR CONTRIBUTIONS 29
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Figure 15: Chernoff information of a single sensor designed usingBenitz and Bucklews method and using the numerical method fordifferent SNRs, E .
we have proved, at high rates, the concavity condition holds. Also, usingsimulation results we have shown that the Chernoff information of a sensordesigned using Benitz and Bucklews method is a concave function of itsrate. Then, we have concluded that for a network of sensors whose decisionrules are designed using Benitz and Bucklews method, an optimal rateallocation to the sensors is a uniform rate allocation.
Next, we have proposed a numerical method for the design of a sensor through a numerical optimization. In the proposed method, for a raterquantizer, the real interval was divided into 2r nonoverlapping intervals randomly, where each interval corresponds to an output message. Then, in aniterative fashion, the intervals are modified (in a personbyperson methodology) to maximize the Chernoff information, until a stopping criterion wassatisfied. Our simulation results show that the Chernoff information resulting from the proposed numerical method is also a concave function ofsensor rate. Figure 15 illustrates resulting Chernoff information curves ofsensors whose decision rules are designed using the aforementioned methodsfor different SNRs (E = a2). As we can see from this figure, the Chernoffinformation resulting from both methods is a discrete concave function of
30 Summary
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Figure 16: Error probability performance of a network of N = 6sensors for different rate allocation schemes.
rate.
According to concavity of the Chernoff information, we can concludethat for a network of N sensors, where the sensors share a common MACchannel with sum rate capacity R (where R divides N), an optimal rateallocation to the sensors is a uniform rate allocation. To exemplify this, wehave considered a network of N = 6 sensors and sum rate capacity R = 12and compared their performance in terms of error probability at the FC(note that the FC applies the MAP rule to make the final decision). Itcan be observed from Figure 16 that the uniform rate allocation results inthe best error probability performance, which is consistent with the resultsobtained using the Chernoff information.
In this paper, we have found sufficient conditions for the optimalityof uniform rate allocation in a decentralized detection network. We havenumerically verified that these conditions hold true for some sensor decisionrule design methods. However, it is hard to stringently prove the requiredconcavity property for Chernoff information. This difficulty arises mainlybecause of the optimization problem over parameter in the definition ofthe Chernoff information. In Paper E, using the Bhattacharyya distance,we have generalized the results.
2. OUR CONTRIBUTIONS 31
Paper E: Optimality of rate balancing in wireless sensor networks[62]This paper generalizes the results in Paper D, in which we have consideredthe problem of rate allocation for decentralized detection in a network ofsensors arranged as in Figure 2. In this configuration, the sensors share aMAC channel to send their observations towards the FC. The MAC channelis modeled as a sum rate constrained channel of capacity R bits per unittime.
This problem was previously studied in [21] by Chamberland and Veeravalli, where they argued that when an unbounded number of sensors withi.i.d. observations compete for rates under sum rate constraints at the inputof the FC, it is often optimal to use as many sensors as possible and leteach sensor communicate with the FC over a one bit link. They proposedthe Chernoff information at the input of the FC as a measure of optimalitygiven the intractability of the Bayesian probability of error. They provedthat if there exists a one bit sensor rule with a Chernoff information of thesensor output that is at least half of the Chernoff information of the originalobservation, having as many one bit sensors as possible is optimal. Theyalso proved that such a sensor rule exists when the observations are drawnfrom particular Gaussian and exponential observation models.
Our contribution is, in the same vein, to study when equal rate allocation or rate balancing is an optimal solution for a fixed number of sensorsoperating in a network under a common sum rate constraint. Concretely,we have driven a sufficient condition for when rate balancing is an optimalstrategy in the sense that one can, without loss of optimality, assume thatrates of any two sensors differ by at most one bit.
One problem with extending the results in Paper D to a general case(when R does not divide N) is that parameter which optimizes the Chernoff information of a rate r1 sensor is not necessarily equal to that of asensor of rate r2 for r1 6= r2. In order to circumvent this difficulty we haveinstead of using the Chernoff information, used the Bhattacharyya distance,as the performance measure of the network. The Bhattacharyya distancehas another benefit over the Chernoff information which is captured in thefollowing lemma:
Lemma: The Bhattacharyya distance of a network of sensors, arranged asin Figure 2 and with independent observations, is equal to the sum of theBhattacharyya distances of individual sensors, i.e.,
Br() =N
n=1
Brn(n),
where Brn(n) is the Bhattacharyya distance of a sensor of rate rn anddecision rule n.
32 Summary
This lemma implies that Bhattacharyya distances of individual sensorscompletely describe the Bhattacharyya distance of the network, and a network which optimizes the Bhattacharyya distance is a network with individually optimized sensors. Then by defining Brn as the maximum Bhattacharyya distance of a sensor of rate rn, we have found the following theorem for an optimal rate allocation:
Theorem: If, for a given observation distribution at the sensors, Brn is adiscrete concave function in the rate rn, then rate balancing is an optimalrate allocation.
Although this theorem provides sufficient conditions for the optimalityof rate balancing, it however is difficult to analytically characterize Brn ingeneral. To circumvent this difficulty and obtain a mathematically tractablecriterion, we have found a sufficient condition under which Brn is a concavefunction of rate rn, and this sufficient condition is described in the followingremark:
Remark: Conditioned on an observation X being in any given interval ofthe real line (or an interval of an optimum monotone quantizer), if there isa one bit quantization of X with a Bhattacharyya distance more than halfof the Bhattacharyya distance of X itself, then having rate balanced sensorsis optimal.
This is in agreement with Chamberland and Veeravallis result. Theconditioning on X being in an interval of an optimum monotone quantizer,is in part what generalizes the result to higher rates. However, verifyingthis condition is considerably harder than verifying the condition of [21] asit needs to be established for all possible optimal intervals for any arbitraryrate. Nevertheless, we proceed to discuss the Laplacian and the Gaussianobservation models where the conditions of the remark can be establishedin practice. Then we have shown that rate balancing is an optimal strategy when the observations at the sensors are distributed as Gaussian orLaplacian.
Our next contribution was then to show how the performance of different rate allocation schemes can be partially compared using majorizationtheory [108] and also the concavity properties of the Bhattacharyya distance. Concretely, we have shown that the optimal Bhattacharyya distanceof a network of sensors is Schurconcave and consequently, if a rate allocation vector is majorized by another rate allocation vector, it has higherBhattacharyya distance. This is in agreement with our previous result in thesense that the rate allocation vector of a balanced rate network is majorizedby any
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