DISTRIBUTED SUBSPACE PROJECTION OVER WIRELESS SENSOR NETWORKS WITHUNRELIABLE LINKS

Daniel Alonso-Rom´an, Fernando Camar´o-Nogu´es, C´esar Asensio-Marco, Baltasar Beferull-Lozano

Group of Information and Communication SystemsInstituto de Robotica y Tecnologıas de la Informacion & las Comunicaciones (IRTIC)

Universidad de Valencia46980, Paterna (Valencia), Spain

Email: {Daniel.Alonso,Fernando.Camaro,Cesar.Asensio, Baltasar.Beferull}@uv.es

ABSTRACTThe inaccuracy of the measurements collected by sensor

nodes in a Wireless Sensor Network motivates the use of ap-propriate techniques for reducing this observation error. Oneof the most widely used method consists of projecting orthog-onally the observation signal onto a subspace of interest. Thein-network processing nature of these architectures suggeststhat such a method should be performed in a distributed man-ner, through successive local communications between neigh-bors and under a time-varying network topology. Existingapproaches to this technique assume an ideal scenario, wherethese communications are not subject to interferences, packetlosses or fading. In this work, we propose a new technique,which under unreliable communications, projects the initialobservation onto a proper subspace in a totally distributedfashion. We analyze the deviation from the optimal projec-tion due to packet losses, and compare it with the initial ob-servation error. We also consider the additional error due toan inaccurate choice of the subspace. Simulation results arepresented to show the efficiency and validity of our approach.

Index Terms— Wireless sensor networks, Distributed es-timation, Orthogonal projection, Average consensus

1. INTRODUCTION

Most of the existing applications based on Wireless SensorNetworks (WSNs) rely on the correctness of the data sensedin the area of interest [1-3]. Sensing inaccuracy due to nodemalfunctioning or noise may spoil the entire solution. A gen-eral technique to mitigate this problem consists in orthogo-nally projecting the signal observation onto a subspace of in-terest. Some existing works [4][5] compute this projection ina distributed manner, through an iterative process guided bya weight matrix. However, the computation of this matrix,

This work was supported by the Spanish MICINN Grants TEC2010-19545-C04-04 “COSIMA”, CONSOLIDER-INGENIO 2010 CSD2008-00010 “COMONSENS”, the European STREP Project “HYDROBIONETS”Grant no. 287613 within the FP 7 and by a Telefonica Chair.

and thus the applicability of these methods, is strongly con-strained by the network connectivity and the limited compu-tational resources of the network nodes. Although this impor-tant drawback is partially solved in the work presented in [6],where having a strongly connected graph is the only require-ment to be satisfied, none of these proposals [4-6] take into ac-count the constraints present in a real wireless environment.In fact, these works assume perfect communications so thatthe resulting underlying graph is undirected and strongly con-nected. However, in a real scenario, the communications be-tween network nodes are constrained by the interferences andother environmental factors. Consequently, these communi-cations are asynchronous and asymmetric in general. This im-plies that the underlying instantaneous graphs do not satisfythe required convergence conditions, leading to the inapplica-bility of the proposed methods under realistic situations. Inthis work, we extend the method proposed in [6] to the caseof random asymmetric graphs and, consequently, we makepossible its application under real wireless constraints. Theapproach is based on the parallel execution of several averageconsensus processes. Each process obtains, with a certain er-ror due to packet losses, the coefficient corresponding to thecomponent of the projection in each subspace dimension. Ourwork analyzes the error between the final obtained projectionand the initial observation, and compares it with the initialobservation error. This allows us to analyze the applicabil-ity of our distributed method, such that it is worth applying itwhenever the resulting projection approximates the field bet-ter than the initial observation. We present some simulationswhere the relation between the different errors is explainedand the validity of our method is verified.

The remainder of this paper is structured as follows: theproblem formulation and the motivation of our work are givenin Section 2. An explanation of our distributed projectionmethod over directed random graphs is covered in Section 3.In Section 4, the error of our process is characterized in orderto decide when it is worth projecting. Finally, the conclusionsare summarized in Section 5.

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

345

2. PROBLEM FORMULATION

The communications between the nodes of a WSN over awireless medium can be modeled as a time-varying graphG(k) = (V,E(k)), where V is a constant set of N nodesand E(k) is the set of active links at time instant k. The linkfrom node i to node j is denoted by eij , and the probability ofsuccessful transmission from i to j is given by 0 pij 1,being pij = 0 if i and j are not neighbors or i = j. Given atime-varying graph G(k), we can assign an N⇥N adjacencymatrix A(k) where an entry [A(k)]ij is equal to 1 with prob-ability pij . The matrix A(k) is non symmetric random and itsexpected value is denoted by A, with entries [A(k)]ij = pij .Finally, the Laplacian of a graph G(k) is a matrix definedas L(k) = D(k) � A(k) whose smallest eigenvalue can beshown to be equal to zero and the in-degree matrix D(k) is adiagonal matrix whose entries are di(k) =

PNj=1[A(k)]ij .

Let us denote by x = [x1, . . . , xN ]T 2 RN the vectorcontaining the measurements of all sensors, corrupted by anobservation error. Thus, the vector x can be expressed as:

x = s+w

where s is the vector containing the values of the field, andw is the vector containing the observation errors for all thesensors.

Let M N be the dimension of the chosen subspace andU = [u1u2 . . .uM]T a M ⇥ N matrix, whose rows forman orthonormal basis of the mentioned subspace, denoting byuj the j-th vector of the basis and by uji its i-th component.The projection matrix P over the subspace defined by U is aN ⇥N matrix, computed as P = U

TU. The projection of x

over the subspace is:

x? = Px = Ps+Pw = s+(Ps�s)+(Pw) = s+v (1)

where v is the error of the projection with respect to the field,composed of two terms: the error introduced by the choice ofa subspace that not exactly contains the field, and the orthog-onal projection of the error over the subspace. Our problem ishow to compute (1) in a decentralized fashion under random

s

x

x

w

q

e

v

x?

Fig. 1: Projection example including all the errors involved.

asymmetric communications, thus making our approach di-rectly applicable to real environments. Previous works [4][5]have addressed the distributed projection problem by meansof an iterative process. Starting from the initial observation,denoted by x(0), and assuming perfect communications, thenodes are able to converge to the projection, that is, x =limk!1

x(k) = x?. This holds for symmetric communicationsand it is not generally true for the asymmetric case. The iter-ative process proposed in [4][5], is based on a weight matrixx(k + 1) = Wx(k), which determines how the informationis mixed in each iteration. Necessary and sufficient condi-tions for this expression to converge to the projection are [4]WP = P, PW = P and ⇢(W � P) < 1. However, inpractice, the existence of interferences causes that, at each it-eration of the process, an instantaneous sparse matrix A(k) isgenerated. Since this matrix determines the zero entries of theweight matrix, a different W(k) must be generated at each it-eration k, such that [W(k)]ij = 0 if [A(k)]ij = 0. In thiscase, the system evolution becomes x(k + 1) = W(k)x(k)and the convergence to the projection is ensured if each in-stantaneous matrix W(k) satisfies W(k)P = P, PW(k) =P and ⇢(W(k) � P) < 1, 8k. The first condition requires aminimum connectivity for every node in each iteration, whichis not always guaranteed. Moreover, the methods proposed in[4] and [5] are too complex to build a different matrix W(k)each iteration. Finally, since the matrix A(k) is not symmet-ric in general, the second condition PW(k) = P is not gen-erally satisfied either.

3. PROJECTION OVER RANDOM GRAPHS

Alternatively to what is done in [4] and [5], the work pre-sented in [6] splits the distributed projection problem overundirected static graphs in M consensus processes, which areguided by the same weight matrix W. Therefore, the conver-gence to the projection is ensured as long as the conditionsconcerning the matrix W for average consensus [7] are satis-fied. In this work, we extend the method in [6] to the case ofdirected random graphs.

The projection of the observation x can be expressed asa linear combination of the vectors of the basis U, with thecoefficients ↵j , j = 1, ...,M :

x? = Px = ↵1u1 + . . .+ ↵MuM =MX

j=1

↵juj (2)

The contribution of node i to the aforementioned projec-

tion is the component i of vector x?, given byMPj=1

↵juji;

i = 1, . . . , N . We assume that, by using some subspace track-ing method [10], every node i knows the i-th component ofevery vector of the basis, uji, j = 1, ...,M , and the value ofN . Then, to obtain the value of the projection x?, all nodes

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

346

0 100 200 300 400 50010

15

20

25

30

35

40

Iteration (k)

MSE

(dBs

)

q: field − distributed projection e: distributed projection − final projection w: field − observation v: field − projection 1/N * sum(ei)

(a)

0 100 200 300 400 500−15

−5

5

15

25

35

Iteration (k)

MSE

(dBs

)

q: field − distributed projection e: distributed projection − final projection w: field − observation v: field − projection 1/N * sum(ei)

(b)

Fig. 2: Mean square of the different errors involved in the projection, as a function of the evolution of the consensus processes.The larger the gap between the observation error w and the projection error v, the greater the range of possible values for theerror e. (a) The M consensus processes are performed over a graph with asymmetric connection probabilities, thus we obtaina large error e. However, q improves w because of the large gap between w and v. (b) Since the gap has narrowed, we havebalanced the connection probabilities to reduce the consensus error. Note that the of the MSE of the individual consensusprocesses divided by N converge to the MSE of e. Experiment performed for N = 50, M = 10 and ✏ = 1

N .

only need to compute the values of the coefficients, {↵j}Mj=1

in (2). Each coefficient ↵j is obtained from the initial vector

x, through the inner product ↵j = hx,uji =NPi=1

xiuji j =

1, . . . ,M . For each dimension j, each of the N nodes i startsa consensus process with N · xiuji as initial value, and itera-

tively converges toNPi=1

xiuji. If we aggregate the estimation

of every sensor for the coefficient ↵j at time k in the vector↵j(k), the system evolves according to the following linearequation:

↵j(k) = W(k)↵j(k � 1) j = 1, . . . ,M

where ↵j(0) = [N ·x1uj1, N ·x2uj2, . . . , N ·xNujN ] is thevector with the states corresponding to the N nodes at itera-tion k = 0 and W(k) is a weight matrix, common to all di-mensions, that changes at each iteration. The aim of this pro-cess is to obtain the convergence of every node to ↵j , that is,limk!1

↵j(k) = ↵j1. At any step k of the iterative process, theinstantaneous value of the estimation of the projection vectoris:

x(k) =

2

4MX

j=1

↵j1(k)uj1, . . . ,

MX

j=1

↵jN (k)ujN

3

5T

(3)

and the error at this step k is given by e(k) = x(k)�x, whichcan be expressed as:

e(k) =MX

j=1

diag(uj) ej(k) (4)

where diag(u) is the diagonal matrix formed by the elementsof the vector u and ej(k) = ↵j(k) � ↵j1 is the error of thesingle consensus process corresponding to the dimension j atstep k. Hence, the total error depends on the error of everysingle consensus process and the basis of the subspace.

Necessary and sufficient conditions for the consensus pro-cesses to converge to the average are W(k)1 = 1, 1

TW(k) =

1

T, ⇢

⇣W(k)� 11T

N

⌘< 1, 8 W(k). First and third con-

ditions guarantee that every node reaches a common value,while second condition guarantees that this value is the aver-age. In this work, we assume the matrix W(k) to be given byW(k) = I � ✏L(k), where I is the identity matrix and ✏ is aconstant chosen from a specific interval [7], such that the thirdcondition is satisfied on average. Besides, this matrix struc-ture directly satisfies W(k)1 = 1. First and third conditionsare enough for ensuring the convergence to a random value[9]. However, since the second condition 1

TW(k) = 1

T

can not be guaranteed for every iteration, the convergencevalue may be different from the initial average. Therefore,the error e(k) in (4) is not asymptotically zero, and revealsthe distance between the correct projection x? and the finalobtained solution x (see Fig. 1).

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

347

0 10 20 30 40!40

!30

!20

!10

0

10

20

30

40

||e||�

||v|

| !

||w

||

||q|| >= ||w|| ||q|| < ||w|| ||e|| = ||v|| ! ||w|| !||e|| = ||v|| ! ||w||

Fig. 3: Realizations below the continuous line always im-prove the initial observation. Realizations above the dashedline never improve it. Realizations between both lines im-prove it depending on the direction of the error vector. Exper-iment performed for N = 50, M = 10 and ✏ = 1

N

4. PROJECTION ERROR

The different errors involved in our process are shown inFig.1. The deviation q of the final projection with respect tothe field involves two components: the error of the projectionitself v, as is given in (1), and the error e introduced by theM consensus processes. Then, it is worth projecting as longas the final error q reduces the initial observation error w:

kqk = kv + ek < kwk

where k · k stands for the Euclidean norm of the vector. If weapply the triangular inequality, that is, kv+ ek kvk+ kek,we have that if the norm of the error kek satisfies that:

kek < kwk � kvk , �kek > kvk � kwk (5)

the final achieved vector is closer to the initial field than theobservation, and therefore it is worth projecting despite theerror introduced by the M consensus processes (realizationsbelow the continuous line in Fig. 3). Moreover, as the normof the observation error kwk increases, or the norm of theprojection error kvk decreases, we can tolerate higher valuesof kek (see Fig. 2). On the other hand, if the norm of the errorkek is such that:

kek kvk � kwk (6)

the projection process never improves over the initial obser-vation (realizations above the dashed line in Fig. 3).

Finally, if the following expression holds:

kvk � kek < kwk kvk+ kek (7)

the decision depends on the the direction of the error vector e(realizations between both lines in Fig. 3). Since we are deal-ing with random graphs, the error must be studied in proba-bilistic terms along different realizations. The mean squareerror (MSE) of the consensus process for the j-th dimensionis given by:

MSEj(k) =1

N

E⇥||↵j(k)� ↵j1||22

⇤(8)

The MSE corresponding to the projection is given by:

MSEtot(k) =1

N

E⇥||x(k)� x?||22

⇤=

1

N

E⇥||e(k)||22

⇤(9)

It can be shown that expressions (8) and (9) are related asfollows:

1

N

E⇥||e||22

⇤=

1

N

limk!1

E⇥||e(k)||22

⇤=

1

N

limk!1

MX

j=1

MSEj(k)

that is, the MSE of the projection process asymptotically con-verges to the sum of the MSE of the M consensus processesdivided by the number of nodes N (see Fig. 2). In addition,the inequality in (5) can be converted into the weaker stochas-tic condition E

⇥||e||22

⇤< E

⇥||w||22

⇤� E

⇥||v||22

⇤ 4= �. Con-

sequently, the design rule (5) can be expressed as a functionof the asymptotic MSE of every single consensus process asfollows:

limk!1

MX

j=1

MSEj(k) < � (10)

From the expression in (10), we can easily derive the ap-plicability rule for different types of graphs. For example,assuming an Erdos-Renyi graph with equal probabilities ofconnection pij = pji = p, 8 i, j 2 V, we have [8]:

�

2

✓N � 1

N

(1� p) ✏

◆<

N

M

�

where it is assumed that all the coefficients ↵ have the sameprobability distribution, with variance �

2. For a graph withdifferent but symmetric probabilities, we have [8]:

�

2

✓(tr(Cw)�N)

1� �

kN (Cw)

1� �N (Cw)

◆<

�

M

where Cw = E⇥W(k)WT (k)

⇤8k, and �N (Cw) denotes its

smallest eigenvalue. Thus, for a given observation error and aspecific subspace, which determine the vectors w and v, wecan choose ✏, and try to tune the link activation probabilities,in such a way that the accuracy of the initial observation isimproved by projecting it onto such subspace (see Fig. 2).

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

348

5. CONCLUSIONS

In this work, we propose a new technique to project the initialobservation onto a proper subspace. It is performed in a dis-tributed fashion, and takes into account the existence of unre-liable communications. We characterize the error of the pro-jection process due to packet loses, and compare it with theinitial error of the measurements, in order to find out whetherthe projection is worthwhile or not.

6. REFERENCES

[1] Beferull-Lozano, B. and Konsbruck, R.L., “On SourceCoding for Distributed Temperature Sensing With Shift-Invariant Geometries, IEEE Transactions on Communi-

cations, Vol. 59, no. 4, April 2011.

[2] Zhi Quan; Shuguang Cui; Poor, H.; Sayed, A.; , ”Col-laborative wideband sensing for cognitive radios,” IEEE

Signal Processing Magazine, vol.25, no.6, pp.60-73,November 2008.

[3] P. Park, C. Fischione, A. Bonivento, K. H. Johansson,A. Sangiovanni-Vincentelli, Breath: a Self-Adapting Pro-tocol for Industrial Control Applications Using WirelessSensor Networks, IEEE Transactions on Mobile Comput-

ing, Vol. 6, No. 6, pp. 821-838, June 2011.

[4] Barbarossa, S., Scutari, G., Battisti, T.; “DistributedSignal Subspace Projection Algorithms with MaximumConvergence Rate for Sensor Networks with Topolog-ical Constraints”, IEEE International Conference on

Acoustics, Speech and Signal Processing Taipei, Taiwan(2009).

[5] Insausti, X.; Crespo, P.M.; Beferull-Lozano, B.; , “In-Network Computation of the Transition Matrix for Dis-tributed Subspace Projection,” IEEE International Con-

ference on Distributed Computing in Sensor Systems.DCOSS 2012 , vol., no., pp.124-131, 16-18 May 2012.

[6] Camaro-Nogues F., Alonso-Roman D., Asensio-MarcoC., Beferull-Lozano B. “Reducing the observation errorin a WSN through a consensus-based subspace projec-tion,” IEEE Wireless Communications and Networking

Conference. WCNC 2013.

[7] L. Xiao; S. Boyd, “Fast linear iterations for distributedaveraging,” IEEE Conference on Decision and Control,CDC 2003., vol.5, no., pp. 4997-5002 Vol.5, 9-12, De-cember 2003.

[8] Pereira, S.S.; Pages-Zamora, A.; , “Mean Square Con-vergence of Consensus Algorithms in Random WSNs,”IEEE Transactions on Signal Processing, vol.58, no.5,pp.2866-2874, May 2010.

[9] Porfiri, M.; Stilwell, D.J.; , ”Consensus Seeking OverRandom Weighted Directed Graphs,” Automatic Control,IEEE Transactions on , vol.52, no.9, pp.1767-1773, Sept.2007.

[10] Reyes, C.; Hilaire, T.; Mecklenbrauker, C.F.; , ”Dis-tributed Projection Approximation Subspace Trackingbased on consensus propagation,” Computational Ad-vances in Multi-Sensor Adaptive Processing (CAMSAP),2009 3rd IEEE International Workshop on , vol., no.,pp.340-343, 13-16 Dec. 2009

978-1-4673-5577-3/13/$31.00 ©2013 IEEE

2013 IEEE 14th Workshop on Signal Processing Advances in Wireless Communications (SPAWC)

349