Research Article A Low-Complexity Approach for Improving

Research ArticleA Low-Complexity Approach for Improvingthe Accuracy of Sensor Networks

Angelo Coluccia

Dipartimento di Ingegneria dellrsquoInnovazione Universita del Salento Via Monteroni 73100 Lecce Italy

Correspondence should be addressed to Angelo Coluccia angelocolucciaunisalentoit

Received 31 March 2015 Revised 3 June 2015 Accepted 10 June 2015

Academic Editor Federico Barrero

Copyright copy 2015 Angelo Coluccia This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The paper addresses the problem of improving the accuracy of the measurements collected by a sensor network where simplicityand cost-effectiveness are of utmost importance An adaptive Bayesian approach is proposed to this aimwhich allows improving theaccuracy of the delivered estimates with no significant increase in computational complexity Remarkably the resulting cooperativealgorithm does not require prior knowledge of the (hyper)parameters and is able to provide a ldquodenoisedrdquo version of the monitoredfield without losing accuracy in detecting extreme (less frequent) values which can be very important for a number of applicationsA novel performance metric is also introduced to suitably quantify the capability to both reduce the measurement error and retainhighly-informative characteristics at the same time The performance assessment shows that the proposed approach is superior toa low-complexity competitor that implements a conventional filtering approach

1 Introduction and Motivations

In the last years sensor networks have started to be deployedfor an increasing number of different applications [1] Theavailability of low-cost commercial off-the-shelf nodes fos-tered by significant advances in wireless communicationtechnologies and size scaling of integrated circuits hasenabled the deployment of small low-cost sensor nodeswith increased lifetime [2] Typical applications are sensingestimation of some parameters [3 4] such as temperaturepollution level [5] electromagnetic exposure [6 7] or fieldreconstruction [8 9] Such problems are particularly impor-tant in environmental monitoring [10] ecology [11] meteo-rology agriculture and related fields as reported in a numberof case studies [12 13] see also [14] and references thereinMore in general sensing capabilities are currently regardedas a key enabler for smart applications in contexts as diverseas transportation systems [15ndash17] cyber-physical systems[18ndash20] and ad hoc networks [21] and in (opportunistic)applications like position estimation for location awareness[22ndash25] Finally interconnection of standalone systems canlead to advanced sensing capabilities for example in radarapplications [26]

Both centralized and distributed approaches can beadopted to process the information collected through a sen-sor network [27] In the centralized approach data are sent toa fusion center (FC) performing the whole computation [28]while in the distributed one neighboring nodes cooperate ina peer-to-peer fashion until convergence [29 30] Regardlessof implementation aspects the goal can be formalized as aninference problem based on sensor observations about anunderlying physical phenomenon Clearly observations areaffected by errors introduced in the sensingmeasurementprocess for instance due to thermal noise atmosphericeffects and residual sensor calibration errors This is espe-cially true when data are collected through general-purposedevices even smartphones [31] instead of dedicated (expen-sive) sophisticated sensors Therefore techniques aimed atimproving the accuracy of sensor measurements are highlydesirable [32 33]

Unfortunately a peculiarity of sensor networks is thateach sensor has quite limited power supply and computationcapabilities since advanced processing techniques cannot beoften implementedwith reduced effort novel low-complexityapproaches are needed to actually make sensor networkapplications feasible in most real-world contexts As a matter

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015 Article ID 521948 12 pageshttpdxdoiorg1011552015521948

2 International Journal of Distributed Sensor Networks

of fact systems designed to be effectively deployable have toface a number of practical difficulties thus things are kept assimple and cheap as possible this however may negativelyhave an impact on the final accuracy [12 34] In particular acoarse granularity may be reported on reconstructed sensingmaps with resulting sharp edges between contiguous levels(as eg in [16]) In other cases values are averaged [5]or if their spatial distribution is of concern smoothed viasuitable low-pass filters [35] A simple moving average isoften used in practice which can be easily implemented asa correlation with a weight mask through a sliding windowapproach

In this work we propose a low-complexity Bayesianapproach for improving the accuracy of the measurementsso that amore reliable value of themonitored field is obtainedwithout the need for a sophisticated processingWe will showthat with other things being equal few lines of code canbe effective to improve the final accuracy if the informationcoming from other sensors is exploited in a cooperativeway The advantage of this approach is that the ldquofilteringrdquoprocedure takes into account the statistical relationships inthe data at hand We consider a quite general observationmodel where the measurement error is modeled through aGaussian lawThe latter is a versatile model for measurementerrors and other randomeffects [36] supported by the centrallimit theorem (CLT) A Bayesian approach is used to estimatethe value of the field byminimizing themean squared error ateachmonitoring point Different from conventional Bayesiantechniques which require full prior knowledge of the datadistribution we follow an Empirical Bayes approach [37]where the parameters of the prior (hyperparameters) areunknown We derive their Maximum Likelihood (ML) esti-mator and show that it has a simple closed form amenable topractical implementation in low-cost devices An applicationto spatial field monitoring and reconstruction is reportedto highlight the performance improvement compared to aconventional ldquodenoisingrdquo technique based on low-pass two-dimensional filtering (moving average)

The rest of the paper is organized as follows In Section 2we formulate the problem within the reference scenarioThen in Section 3 we introduce the proposed filteringapproach which includes the derivation of the MinimumMean Square Error (MMSE) estimator of the field and theMaximumLikelihood (ML) estimator of the hyperparemtersBesides the mathematical derivation we provide also ascheme (and the corresponding algorithmrsquos pseudocode) forpractical application In Section 4 we evaluate the proposedapproach showing that it can improve the estimation accu-racywithout significantly increasing the complexity To betterspotlight the ability to represent correctly specific character-istics of the monitored field a novel metric is preliminarilyintroduced Finally Section 5 contains the conclusions of thework

2 Problem Formulation

A general scenario is considered where 119873 sensor nodesobserve a given phenomenon We denote by 119909

119894 119894 = 1 119873

Possible linksFusion center (if any) Sensor node (monitoring point)

Monitored field (shades = intensity)

Figure 1 Reference scenario of a monitoring sensor network

the measurement at sensor (equivalently location) 119894 relativeto an unknown local parameter119898

119894 that is

119909119894= 119898119894+ 120598119894 (1)

where 120598119894is a zero-mean error term with variance 1205902 Lacking

specific models it is customary to consider 120598119894as normally

distributed result of the CLT The reference scenario isdepicted in Figure 1 The monitored field is represented withshades proportional to the intensity of the parameter underinvestigation Circles indicate the locations of the deployedsensors with (some of the) links reported as edges of a(time-varying) graph which depends on the communicationpolicies of the deployed network and on environmentalconditions Measurements may be sent to a fusion centerif the processing is centralized in a distributed setup con-versely nodes will exchange status updates by combiningtheir measurement with the status updates of neighboringnodes until convergence

One can exploit the fact that sensors are deployed overa ldquocontinuousrdquo field for monitoring purposes hence theymeasure a variable which represents a ldquosamplingrdquo of theunderlying whole process As a consequence some correla-tion can be expected according to the spatial proximity (dis-tance) between sensors However correlations are difficultto model exactly since they require deep knowledge of theprocess at hand Moreover they are very site-specific andchange with time Complicated models if available requirein turn computational-intensive techniques conversely asmentioned simple approaches are needed in low-cost sensornetworks

To this aim we propose modelling the relationshipsbetween the sensed points of the field by means of a priordistribution on the measurements with unknown hyperpa-rameters In particular given the Gaussian model for 119909

119894

International Journal of Distributed Sensor Networks 3

we consider a conjugate prior distribution for the mean 119898119894

which is again a Gaussian distribution Thus the model canbe written as

119909119894| 119898119894sim N (119898

119894 120590

2)

119898119894sim N (120583 ]2)

(2)

The goal is to recover each 119898119894 based on the statistic 119909

119894

through the lens of the Bayesian hierarchy that is to enhancethe estimation quality by considering the probability ofobserving specific field values and computing the estimatorthat minimizes the mean squared error (MMSE) Howeverthe fact that the hyperparameters (120583 ]2) are unknownmakesthis approach different from the classical Bayesian frame-work where the prior distribution is completely known Inparticular in Section 32 we derive in closed form the Max-imum Likelihood (ML) estimators for (120583 ]2) following theso-called Empirical Bayes (EB) rationale [37] To this aim weconsider the whole statistic x = [1199091 sdot sdot sdot 119909

119873]119879 and exploit the

fact that observations share the same probability distributionThis allows fitting the Bayesian model by the most suitable(hyper)parameters which are ldquolearnedrdquo adaptively from thedata thus allowing for generality Beforehand we derivebelow the MMSE estimator for the value of the field at thesensor locations

3 Empirical Bayes-BasedMeasurement Filtering

31 MinimumMean Square Error Field Estimation The con-ditional distribution of 119909

119894given119898

119894is

119901 (119909119894| 119898119894) =

1radic21205871205902

expminus(119909119894minus 119898119894)2

21205902 (3)

and as mentioned we consider a normal prior for119898119894so that

119901 (119898119894) =

1radic2120587]2

expminus(119898119894minus 120583)

2

2]2 (4)

We can derive the joint probability distribution of 119909119894and119898

119894

119901 (119909119894 119898119894) = 119901 (119909

119894| 119898119894) 119901 (119898

119894) =

12120587120590]

sdot expminus12[(

1]2

+11205832)119898

2119894minus 2(

119909119894

1205902+120583

]2)119898119894+1205832

]2

+1199092119894

1205902]

(5)

TheMinimumMean Square Error (MMSE) estimator of119898119894is

obtained by computing the conditional mean [37] thereforeit is first necessary to calculate the posterior distribution of119909119894 according to Bayesrsquo theorem

119901 (119898119894| 119909119894) =

119901 (119909119894 119898119894)

119901 (119909119894)

(6)

The unconditional distribution of 119909119894 119901(119909119894) can be obtained

from the joint distribution 119901(119909119894 119898119894) by integrating119898

119894out

119901 (119909119894) = int

+infin

minusinfin

119901 (119909119894 119898119894) d119898119894

=1

2120587120590]expminus1

2(1205832

]2+1199092119894

1205902) 119868119909119894

(7)

where

119868119909119894

= int

+infin

minusinfin

exp minus12(1]2

+11205902)119898

2119894+(

120583

]2+119909119894

1205902)119898119894 d119898119894

(8)

The integral above can be computed by completing the squareto a Gaussian kernel After some calculation the result is

119868119909119894= exp

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)

radic2120587

radic(1205902 + ]2) ]21205902 (9)

Substituting (9) into (7) yields

119901 (119909119894) =

1

radic2120587 (1205902 + ]2)

sdot exp

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)

(10)

Using (6) (together with (5) and (10)) we obtain the finalexpression of the posterior distribution

119901 (119898119894| 119909119894) =

1

radic2120587 (1205902]2 (1205902 + ]2))

sdot exp

minus

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)minus12(1]2

+11205902)119898

2119894

+(120583

]2+119909119894

1205902)119898119894

=1

radic2120587 (1205902]2 (1205902 + ]2))

sdot exp

minus

(119898119894minus (120583120590

2+ 119909119894]2) (1205902 + ]2))

2

2 (1205902]2 (1205902 + ]2))

(11)

which can be recognized as a Gaussian distribution withmean (1205902120583+]2119909

119894)(120590

2+]2) and variance1205902]2(1205902+]2) Such a

result is important because it yields a very tractable posteriordistribution

Given the result above the MMSE is obtained in a simpleclosed form as

119894= E [119898

119894| 119909119894] =

1205902120583 + ]2119909

119894

1205902 + ]2= 120574120583+ (1minus 120574) 119909

119894 (12)


where

120574 =1205902

1205902 + ]2isin (0 1) (13)

Equation (12) has a revealing interpretation as convex combi-nation of the measurement 119909

119894and the prior mean 120583 accord-

ing to the parameter 120574 The latter depends on the relativeimportance of the measurement uncertainty (quantified bythe variance 1205902) with respect to the overall uncertainty (ieincluding also the prior variance ]2) It is obvious thereforethat the hyperparameters (120583 ]2) cannot be assumed known apriori as in the conventional Bayesian approach we proposeinstead fitting them to the observed data by leveraging coop-eration between the nodes that is using the whole statisticx In particular in the next section we derive the MaximumLikelihood (ML) estimator of the hyperparameters finallyobtaining the so-called Empirical Bayes (MMSE) estimatorfor119898119894by replacing the estimates (120583 ]2) into (12)

32 Maximum Likelihood Hyperparameter Estimation Thejoint probability distribution of the sensor measurements isobtained from (10) as

119901 (x | 120583 ]2) =119873

prod

119894=1119901 (119909119894| 120583 ]2) = [2120587 (1205902 + ]2)]

minus1198732

sdot exp

119873

sum

119894=1

[

[

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)]

]

(14)

where the conditioning stresses the fact that the hyperparam-eters are the unknown quantities to be estimated

It is easy to verify that the expression between squarebrackets above is simplified as minus(12(1205902 + ]2))(119909

119894minus 120583)

2 sothat (14) can be rewritten as

119901 (x | 120583 ]2)

= [2120587 (1205902 + ]2)]minus1198732

expminussum119873

119894=1 (119909119894 minus 120583)2

2 (1205902 + ]2)

(15)

to be maximized with respect to (120583 ]2) It is convenient torewrite this optimization problem after a logarithmic trans-formation negating the result and omitting an irrelevantterm we obtain the equivalent problem

(120583 ]2)

= argmin120583isinR]2ge0

119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2

(16)

We can write the Karush-Kuhn-Tucker conditions [38] forthis constrained problem as follows

2119873

sum

119894=1(119909119894minus120583) = 0

119873

1205902 + ]2minussum119873

119894=1 (119909119894 minus 120583)2

(1205902 + ]2)2minus120582 = 0

minus ]2 le 0 120582]2 = 0 120582 ge 0

(17)

where the first two equations are the derivative of theLagrangian function

L (120583 ]2 120582) = 119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2minus120582]2 (18)

with respect to 120583 and ]2 the third line is the positivityconstraint on the variance (primal feasibility) the fourthequation is the complementary slackness condition and thelast inequality is the dual feasibility condition From the firstequation we obtain

120583 =1119873

119873

sum

119894=1119909119894 (19)

that is the sample mean denoted by 119909 From the feasibilityconditions it is easy to realize that the Lagrange multiplier 120582is null at the stationary point if ]2 gt 0 in that case from thesecond equation we obtain ]2 = 119904

2minus 120590

2 where

1199042=

1119873

119873

sum

119894=1(119909119894minus120583)

2 (20)

denotes the sample variance Any 120582 gt 0 necessarily implies]2 = 0 in force of the complementary slackness The MLestimator of ]2 is therefore given by the following expression

]2 = max (0 1199042 minus1205902) (21)

This automatically accounts for the case 1199042 minus 1205902lt 0 which

would lead to a (unfeasible) negative value for ]2

33 Practical Application of the Algorithm The approachdeveloped above can be applied in a straightforward way toa monitoring network deployed in a given area A schematicrepresentation is depicted in Figure 2 which details thescheme of Figure 1 Some of the nodes are exploded toindicate the local processing which can be summarized as afunction119891 of the local measurement and of the ldquostaterdquo (119909 1199042)The latter is the result of the cooperation in the network andas mentioned can be computed by a fusion center by meansof a distributed processing Clearly we have that

119891 (119909119894 119909 119904

2) = 120574119909+ (1minus 120574) 119909

119894 120574 = min(1 120590

2

1199042) (22)

where the expression for 120574 has been obtained by simplemanipulations of (13)


Centralized or distributed

computation

x s2

x s2

x s2

x1

x2

xN

f(x1 x s2)

f(xN x s2)

f(x2 x s2)

Figure 2 Schematic representation of the proposed estimationapproach

It is reasonable to expect that except for small-areanetworks nodes may not be able to communicate in a com-pletely meshed way but rather they have limited connectivitydictated by their communication range For each node 119894we can define the set of neighbors 119873

119894(119905) according to the

coverage area allowed by the communication technologyavailable in the network then the proposed algorithm can beapplied locally with each node exchanging a limited numberof communication packets to neighboring nodes in orderto calculate the value of the ldquostaterdquo (119909 1199042) for its local areathat is by interacting with nodes 119896 isin 119873

119894(119905) Clearly each

node may contribute (as neighbor) to the calculation ofdifferent nodes according to the extent of the coverage areasAt limit one could even have a single area enclosing all 119873nodes otherwise a certain number of ldquoclustersrdquo will arisewhere the (possibly distributed) computation is performedindependently though it may share some information withgateway nodes This means that the approach is consistentand very scalable

Summing up despite the lenghty calculation in Sections31 and 32 (necessary for a rigorous derivation) the result isvery handy and can be easily implemented in a few lines ofcode for a generic cluster as follows

Pseudocode of the Proposed Algorithm

compute 119909=(1119873)sum119873

119894=1 119909119894 and 1199042=(1119873)sum

119873

119894=1(119909119894minus

119909)2

compute 120574 = min(1 12059021199042)compute

119894= 120574119909 + (1 minus 120574)119909

119894at each node 119894

4 Performance Assessment

In this section we show how the proposed approach can beused to improve the accuracy of a sensor network withoutsignificantly increasing the computational cost We resort tosimulations to control the ground truth that is we simulatethe true value of the field (of some physical quantity namelytemperature) plus additive noise that models measurement

errors Performance are assessed as function of the power ofthe noise that is the variance 1205902 of the Gaussian disturbance

41 A Novel Metric for Accuracy Evaluation In order toreduce the error introduced in the measurement pro-cess a smoothing filter is typically used However simpleapproaches actually used in real networks just rely ontechniques that ignore the statistical properties of the datain particular data are often processed through a slidingwindow where measurements are low-pass filtered to reducethe disturbance similarly to a basic image denoising algo-rithm Although this approach provides reasonable results onaverage it may negatively have an impact on the values thatdeviate from the mean The latter are conversely the mostinteresting data in a number of monitoring applications forinstance to detect extreme events As a result it is importantto measure the ability of a smoothing approach to retain thelow-probability characteristics of the monitored field

On the other hand an algorithm that focuses too muchon extreme events tends to lose its ability to ldquodenoiserdquo areaswhere the field is almost stationary that is plateaux withvery similar values that fluctuate just because of the intrinsicuncertainty introduced by the measurement process To cor-rectly evaluate the overall performance thus it is necessarythat the performance metric takes into account both theseconflicting objectives that is ability to retain low-probabilityvalues and ability to simultaneously ensure a satisfactorysmoothing

To this aim in the following we propose a novel com-pound metric based on a weighted version of the Frobeniusnorm More precisely denoting by M the matrix of the truefield values 119898

119894119895 where (119894 119895) indicates a possible location of

the monitored area (in this section we use matrix notation tolink more clearly the measured value to the correspondinglocation where it has been measured therefore we need apair of indices instead of the single index 119894 formerly usedto denote node-119894-related quantities) the overall error in thereconstructed field M (which is function of the measurementmatrix X) can be evaluated as

1198631 (M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

(23)

where sdot 119865is the Frobenius norm that is the sum of the

squares of all elements of the matrix argument In order tospotlight deviations in particular points however aweightingmatrixmust be introducedWe therefore define the followingmetric

119863119882(M) =

10038171003817100381710038171003817W ∘ (MminusM)

10038171003817100381710038171003817119865 (24)

where ∘ is the Hadamard (entry-wise) matrix product andWis a matrix of weights Clearly the metric1198631(sdot) is obtained asparticular case of119863

119882(sdot) for all-ones matrixW

Based on the twometrics above we propose the followingcompound metric

119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817I (M) ∘ (MminusM)

10038171003817100381710038171003817119865 (25)

where I(M) denotes the matrix of self-information for thefield that is 119868

119894119895= log(1119875

119894119895) = minuslog119875

119894119895 where P = [119875

119894119895] is


2

4

6

8

10

12

14

16

18

20

(a)

0 5 10 15 20 250

010203040506070809

1

Prob

abili

ty

0 5 10 15 20

0

001

002

003

004

minus5

(b)

0

2

4

6

8

10

12

14

16

18

20

(c)

Figure 3 Example of monitoring application scenario (a) ground truth (matrix M) (b) histogram of the values (leading to matrix P) and(c) noisy version (measurement matrix X) for 1205902

= 1

(an estimate of) the probability distribution of the field Thismeans that P is such thatsum119873

119894=1sum119873

119895=1 119875119894119895 = 1 More precisely Pcan be taken as the normalized histogram of the true valuesof the field M resulting from some binning rule (Since thisinformation is not available to the algorithm it is not possibleto design an estimation scheme that minimizes the metric119863 The use of 119863 here is only for the purpose of performanceassessment)

The definition in (25) can be rewritten as the product ofthe distance between the real field and the reconstructed onein both the original and the transformed (weighted) spaces

119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817M1015840 minusM101584010038171003817100381710038171003817119865 (26)

where M1015840 = I(M) ∘ M and M1015840 = I(M) ∘ M can beinterpreted as the transformed versions (through I(M)) of Mand M respectively this highlights the ability of the metricto account for both aspects discussed above

In Figure 3(a) we show a clarifying example the truevalues of the fieldM change very slowly but for a region near

the left-bottom corner where an event has occurred that per-turbs the field (eg accidental overheating) The histogramof the field values is reported in Figure 3(b) it shows thatmore than 90 of the values fall in the first bin (rangingfrom 0 to about 25) while only few points are distributedover the rest of the span (the inset reports a zoomed versionof the distribution tail for a better visualization) In thisexample an algorithm that completely misses to accuratelyrepresent higher values but retains the lower ones wouldhave a very good score for the metric 1198631 however it wouldperformbadly through the lens of themetric119863

119868(ie119863

119882with

W = I(M)) which weighs more heavily the points where theinformation is high Accurate estimation of values deviatingfrom the expected level is in fact of utmost importance inmany cases namely for early detection of extreme eventswith alarms typically triggered when exceeding a thresholdOn the other hand what we can measure is not M butrather a noisy version X as the one reported in Figure 3(c)the algorithm thus must be able to also smooth out thevariations due to noise so as to not increase the false alarm


Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20

Figure 4 Result comparison for the proposed algorithm versus the moving average 1205902= 1

rate due to values accidentally exceeding the threshold Tothis aim the metric 1198631 is more appropriate as indicatorof the goodness of the smoothing process As mentionedthese aspects are both taken into account in 119863 resulting ina compound metric that reflects both the estimation rootmean square error and the accuracy in representing less likely(hence more informative) values

We have applied the proposed Empirical Bayes approachto a sensor network deployed on the field shown in Fig-ure 3(a) Following the cluster-based approach describedabove we have run the algorithm on the noisy version result-ing from the measurement process as function of the noiselevel 1205902 We use normalized units in order to have a directmapping between locations and index pairs in the matrixnotation Nodes embed a technology with communicationrange limited to a few units in particular we consideredmeasurements from nodes closer than 3 units in range whichresulted in a minimum number of neighbors equal to 11 thatis119873119894(119905) ge 11

As mentioned as competitor we consider the conven-tional moving-average filter which is a typical smoothing(ldquodenoisingrdquo) technique to improve the quality of sensedimages [39] Such an algorithm uses a spatial mask centeredin each location (119909 119910) with constant weights and performsthe local average by moving across the whole field thiscorresponds to the two-dimensional convolution betweenthe matrix X representing the measured field and the maskand produces a low-pass filtered output where noise hasbeen reduced while sharp edges have been better preservedcompared to nonconstant weight masks

The result of the filtering for the case of Figure 3(c)which is related to 120590

2= 1 is reported in Figure 4 together

with the ground truth and the competitor moving average(on the same data and neighbor sets) The figure clearlyshows that the proposed approach is able to smooth themeasured data reconstructing (an estimate of) the field in away that also preserves the information about low-probabilityevents that is the ones with higher values but occurring


0

01

02

03

04

05

06

07

08

09

1

Figure 5 Values of 120574 for the different points of the field 1205902= 1

much less frequently On the contrary the moving averagealgorithm treats all locations in the same way smoothing outtoo much the left-bottom corner as a result it completelymisses the higher values of the fieldThis behavior is reflectedby the metric 119863 which takes on a much smaller value forthe Empirical Bayes than for the competitor The proposedapproach is able to adaptively ldquolearnrdquo from the data how thefield varies in the local region from a statistical point of viewan information that is ultimately condensed in the parameter120574 The values of the latter at the different locations reportedin Figure 5 reveal that the algorithm can adaptively identifythe points where the measurement must be prevalent (ie120574 asymp 0 dark locations) treating them differently from theones where a smoothing can be safely performed due to thelimited variability (ie 120574 asymp 1 light locations) Intermediateconditions are automatically accommodated (shades of gray)

42 Numerical Results It is worth noticing that the proposedEmpirical Bayes approach leads to a better value of119863 but it isalso superior on 1198631 and 119863119868 individually as it can be noticedby comparing the individual scores in Figure 4

By increasing the noise level it turns out that the pro-posed algorithm still outperforms the competitor and fur-thermore remains able to retain the highly informative (low-probability) valueswhile ensuring a satisfactory smoothing ofthe noise Results for 1205902 = 4 are reported in Figures 6 and 7Thevalue of119863has increased of course but is still significantlylower than (the noisy version and) the competitor one Wenotice however that the gap with respect to the latter isslightly less

To investigate more thoroughly this point we haveevaluated how 119863 varies with 1205902 For a better understandingthe metrics 1198631 and 119863

119868are also analyzed We compare the

proposed approach against the moving average consideringas reference the values of the metrics calculated on theraw measurements (X) In Figure 8 we report the resultingcurves (from (a) to (c) 119863 119863

119868 and 1198631) as clearly visible

the proposed approach outperforms the competitor in thewhole span of 1205902 The two algorithms are comparable only in

terms of 1198631 at very high noise level when the measurementsare unreliable hence no great improvement is possiblein terms of denoising compared to a low-pass filteringHowever it is also evident that the Empirical Bayes approachalways exhibits the best accuracy in terms of 119863

119868 In other

words while the overall performance obviously degrade as1205902 increases the moving average algorithm shows a roughly

constant 119863119868 meaning that even for very accurate measure-

ments (low1205902) such an algorithm is not able to retain the low-

probability values Therefore it comes as little surprise thatwhenmeasurements are accurate enough themoving averageapproach is worse (in terms of119863) than simply taking the rawmeasurements The proposed approach conversely yieldsbest performance by automatically adapting its parameter 120574to the different contexts

Finally the case of multiple ldquohot spotsrdquo that is multiplesources of far-from-average spikes is analyzed in Figure 9 Inaddition to the already observed properties of the proposedapproach which remain valid here an additional featurecan be observed in terms of resilience to masqueradingeffects Indeed when multiple sources of far-from-averageevents are present conventional filtering techniques like themoving average do not have enough resolution power evenformoderate noise levelThis is due to the averaging of pointsin spatial proximity so that sources not sufficiently separatedbecome indistinguishable and collapse onto a same blurredblob (as in the top right corner of the field in Figure 9)Conversely the proposed approach is able to detect all sourcesand also to maintain the proportionality in their intensitywhile ensuring denoising

5 Conclusions

The problem of improving the accuracy of the measurementscollected by a sensor network has been addressed Aimingat simplicity and cost-effectiveness which are of utmostimportance in application contexts where sensor networksare often deployed a low-complexity automatic approachhas been proposed which does not require manual setting


Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60

Raw measurementsMoving averageEmpirical Bayes

D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)

Figure 8 Comparison between the rawmeasurements proposed Empirical Bayes algorithm and the competitor algorithm (moving average)from (a) to (c) the metrics119863119863

119868 and1198631 for varying 120590

2

of parameters nor recalibration By following an adaptive(Empirical Bayes) rationale the algorithm is able to improvethe estimation accuracy by leveraging cooperation betweennodes Remarkably it can provide a ldquodenoisedrdquo version of

the monitored field without losing accuracy in detecting lessprobable values Using a novel performance metric the capa-bility to both reduce themeasurement error and retain highlyinformative characteristics has been quantified revealing that


Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517

Figure 9 Result comparison for the proposed algorithm versus the moving average in case of multiple far-from-average events 1205902= 1

the proposed approach can outperform conventional low-pass filtering

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] I F Akyildiz W Su Y Sankarasubramaniam and E Cayirci ldquoAsurvey on sensor networksrdquo IEEE Communications Magazinevol 40 no 8 pp 102ndash114 2002

[2] Y Dingcheng W Zhenghai X Lin and Z Tiankui ldquoOnlinebayesian data fusion in environment monitoring sensor net-worksrdquo International Journal of Distributed Sensor Networksvol 2014 Article ID 945894 10 pages 2014

[3] Y-R Tsai and C-J Chang ldquoCooperative information aggre-gation for distributed estimation in wireless sensor networksrdquoIEEE Transactions on Signal Processing vol 59 no 8 pp 3876ndash3888 2011

[4] A Agarwal and A K Jagannatham ldquoDistributed estimation inhomogenous poisson wireless sensor networksrdquo IEEE WirelessCommunications Letters vol 3 no 1 pp 90ndash93 2014

[5] A Kadri E Yaacoub M Mushtaha and A Abu-Dayya ldquoWire-less sensor network for real-time air pollution monitoringrdquo inProceedings of the 1st International Conference on Communi-cations Signal Processing and Their Applications (ICCSPArsquo 13)February 2013

[6] N Djuric M Prsa and K Kasas-Lazetic ldquoInformation networkfor continuous electromagnetic fields monitoringrdquo Interna-tional Journal of Emerging Sciences vol 1 no 4 pp 516ndash5252011

[7] D Hasenfratz S Sturzenegger O Saukh and L Thiele ldquoSpa-tially resolved monitoring of radio-frequency electromagneticfieldsrdquo in Proceedings of the 1st International Workshop onSensing and Big Data Mining (SenseMine rsquo13) pp 1ndash6 ACMRome Italy November 2013

[8] I Nevat G W Peters and I B Collings ldquoRandom fieldreconstruction with quantization in wireless sensor networksrdquoIEEE Transactions on Signal Processing vol 61 no 23 pp 6020ndash6033 2013


[9] I Bahceci and A K Khandani ldquoLinear estimation of correlateddata in wireless sensor networks with optimum power alloca-tion and analog modulationrdquo IEEE Transactions on Communi-cations vol 56 no 7 pp 1146ndash1156 2008

[10] T Wark D Swain C Crossman P Valencia G Bishop-Hurleyand R Handcock ldquoSensor and actuator networks protectingenvironmentally sensitive areasrdquo IEEE Pervasive Computingvol 8 no 1 pp 30ndash36 2009

[11] P W Rundel E A Graham M F Allen J C Fisher andT C Harmon ldquoEnvironmental sensor networks in ecologicalresearchrdquo New Phytologist vol 182 no 3 pp 589ndash607 2009

[12] G Barrenetxea F Ingelrest G Schaefer and M VetterlildquoWireless sensor networks for environmental monitoring theSensorScope experiencerdquo in Proceedings of the InternationalZurich Seminar on Communications (IZS rsquo08) pp 98ndash101 IEEEZurich Switzerland March 2008

[13] X Jiang G Zhou Y Liu and Y Wang ldquoWireless sensornetworks for forest environmental monitoringrdquo in Proceedingsof the IEEE 7th International Conference onE-Science December2011

[14] P Corke T Wark R Jurdak W Hu P Valencia and D MooreldquoEnvironmental wireless sensor networksrdquo Proceedings of theIEEE vol 98 no 11 pp 1903ndash1917 2010

[15] J Guevara F Barrero E Vargas J Becerra and S Toral ldquoEnvi-ronmental wireless sensor network for road traffic applicationsrdquoIET Intelligent Transport Systems vol 6 no 2 pp 177ndash186 2012

[16] P McDonald D Geraghty I Humphreys and S FarrellldquoAssessing environmental impact of transport noise with wire-less sensor networksrdquoTransportation Research Record no 2058pp 133ndash139 2008

[17] A Coluccia and G Notarstefano ldquoDistributed Bayesian estima-tion of arrival rates in asynchronous monitoring networksrdquo inProceedings of the IEEE International Conference on AcousticsSpeech and Signal Processing (ICASSPrsquo 14) pp 5050ndash5054Florence Italy May 2014

[18] T-Y Wang L-Y Chang and P-Y Chen ldquoA Collaborativesensor-fault detection scheme for robust distributed estimationin sensor networksrdquo IEEETransactions onCommunications vol57 no 10 pp 3045ndash3058 2009

[19] A Coluccia and G Notarstefano ldquoA hierarchical Bayesapproach for distributed binary classification in cyber-physicaland social networksrdquo in Proceedings of the 19th World Congressof the IFAC August 2014

[20] G Lo Re F Milazzo and M Ortolani ldquoA distributed Bayesianapproach to fault detection in sensor networksrdquo in Proceedingsof the IEEE Global Communications Conference (GLOBECOMrsquo12) pp 634ndash639 Anaheim Calif USA December 2012

[21] A Coluccia and G Notarstefano ldquoDistributed estimation ofbinary event probabilities via hierarchical Bayes and dualdecompositionrdquo in Proceedings of the 52nd Annual Conferenceon Decision and Control (CDC rsquo13) pp 6753ndash6758 IEEEFirenze Italy December 2013

[22] J J Cho Y Ding Y Chen and J Tang ldquoRobust calibrationfor localization in clustered wireless sensor networksrdquo IEEETransactions on Automation Science and Engineering vol 7 no1 pp 81ndash95 2010

[23] A Coluccia ldquoReduced-bias ML-based estimators with lowcomplexity for self-calibrating RSS rangingrdquo IEEE Transactionson Wireless Communications vol 12 no 3 pp 1220ndash1230 2013

[24] A Coluccia and F Ricciato ldquoRSS-Based localization viabayesian ranging and iterative least squares positioningrdquo IEEECommunications Letters vol 18 no 5 pp 873ndash876 2014

[25] A Coluccia and G Ricci ldquoA tunable W-ABORT-like detectorwith improved detection vs rejection capabilities trade-offrdquoIEEE Signal Processing Letters vol 22 no 6 pp 713ndash717 2015

[26] A Coluccia and G Ricci ldquoABORT-like detection strategies tocombat possible deceptive ECM signals in a network of radarsrdquoIEEE Transactions on Signal Processing vol 63 no 11 pp 2904ndash2914 2015

[27] J Li and G AlRegib ldquoDistributed estimation in energy-constrained wireless sensor networksrdquo IEEE Transactions onSignal Processing vol 57 no 10 pp 3746ndash3758 2009

[28] G Quer R Masiero G Pillonetto M Rossi and M ZorzildquoSensing compression and recovery for WSNs sparse signalmodeling and monitoring frameworkrdquo IEEE Transactions onWireless Communications vol 11 no 10 pp 3447ndash3461 2012

[29] I D Schizas A Ribeiro and G B Giannakis ldquoConsensus inad hoc WSNs with noisy links I Distributed estimation ofdeterministic signalsrdquo IEEE Transactions on Signal Processingvol 56 no 1 pp 350ndash364 2008

[30] S Barbarossa S Sardellitti and P Di Lorenzo ldquoDistributeddetection and estimation in wireless sensor networksrdquo inCommunications and Radar Signal Processing R Chellappa andS Theodoridis Eds vol 2 pp 329ndash408 Academic Press 2014

[31] D Hasenfratz O Saukh S Sturzenegger and L ThieleldquoParticipatory air pollution monitoring using smartphonesrdquoin Proceedings of the 2nd International Workshop on MobileSensing p 1620 Beijing China August 2012

[32] Y Zhang N Meratnia and P Havinga ldquoOutlier detectiontechniques for wireless sensor networks a surveyrdquo IEEE Com-munications Surveys and Tutorials vol 12 no 2 pp 159ndash1702010

[33] Y Ma M Richards M Ghanem Y Guo and J Hassard ldquoAirpollution monitoring and mining based on sensor Grid inLondonrdquo Sensors vol 8 no 6 pp 3601ndash3623 2008

[34] L M L Oliveira and J J P C Rodrigues ldquoWireless sensornetworks a survey on environmental monitoringrdquo Journal ofCommunications vol 6 no 2 pp 143ndash151 2011

[35] S De Vito and G Fattoruso ldquoWireless chemical sensor net-works for air quality monitoringrdquo in Proceedings of the 14thInternational Meeting on Chemical Sensors (IMCS rsquo12) 2012

[36] M A Osborne S J Roberts A Rogers and N R JenningsldquoReal-time information processing of environmental sensornetwork data using Bayesian Gaussian processesrdquo ACM Trans-actions on Sensor Networks vol 9 article 1 2012

[37] E L Lehmann and G Casella Theory of Point EstimationSpringer Berlin Germany 2nd edition 2008

[38] S Boyd andLVandenbergheConvexOptimization CambridgeUniversity Press Cambridge UK 2004

[39] PMather andMKochComputer Processing of Remotely-SensedImages An Introduction John Wiley amp Sons 4th edition 2011

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



of fact systems designed to be effectively deployable have toface a number of practical difficulties thus things are kept assimple and cheap as possible this however may negativelyhave an impact on the final accuracy [12 34] In particular acoarse granularity may be reported on reconstructed sensingmaps with resulting sharp edges between contiguous levels(as eg in [16]) In other cases values are averaged [5]or if their spatial distribution is of concern smoothed viasuitable low-pass filters [35] A simple moving average isoften used in practice which can be easily implemented asa correlation with a weight mask through a sliding windowapproach

In this work we propose a low-complexity Bayesianapproach for improving the accuracy of the measurementsso that amore reliable value of themonitored field is obtainedwithout the need for a sophisticated processingWe will showthat with other things being equal few lines of code canbe effective to improve the final accuracy if the informationcoming from other sensors is exploited in a cooperativeway The advantage of this approach is that the ldquofilteringrdquoprocedure takes into account the statistical relationships inthe data at hand We consider a quite general observationmodel where the measurement error is modeled through aGaussian lawThe latter is a versatile model for measurementerrors and other randomeffects [36] supported by the centrallimit theorem (CLT) A Bayesian approach is used to estimatethe value of the field byminimizing themean squared error ateachmonitoring point Different from conventional Bayesiantechniques which require full prior knowledge of the datadistribution we follow an Empirical Bayes approach [37]where the parameters of the prior (hyperparameters) areunknown We derive their Maximum Likelihood (ML) esti-mator and show that it has a simple closed form amenable topractical implementation in low-cost devices An applicationto spatial field monitoring and reconstruction is reportedto highlight the performance improvement compared to aconventional ldquodenoisingrdquo technique based on low-pass two-dimensional filtering (moving average)

The rest of the paper is organized as follows In Section 2we formulate the problem within the reference scenarioThen in Section 3 we introduce the proposed filteringapproach which includes the derivation of the MinimumMean Square Error (MMSE) estimator of the field and theMaximumLikelihood (ML) estimator of the hyperparemtersBesides the mathematical derivation we provide also ascheme (and the corresponding algorithmrsquos pseudocode) forpractical application In Section 4 we evaluate the proposedapproach showing that it can improve the estimation accu-racywithout significantly increasing the complexity To betterspotlight the ability to represent correctly specific character-istics of the monitored field a novel metric is preliminarilyintroduced Finally Section 5 contains the conclusions of thework

2 Problem Formulation

A general scenario is considered where 119873 sensor nodesobserve a given phenomenon We denote by 119909

119894 119894 = 1 119873

Possible linksFusion center (if any) Sensor node (monitoring point)

Monitored field (shades = intensity)

Figure 1 Reference scenario of a monitoring sensor network

the measurement at sensor (equivalently location) 119894 relativeto an unknown local parameter119898

119894 that is

119909119894= 119898119894+ 120598119894 (1)

where 120598119894is a zero-mean error term with variance 1205902 Lacking

specific models it is customary to consider 120598119894as normally

distributed result of the CLT The reference scenario isdepicted in Figure 1 The monitored field is represented withshades proportional to the intensity of the parameter underinvestigation Circles indicate the locations of the deployedsensors with (some of the) links reported as edges of a(time-varying) graph which depends on the communicationpolicies of the deployed network and on environmentalconditions Measurements may be sent to a fusion centerif the processing is centralized in a distributed setup con-versely nodes will exchange status updates by combiningtheir measurement with the status updates of neighboringnodes until convergence

One can exploit the fact that sensors are deployed overa ldquocontinuousrdquo field for monitoring purposes hence theymeasure a variable which represents a ldquosamplingrdquo of theunderlying whole process As a consequence some correla-tion can be expected according to the spatial proximity (dis-tance) between sensors However correlations are difficultto model exactly since they require deep knowledge of theprocess at hand Moreover they are very site-specific andchange with time Complicated models if available requirein turn computational-intensive techniques conversely asmentioned simple approaches are needed in low-cost sensornetworks

To this aim we propose modelling the relationshipsbetween the sensed points of the field by means of a priordistribution on the measurements with unknown hyperpa-rameters In particular given the Gaussian model for 119909

119894




119909119894| 119898119894sim N (119898

119894 120590

2)

119898119894sim N (120583 ]2)

(2)


119894






119894given119898

119894is

119901 (119909119894| 119898119894) =

1radic21205871205902

expminus(119909119894minus 119898119894)2

21205902 (3)


119901 (119898119894) =

1radic2120587]2

expminus(119898119894minus 120583)

2

2]2 (4)


119894

119901 (119909119894 119898119894) = 119901 (119909

119894| 119898119894) 119901 (119898

119894) =

12120587120590]

sdot expminus12[(

1]2

+11205832)119898

2119894minus 2(

119909119894

1205902+120583

]2)119898119894+1205832

]2

+1199092119894

1205902]

(5)



119901 (119898119894| 119909119894) =

119901 (119909119894 119898119894)

119901 (119909119894)

(6)



119894out

119901 (119909119894) = int

+infin

minusinfin

119901 (119909119894 119898119894) d119898119894

=1

2120587120590]expminus1

2(1205832

]2+1199092119894

1205902) 119868119909119894

(7)

where

119868119909119894

= int

+infin

minusinfin

exp minus12(1]2

+11205902)119898

2119894+(

120583

]2+119909119894

1205902)119898119894 d119898119894

(8)


119868119909119894= exp

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)

radic2120587

radic(1205902 + ]2) ]21205902 (9)


119901 (119909119894) =

1

radic2120587 (1205902 + ]2)

sdot exp

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)

(10)


119901 (119898119894| 119909119894) =

1

radic2120587 (1205902]2 (1205902 + ]2))

sdot exp

minus

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)minus12(1]2

+11205902)119898

2119894

+(120583

]2+119909119894

1205902)119898119894

=1

radic2120587 (1205902]2 (1205902 + ]2))

sdot exp

minus

(119898119894minus (120583120590

2+ 119909119894]2) (1205902 + ]2))

2

2 (1205902]2 (1205902 + ]2))

(11)


119894)(120590




119894= E [119898

119894| 119909119894] =

1205902120583 + ]2119909

119894

1205902 + ]2= 120574120583+ (1minus 120574) 119909

119894 (12)


where

120574 =1205902

1205902 + ]2isin (0 1) (13)





119901 (x | 120583 ]2) =119873

prod

119894=1119901 (119909119894| 120583 ]2) = [2120587 (1205902 + ]2)]

minus1198732

sdot exp

119873

sum

119894=1

[

[

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)]

]

(14)



119894minus 120583)


119901 (x | 120583 ]2)

= [2120587 (1205902 + ]2)]minus1198732

expminussum119873

119894=1 (119909119894 minus 120583)2

2 (1205902 + ]2)

(15)


(120583 ]2)


119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2

(16)


2119873

sum

119894=1(119909119894minus120583) = 0

119873

1205902 + ]2minussum119873

119894=1 (119909119894 minus 120583)2

(1205902 + ]2)2minus120582 = 0

minus ]2 le 0 120582]2 = 0 120582 ge 0

(17)


L (120583 ]2 120582) = 119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2minus120582]2 (18)


120583 =1119873

119873

sum

119894=1119909119894 (19)


2minus 120590

2 where

1199042=

1119873

119873

sum

119894=1(119909119894minus120583)

2 (20)


]2 = max (0 1199042 minus1205902) (21)




119891 (119909119894 119909 119904

2) = 120574119909+ (1minus 120574) 119909

119894 120574 = min(1 120590

2

1199042) (22)




computation

x s2

x s2

x s2

x1

x2

xN

f(x1 x s2)

f(xN x s2)

f(x2 x s2)









compute 119909=(1119873)sum119873

119894=1 119909119894 and 1199042=(1119873)sum

119873

119894=1(119909119894minus

119909)2


119894= 120574119909 + (1 minus 120574)119909










1198631 (M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

(23)



119863119882(M) =

10038171003817100381710038171003817W ∘ (MminusM)

10038171003817100381710038171003817119865 (24)




119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817I (M) ∘ (MminusM)

10038171003817100381710038171003817119865 (25)


119894119895= log(1119875

119894119895) = minuslog119875

119894119895 where P = [119875

119894119895] is


2

4

6

8

10

12

14

16

18

20

(a)

0 5 10 15 20 250

010203040506070809

1

Prob

abili

ty

0 5 10 15 20

0

001

002

003

004

minus5

(b)

0

2

4

6

8

10

12

14

16

18

20

(c)


= 1


119894=1sum119873



119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817M1015840 minusM101584010038171003817100381710038171003817119865 (26)




119868(ie119863

119882with



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20









0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119909119894| 119898119894sim N (119898

119894 120590

2)

119898119894sim N (120583 ]2)

(2)


119894






119894given119898

119894is

119901 (119909119894| 119898119894) =

1radic21205871205902

expminus(119909119894minus 119898119894)2

21205902 (3)


119901 (119898119894) =

1radic2120587]2

expminus(119898119894minus 120583)

2

2]2 (4)


119894

119901 (119909119894 119898119894) = 119901 (119909

119894| 119898119894) 119901 (119898

119894) =

12120587120590]

sdot expminus12[(

1]2

+11205832)119898

2119894minus 2(

119909119894

1205902+120583

]2)119898119894+1205832

]2

+1199092119894

1205902]

(5)



119901 (119898119894| 119909119894) =

119901 (119909119894 119898119894)

119901 (119909119894)

(6)



119894out

119901 (119909119894) = int

+infin

minusinfin

119901 (119909119894 119898119894) d119898119894

=1

2120587120590]expminus1

2(1205832

]2+1199092119894

1205902) 119868119909119894

(7)

where

119868119909119894

= int

+infin

minusinfin

exp minus12(1]2

+11205902)119898

2119894+(

120583

]2+119909119894

1205902)119898119894 d119898119894

(8)


119868119909119894= exp

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)

radic2120587

radic(1205902 + ]2) ]21205902 (9)


119901 (119909119894) =

1

radic2120587 (1205902 + ]2)

sdot exp

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)

(10)


119901 (119898119894| 119909119894) =

1

radic2120587 (1205902]2 (1205902 + ]2))

sdot exp

minus

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)minus12(1]2

+11205902)119898

2119894

+(120583

]2+119909119894

1205902)119898119894

=1

radic2120587 (1205902]2 (1205902 + ]2))

sdot exp

minus

(119898119894minus (120583120590

2+ 119909119894]2) (1205902 + ]2))

2

2 (1205902]2 (1205902 + ]2))

(11)


119894)(120590




119894= E [119898

119894| 119909119894] =

1205902120583 + ]2119909

119894

1205902 + ]2= 120574120583+ (1minus 120574) 119909

119894 (12)


where

120574 =1205902

1205902 + ]2isin (0 1) (13)





119901 (x | 120583 ]2) =119873

prod

119894=1119901 (119909119894| 120583 ]2) = [2120587 (1205902 + ]2)]

minus1198732

sdot exp

119873

sum

119894=1

[

[

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)]

]

(14)



119894minus 120583)


119901 (x | 120583 ]2)

= [2120587 (1205902 + ]2)]minus1198732

expminussum119873

119894=1 (119909119894 minus 120583)2

2 (1205902 + ]2)

(15)


(120583 ]2)


119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2

(16)


2119873

sum

119894=1(119909119894minus120583) = 0

119873

1205902 + ]2minussum119873

119894=1 (119909119894 minus 120583)2

(1205902 + ]2)2minus120582 = 0

minus ]2 le 0 120582]2 = 0 120582 ge 0

(17)


L (120583 ]2 120582) = 119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2minus120582]2 (18)


120583 =1119873

119873

sum

119894=1119909119894 (19)


2minus 120590

2 where

1199042=

1119873

119873

sum

119894=1(119909119894minus120583)

2 (20)


]2 = max (0 1199042 minus1205902) (21)




119891 (119909119894 119909 119904

2) = 120574119909+ (1minus 120574) 119909

119894 120574 = min(1 120590

2

1199042) (22)




computation

x s2

x s2

x s2

x1

x2

xN

f(x1 x s2)

f(xN x s2)

f(x2 x s2)









compute 119909=(1119873)sum119873

119894=1 119909119894 and 1199042=(1119873)sum

119873

119894=1(119909119894minus

119909)2


119894= 120574119909 + (1 minus 120574)119909










1198631 (M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

(23)



119863119882(M) =

10038171003817100381710038171003817W ∘ (MminusM)

10038171003817100381710038171003817119865 (24)




119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817I (M) ∘ (MminusM)

10038171003817100381710038171003817119865 (25)


119894119895= log(1119875

119894119895) = minuslog119875

119894119895 where P = [119875

119894119895] is


2

4

6

8

10

12

14

16

18

20

(a)

0 5 10 15 20 250

010203040506070809

1

Prob

abili

ty

0 5 10 15 20

0

001

002

003

004

minus5

(b)

0

2

4

6

8

10

12

14

16

18

20

(c)


= 1


119894=1sum119873



119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817M1015840 minusM101584010038171003817100381710038171003817119865 (26)




119868(ie119863

119882with



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20









0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










where

120574 =1205902

1205902 + ]2isin (0 1) (13)





119901 (x | 120583 ]2) =119873

prod

119894=1119901 (119909119894| 120583 ]2) = [2120587 (1205902 + ]2)]

minus1198732

sdot exp

119873

sum

119894=1

[

[

minus12(1205832

]2+1199092119894

1205902)+

(1205831205902+ 119909119894]2)

2

21205902]2 (1205902 + ]2)]

]

(14)



119894minus 120583)


119901 (x | 120583 ]2)

= [2120587 (1205902 + ]2)]minus1198732

expminussum119873

119894=1 (119909119894 minus 120583)2

2 (1205902 + ]2)

(15)


(120583 ]2)


119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2

(16)


2119873

sum

119894=1(119909119894minus120583) = 0

119873

1205902 + ]2minussum119873

119894=1 (119909119894 minus 120583)2

(1205902 + ]2)2minus120582 = 0

minus ]2 le 0 120582]2 = 0 120582 ge 0

(17)


L (120583 ]2 120582) = 119873 log (1205902 + ]2) +sum119873

119894=1 (119909119894 minus 120583)2

1205902 + ]2minus120582]2 (18)


120583 =1119873

119873

sum

119894=1119909119894 (19)


2minus 120590

2 where

1199042=

1119873

119873

sum

119894=1(119909119894minus120583)

2 (20)


]2 = max (0 1199042 minus1205902) (21)




119891 (119909119894 119909 119904

2) = 120574119909+ (1minus 120574) 119909

119894 120574 = min(1 120590

2

1199042) (22)




computation

x s2

x s2

x s2

x1

x2

xN

f(x1 x s2)

f(xN x s2)

f(x2 x s2)









compute 119909=(1119873)sum119873

119894=1 119909119894 and 1199042=(1119873)sum

119873

119894=1(119909119894minus

119909)2


119894= 120574119909 + (1 minus 120574)119909










1198631 (M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

(23)



119863119882(M) =

10038171003817100381710038171003817W ∘ (MminusM)

10038171003817100381710038171003817119865 (24)




119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817I (M) ∘ (MminusM)

10038171003817100381710038171003817119865 (25)


119894119895= log(1119875

119894119895) = minuslog119875

119894119895 where P = [119875

119894119895] is


2

4

6

8

10

12

14

16

18

20

(a)

0 5 10 15 20 250

010203040506070809

1

Prob

abili

ty

0 5 10 15 20

0

001

002

003

004

minus5

(b)

0

2

4

6

8

10

12

14

16

18

20

(c)


= 1


119894=1sum119873



119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817M1015840 minusM101584010038171003817100381710038171003817119865 (26)




119868(ie119863

119882with



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20









0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











computation

x s2

x s2

x s2

x1

x2

xN

f(x1 x s2)

f(xN x s2)

f(x2 x s2)









compute 119909=(1119873)sum119873

119894=1 119909119894 and 1199042=(1119873)sum

119873

119894=1(119909119894minus

119909)2


119894= 120574119909 + (1 minus 120574)119909










1198631 (M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

(23)



119863119882(M) =

10038171003817100381710038171003817W ∘ (MminusM)

10038171003817100381710038171003817119865 (24)




119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817I (M) ∘ (MminusM)

10038171003817100381710038171003817119865 (25)


119894119895= log(1119875

119894119895) = minuslog119875

119894119895 where P = [119875

119894119895] is


2

4

6

8

10

12

14

16

18

20

(a)

0 5 10 15 20 250

010203040506070809

1

Prob

abili

ty

0 5 10 15 20

0

001

002

003

004

minus5

(b)

0

2

4

6

8

10

12

14

16

18

20

(c)


= 1


119894=1sum119873



119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817M1015840 minusM101584010038171003817100381710038171003817119865 (26)




119868(ie119863

119882with



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20









0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2

4

6

8

10

12

14

16

18

20

(a)

0 5 10 15 20 250

010203040506070809

1

Prob

abili

ty

0 5 10 15 20

0

001

002

003

004

minus5

(b)

0

2

4

6

8

10

12

14

16

18

20

(c)


= 1


119894=1sum119873



119863(M) =10038171003817100381710038171003817MminusM10038171003817100381710038171003817119865

10038171003817100381710038171003817M1015840 minusM101584010038171003817100381710038171003817119865 (26)




119868(ie119863

119882with



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20









0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

D = 308691 lowast 011167 = 34472

Moving average

D = 292711 lowast 061375 = 179652

Empirical Bayes

D = 111428 lowast 0098128 = 10934

0

5

10

15

20

0

5

10

15

20









0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

01

02

03

04

05

06

07

08

09

1











119868 In other






5 Conclusions



Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ground truth

0

5

10

15

20Measurements

0

5

10

15

20

Moving average

0

5

10

15

20Empirical Bayes

0

5

10

15

20

D = 596067 lowast 023059 = 137447

D = 307269 lowast 060576 = 186132 D = 205775 lowast 019642 = 40418


0

01

02

03

04

05

06

07

08

09

1



50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










50 10 15 200

10

20

30

40

50

60


D

1205902

(a)

0 5 10 15 200

01

02

03

04

05

06

07


DI

1205902

(b)

05 10 15 200

20

40

60

80

100

120


D1

1205902

(c)


119868 and1198631 for varying 120590

2




Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Ground truth

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

10

20

30

25

15

5

Measurements

D = 296303 lowast 004182 = 12391

Moving average

D = 64355 lowast 03686 = 237214

Empirical Bayes

D = 177258 lowast 0037525 = 066517





References











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article A Low-Complexity Approach for Improving