Sparse spatial filter via a novel objective function minimization with smooth ℓ1 regularization

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ContentslistsavailableatSciVerseScienceDirect

Biomedical

Signal

Processing

and

Control

jou rn a l h o m e pag e :w w w . e l s e v i e r . c o m / l o c a t e / b s p c

Technical

note

Sparse

spatial

ﬁlter

via

a

novel

objective

function

minimization

with

smooth

1

regularization

Ibrahim

Onaran

a,b,∗

_,

_N.

_Firat

_Ince

a,c

_,

_A.

_Enis

_Cetin

b

a_Department_of_{Neurosurgery,}_University_of_Minnesota,_Minneapolis,_MN_55455,_USA b_Department_of_Electrical_Engineering,_Bilkent_University,_Ankara,_Turkey

c_Department_of_Electrical_and_Computer_Engineering,_University_of_Minnesota,_Minneapolis,_MN_55455,_USA

a

r

t

i

c

l

e

i

n

f

o

Articlehistory: Received19April2012

Receivedinrevisedform7September2012 Accepted8October2012

Available online 8 November 2012 Keywords:

Brainmachineinterfaces Commonspatialpatterns Sparsespatialprojections Rayleighquotient Unconstrainedoptimization

a

b

s

t

r

a

c

t

Commonspatialpattern(CSP)methodiswidelyusedinbrainmachineinterface(BMI)applicationsto extractfeaturesfromthemultichannelneuralactivitythroughasetofspatialprojections.Thesespatial projectionsminimizetheRayleighquotient(RQ)astheobjectivefunction,whichisthevarianceratioof theclasses.TheCSPmethodeasilyoverﬁtsthedatawhenthenumberoftrainingtrialsisnotsufﬁciently largeanditissensitivetodailyvariationofmultichannelelectrodeplacement,whichlimitsits applicabil-ityforeverydayuseinBMIsystems.Toovercometheseproblems,theamountofchannelsthatisusedin projections,shouldbelimitedtosomeadequatenumber.Weintroduceaspatiallysparseprojection(SSP) methodthatexploitstheunconstrainedminimizationofanewobjectivefunctionwithapproximated1

penalty.UnliketheRQ,thisnewobjectivefunctiondependsonthemagnitudeofthesparseﬁlter.The SSPmethodisemployedtoclassifythemulticlassECoGandtwoclassEEGdatasets.Wecomparedour resultswitharecentlyintroducedsparseCSPsolutionbasedon0norm.Ourmethodoutperformsthe

standardCSPmethodandprovidescomparableresultsto0normbasedsolutionanditisassociated

withlesscomputationalcomplexity.Wealsoconductedseveralsimulationstudiesontheeffectofnoisy channelandintersessionvariabilityontheperformanceoftheCSPandsparseﬁlters.

1. Introduction

TheBMItechnologyaimstohelpdisabledpeopletoestablish communicationwiththeirenvironmentsolelybytheirbrain sig-nals.Withtherecentadvancesinelectrodedesignandrecording technology,thenumberofrecordingchannelsusedinBMI appli-cationsisincreasingtocapturesignalsfromalargerareaofthe brainortogetmoreinformationfromsmallerregionsusingdense electrodegrids.Therefore,adimensionreductionalgorithmneeds tobeemployedtodecreasethecorrelationbetweenchannelsand improvethesignaltonoiseratio(SNR).Inthisscheme,theCSP algo-rithmiswidelyusedduetoitssimplicityandlowercomputational complexitytoextractfeaturesfromhigh-densityrecordingsboth usingnoninvasiveandinvasivemodalities[1,2].

DespitethebeneﬁtsoftheCSPmethod,italsohasanumber of drawbacks. One major problemof theCSP is that it gener-allyoverﬁts thedata whenit is recorded froma largenumber of electrodesand when there is limited number of train trials.

∗ Correspondingauthorat:DepartmentofElectricalEngineering,Bilkent Univer-sity,Ankara,Turkey.

E-mailaddresses:[email protected],[email protected](I.Onaran), ﬁ[email protected](N.F.Ince),[email protected](A.E.Cetin).

Moreover,thechancethatCSPusesanoisyorcorruptedchannelis linearlyincreasedwithincreasingnumberofrecordingchannels. RobustnessovertimeisalsoamajordrawbackinCSPapplications [3,4].SinceallchannelsareusedinspatialprojectionsofCSP,the classificationaccuracymayreduceincasetheelectrodelocations slightlychangeindifferentsessions.Thisrequiresalmostidentical electrodepositionsovertime,whichisdifficulttorealize[5].The sparsenessof thespatial filtermighthave animportantrole to increase the robustnessand generalization capacity of theBMI system.

TheCSPmethodminimizestheRayleighQuotient(RQ)ofthe spatial covariance matrices to achieve the variance imbalance betweentheclassesofinterest.TheRQisdeﬁnedas

R(w)= wTAw

wT_Bw (1)

whereAandBarethespatialcovariancematricesoftwodifferent classesandwisthespatialﬁlterthatwewanttoﬁnd.Onewayto reducethenumberofchannelsusedintheprojectionw,isto trans-formtheCSPalgorithmintoaregularizedoptimizationproblemin theformof

L(w)=R(w)+||w|| (2)

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whereR(w)istheobjectivefunction,||w||isthe1 normbased penaltyandisaconstantthatcontrolsthesparsityofthesolution. Inthepastfewyears,thereisgrowinginterestinusing1penalty toconstructsparsesolutions.However,RQdoesnotdependon themagnitudeofthesparseﬁlter.Therefore,RQcannotbedirectly usedinanormbasedminimizationproblem,sincetheoptimizer alwaysminimizesthenormalongthedirectionwhichRQhasbeen minimized.

AnumberofstudiesinvestigatedputtingtheCSPinto alterna-tiveoptimizationformstoobtainasparsesolutionforit.In[6]the authorsconvertedCSPintoaquadraticallyconstrainedquadratic optimizationproblemwith1penalty;othersusedan1/2[3,7] normbasedsolution.Thesestudieshavereportedaslightdecrease ornochangeintheclassificationaccuracywhiledecreasingthe numberofchannelssignificantly.Recently,in[8]quasi0 norm basedcriterionwasusedforobtainingthesparsesolutionwhich resulted an improved classification accuracy. Since 0 norm is non-convex,combinatorialandNP-hard,theyimplementedgreedy solutionssuchasForwardSelection(FS)andBackwardElimination (BE)todecreasethecomputationalcomplexity.Ithasbeenshown thatBEwasbetterthanFS(lessmyopic)intermsofclassification errorandsparsenesslevelbutassociatedwithveryhighcomplexity makingitdifficulttouseinrapidprototypingscenarios.

Inthispaper,weconstructacomputationallyefficientspatially sparseprojection(SSP)basedonanovelobjectivefunctionwith similarcharacteristicstoRQ.Thisnewobjectivefunctioncanbe minimizedintheformof(2)toaddressthedrawbacksof regu-larCSPmethod.Weshowthatournewobjectivefunctionhasthe sameminimizationsolutionasRQanditdependsonthe magni-tudeofthespatialfilter.Themagnitudedependencyofournew objectivefunctionallowsustouseacontinuousanddifferentiable functionapproximating1norm[9]asregularizationterminan unconstrainedoptimizationframeworkandcanbesolvedusing standardalgorithmswithlow complexity.Therestofthepaper isorganizedasfollows.Inthefollowingsection,wedescribeour novelobjectivefunctionanditsrelationtoRQ.Thenweexplain itsuseinanunconstrainedoptimizationproblem.Next,weapply ourmethodontheBCIcompetitionIVECoGdatasetinvolving indi-viduatedmovementsoffivefingers[10]andtheBCIcompetition IIIEEGdatasetIVa[11]involvingimaginaryfootandhand move-ments.WealsocompareourmethodtostandardCSPandthe0 normbasedBEsolutiongivenin[8].Finally,westudiedtheeffect ofadditionalGaussiannoiseandsimulatedchanneldisplacements ontheclassificationaccuracyanddiscussourresultsandprovide futuredirections.

2. Materialsandmethods

2.1. StandardCSPandanewobjectivefunction

IntheCSPframework,thespatialﬁltersareaweightedlinear combinationofrecordingchannels,whicharetunedtoproduce spatialprojectionsmaximizingthevarianceofoneclassand mini-mizingtheother.Thespatialprojectioniscomputedusing

XCSP=WT_X ₍₃₎

wherethecolumnsofWarethevectorsrepresentingeachspatial projectionandXisthemultichannelECoGdata.

MaximizingtheRQ(1)isidenticaltothefollowingoptimization problem.

maximize

w w

T_Aw

subjectto wT_Bw₌_1. (4)

AfterwritingthisoptimizationproblemintheLagrangeform andtakingthederivativewithrespecttow,weobtaintheidentical

problemintheformofAw=BwwhichistheGeneralized Eigen-valueDecomposition(GED).Thesolutionsofthisequationarethe jointeigenvectorsofAandBandistheassociatedeigenvalueof aparticulareigenvector.

ThedrawbacksoftheCSPmethodthataredescribedearlierlead ustofindawaytosparsifythespatialfiltertoincreasethe classi-ficationaccuracyandthegeneralizationcapabilityofthemethod. Weassumethatthediscriminatoryinformationisembeddedina fewchannelswherethenumberofthesechannelsismuchsmaller thantheactualnumberofallrecordingchannels.Sothe discrimi-nationcanbeobtainedwithasparsespatialprojection,whichuses onlyinformativechannels.Inthisschemeassumethatthedatawas recordedfromKchannels.Weareinterestedinobtainingasparse spatialprojectionusinganunconstrainedminimizationproblemin theformof(2),wherewhasonlyknonzeroentries,card(w)=kand kK.WenotethattheR(w)doesnotdependonthemagnitudeof w,asshowninthefollowingequation.Letw∗=˛w,then

R(w∗)=R(˛w)=˛2wTAw

˛2_wT_Bw =R(w) (5)

where˛isanyscalarwhichisnotequaltozero.SinceR(w)does notdependonthegainofw,theoptimizerarbitrarilyreducesthe gainofwtominimizeregularizationterm||w||afterﬁndingthe directionthatminimizesR(w).Thus,thesolutionofthe optimiza-tionproblemthatusesR(w)asanobjectivefunctionisessentially thesameastheGEDsolution.

Toﬁndasparsesolutionweneedtohaveanobjectivefunction thatdependsonthegainofw.Inthisscheme,wereplacedR(w) withthefollowingobjectivefunction.

G(w)=wTAw+ 1

wT_Bw (6)

Thisfunctionisboundedfrombelowandhasinteresting prop-erties.Letusdeﬁnea=wT_Aw_and_b₌_wT_Bw._If_we_deﬁne_RQ_in termsofaandbsuchthatR=a/bthenournewobjectivefunction canbeexpressedas G(w)=a+1 b= ab b + 1 b=Rb+ 1 b (7)

Thederivativeof G(w)withrespecttoR isequaltobwhich isalwayspositive.ThisindicatesthatourobjectivefunctionG(w) decreaseswithadecreaseinRvalue.Aftertakingthederivativeof G(w)withrespecttobandsolvingEq.8,

∂

G(w)

∂

b =R−

1

b2 =0 (8)

wenotethatbisequalto√R−1.ByinsertingbvalueintotheEq. 7weobtaintheminimumvalueofG(w)as2√R.Thisresultshows thatthedirectionthatminimizesRalsominimizesG(w).

WeplugG(w)intounconstrainedoptimizationformulationin (2)astheobjectivefunction.Ratherthanworkingtosolve(2)with anon-differentiable1penalty,wereplaceditwithatwice differ-entiablesmoothversionof1 (epsL1)whichissufﬁcientlyclose tominimizing1[9].Themainadvantageofthisapproachisthat, sinceepsL1andG(w)arebothtwicedifferentiablewecandirectly applyan unconstrained optimizationmethodtominimizeL(w) [12].TheepsL1isdeﬁnedas

||w||= K

i=1

w2 i +

(9)

where

isasufﬁcientlysmallparameterandKisthedimension ofw.TheepsL1approximatesthe1normandtheyareidentical when

isequaltozero.TwicedifferentiabilityoftheepsL1norm allowsustouseitwhenwiisequaltozerounliketheregular1 normwhichisnotdifferentiableatzero.

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ThesolutionwthatminimizesthefunctionL(w)=G(w)+||w|| tendstobecomesparseas getsbigger.The entriesof w gen-erallywerenotexactlyequaltozero,sowenormalizedwtoits maximumabsolutevalueandeliminatedtheweightsconsequently correspondingchannelsthatdonotexceedapredeﬁnedthreshold (=10−2).Weused“fminunc”functionofMatlabtoﬁndthe solu-tionofourunconstrainedminimizationproblem.Wecomputedthe desiredcardinalitywhichisthenumberofchannelstobeselected forthespatialprojectionbyimplementingabisectionsearch[13] onthe.Theupperborderofwasdeterminedinitiallyusingthe G(wc)/||wc||ratiowherewcisthefullCSPsolution.Incasetheinitial upperborderresultsacardinalitylargerthanthedesiredvalue,we keptdoublingtheparameteruntilweobtainedathatresultsa cardinalitywhichislessthanorequaltothetargetvalue.

Followingtheaboveprocedure,wecomputedthefirstspatial fil-terwthatminimizestheG(w)whichalsominimizestheR(w).The solutionthatmaximizesR(w)isalsoausefulspatialfilter. There-fore,weinterchangedthematrixAandBtofindasolutionthat maximizesR(w).Inordertofindmultiplesparsefilterswedeflated thecovariancematriceswithsparsevectorsusingtheSchur com-plementdeflationmethoddescribedin[14].

2.2. ECoG&EEGdatasets

WeappliedtheSSPmethodontwodifferentdatasets,multiclass ECoGandtwoclassEEGofBCIcompetitionsIVandIIIrespectively. TheECoGdatawasrecordedfromthreesubjectsduringfinger flexionsandextensions[10]withasamplingrateof1kHz.The elec-trodegridwasplacedonthesurfaceofthebrain.Eachelectrode arraycontained48(8×6)or64(8×8)platinumelectrodes.The fingerindextobemovedwasshownwithacueonacomputer monitor.Thesubjectsmovedoneoftheirfivefingers3–5times duringthecueperiod.TheECoGdataofeachsubjectwassubband filteredinthegammafrequencyband(65–200Hz)asin[15].We used1sdatafollowingthemovementonsetintheanalysis.The datasetcontainsaround146trialsforeachsubject.

WealsousedtheBCIcompetitionIIIdatasetIVa[11].Thedataset isrecordedfromfivesubjects(aa,al,av,aw,ay)whowereasked toimagineeitherrightfootorrightindexfingermovements.The samplingrateofthedatawas1kHzanddatawasrecordedfrom 118channels.TheEEGsignalwasfilteredintherangeof8–30Hz. Therewere140trialsavailableforeachclass.Onceagain,1sdata followingthecuewasusedintheanalysis.

ForbothECoG andEEGdatasets,thesignalwastransformed intofourspatialfiltersbytakingfirstandlasttwoeigenvectorsfor eachCSPmethods.Aftercomputingthespatialfilteroutputs,we calculatedtheenergyofthesignalandconvertedittologscalefor eachsparsefilterandweusedthemasinputfeaturestolib-SVM classifierwithanRBFkernel[16].Wealsoinvestigatedtheefficacy ofusingLinearDiscriminantAnalysis(LDA)classifier[17]whichis parameterfreedecisionfunction.

SincewearetacklingamulticlassproblemfortheECoGdataset, weusedthepairwisediscriminationstrategyof[2]toapplythe CSPtothefive-classfingermovementdata.Inotherwords,we constructedsparsespatialfilterstunedtocontrastpairsoffinger movementssuchas1vs.2;1vs.3;2vs.4,etc.

Ineachdataset,wecomparedtheSSPtothestandardCSPandto the0normbasedBEmethodof[8]asitprovidedsuperiorresults in terms of classification accuracyand reduced cardinality. We studiedtheclassificationaccuracyasafunctionofcardinality.On thetrainingdatawiththepurposeoffindingoptimumsparseness levelfortheclassification,wecomputedseveralsparsesolutions, withdecreasingcardinality.FortheECoGdataset,thesparseCSP methodswereemployedwithk_∈_{40,30,20,15,10,5,2,1_}.For theEEGdataset,wecomputedthesparsefilterswithk∈{80,60, 40,30,20,15,10,5,2,1}.Foreachcardinality,wecomputedthe

Table1

ECoGdatasetclassiﬁcationerrorrates(%)foreachsubjectusingSVMclassiﬁer.

Cardinality Subject1 Subject2 Subject3 Avg

BE 5 19.8 17.1 16.8 18

SSP 5 18.4 13.4 18 17

CSP All 30.7 26 32.8 30

correspondingRQvalue.WestudiedtheinverseoftheRQ(IRQ) curveanddeterminedtheoptimalcardinalitywhereitsvalue sud-denlydroppedindicatingwestartedtoloseinformativechannels. FortheECoGdataset,halfofthetrialswereusedintrainingand theremaininghalffortesting.Inaverage,weused15±2traintrials perfinger(thethumb,index,middle,ringandlittlefingers respec-tively).TheEEGdatasetcontains140trialsperclassandsubject. We used70trials intraining toestimatethesparsefilters,and 70trialsfortesting.Inbothdatasets,thevalueofthe

inepsL1 regularizationtermwaschosentobe10−6.

3. Results

WeobservedthatfortheSSPmethod,anyparticularvaluecan leadtodifferentcardinalityandnormalizedIRQvaluesfordifferent subjectsasshowninFig.1.Inparticular,thisintersubject variabil-ityofIRQdidnotallowustousethesamevalueforallsubjects (SeeFig.1aandb).However,thevariabilityofIRQvaluesof differ-entsubjectswaslowerwhenweﬁxedthecardinalityasshownin Fig.1eandf.Consequently,duetothisreducedvariabilityandto compareourmethodtotheBEtechnique,westudiedthe classiﬁ-cationerrorasafunctionofcardinality.Inordertodecideonthe optimalcardinalityleveltobeusedonthetestdata,theIRQ val-ueswerecomputedonthetrainingdata,scaledtotheirmaximum valueandaveragedoversubjects.Inthefollowingstep,we com-putedtheslopeoftheIRQcurveandnormalizedittoitsmaximum valuetogetanideaabouttherelativechangeintheIRQ.

WedepictedthechangeinIRQvaluesforeachcardinalityas showninFig.2aandb.Asexpected,decreasingthecardinalityof thespatialprojectionresultedtoadecreaseintheIRQvalue.To determinetheoptimumcardinalitytobeusedinclassificationon thetestdata,weselectedthecardinalitythatisbelow10%ofthe maximumrelativechange(seethedashedlinesinFig.2aandb).For theECoGdataset,thecardinalityvaluewasfoundtobe5andforthe EEGdataset,itwasfoundtobe15fortheSSPmethod.FortheBE methodthesevalueswere5and10respectively.Theseindices per-fectlycorrespondedtotheelbowoftheIRQcurve,whichindicates lossofinformativechannels.InTables1and2,weprovidethe clas-sificationresultsandselectedcardinalitiesfortheECoGandEEG datasetusingdifferentmethodsincludingSSP,CSPand0based greedysolution,BE.Inordertogiveaflavoraboutthechangein errorrateversusthecardinality,weprovidedtherelated classi-ficationerrorcurvesinFig.2candd.FortheECoGdataselected cardinalityprovidedminimumtesterrors.However,fortheEEG data,althoughtheminimumclassificationerrorwasobtainedat cardinality5fortheBEmethod,wenoticedthatweidentifiedthe optimumcardinalityas10inthetrainingdata.

Onallsubjectswestudied,weobservedthattheSSPmethod consistentlyoutperformedtheCSPmethod.Wenotedthatthe min-imumerrorratewasobtainedwithSSPmethodforECoGdata.Both

Table2

EEGdatasetclassiﬁcationerrorrates(%)foreachsubjectusingSVMclassiﬁer.

Cardinality aa al av aw ay Avg

BE 10 13.6 2.9 30.7 2.1 5.0 10.9

SSP 15 19.3 1.4 23.6 4.3 5.7 10.9

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10−5 10−4 10−3 10−2 0.6 0.7 0.8 0.9 1 1.1 Normalized IRQ λ Subject 1 Subject 2 Subject 3 10−5 10−4 10−3 10−2 0.2 0.4 0.6 0.8 1 Normalized IRQ λ aa al av aw ay 10−5 10−4 10−3 10−2 10 20 30 40 50 60 λ Cardinality Subject 1 Subject 2 Subject 3 10−5 10−4 10−3 10−2 0 20 40 60 80 100 120 λ Cardinality aa al av aw ay 0 10 20 30 40 All 0.6 0.7 0.8 0.9 1 1.1 Normalized IRQ Cardinality Subject 1 Subject 2 Subject 3 20 40 60 80 100 0.2 0.4 0.6 0.8 1 Normalized IRQ Cardinality aa al av aw ay

Fig.1.NormalizedIRQvaluesareshownin(a)forECoGandin(b)forEEGdata.ThecardinalityvsvalueoftheminimizationfunctionL(ω)=G(ω)+ω(c)ECoGand(d) EEGdataforeachsubject.Theverticallinesindicatethevaluesthatareinitiallychosenforbisectionsearch.Theinitialdatapointcorrespondsto=0whichproducesthe regularCSPsolution.ThenormalizedIRQvaluesvs.cardinalityforeachsubjectisshownin(e)and(f).

SSPandBEmethodsusedcardinalityof5toachievetheminimum errorrate.AsexpectedthefullCSPsolutiondidnotperformasgood astheothersparsemethodsandlikelyoverfittedthetrainingdata. TheSSPmethodimprovedtheclassificationerrorratewithanerror differenceof13.2%.WeobtainedcomparableresultsonEEGdata usingtheSSPandBEmethods(p-value=0.5,pairedt-test), how-everBEprovidedasignificantimprovementoverSSPmethodon thenumberofchannels(p-value=0.003)usedinspatialprojection. TheerrordifferencebetweenregularCSPandSSPisless appar-entintheEEGdatasetwherethedifferencebetweenclassification accuracieswas3.8%.Thiscouldbeduetothehighnumberof train-ingtrialsusedinEEGdata.Westudiedtheeffectoftheamount oftrainingdataontheclassificationaccuracyandpresentedthe resultsinFig.4a.Whenasmallnumberoftrainingtrials,aslowas 15areusedintheEEGdataset,thedifferencebetweenthesparse andstandardCSPtechniquewasmorethan6%.Interestingly,with increasingnumberoftrainingtrialstheSSPmethodconsistently providedbetterresultsandthedifferenceremainedbetween3and 4%.TherewasnonoticeabledifferencebetweenSSPandBE.

TheclassificationresultsobtainedwithLDAclassifieraregiven inTable3.We observedthat theLDA classifierwhich doesnot involve parameter selection like SVM, provided slightly higher errorratesforthesparsesolutions.Thiscouldbedueto nonlin-eardecisionsurfaceandmaximummarginidentifiedbytheSVM classifier.Interestingly,inbothdatasets,theLDAclassifierresulted inlowererrorrateswiththenon-sparseCSPsolution.

Table3

Averagetesterrorrate(%)andcorrespondingcardinality.

ECoG EEG

BE SSP CSP BE SSP CSP

LDA 19.9 18.9 24.1 11.4 11.1 14.3

SVM 17.9 16.6 29.8 10.9 10.9 14.7

Cardinality 5 5 All 10 15 All

Fig.3illustratesthedistributionofthespatialfiltersobtained usingSSPandCSPalgorithmsforallsubjects.Weobservedthat theSSPfiltercoefficientsarelocalizedonthelefthemisphereand thecentralarea,whichisinaccordancewiththecorticalregions relatedtorighthandandthefootmovementgeneration.

Arvaneh et al. [7] used the 1/2 ratio as a penalty term and they applied their algorithm to the BCI competition III EEG dataset IVa [11] which we used in this paper as well. Theyachieveda meanerrorrateof17.7±15.4%using22.6±11 channels. Here, we compared our method with the study of Arvaneh et al. by extracting one filter from each end of the sparse solutions. The SSP method achieved a mean error rate of 12±11.3% with an average number of channels 25.6±2.3. The obtained results indicated that the SSP method provided a significant improvement (p-value=0.024, paired t-test) over the1/2basedalgorithmontheclassificationaccuracywithout anysignificantdifferencebetweennumberofchannelsused (p-value=0.28).

In order to compare the computational complexity of SSP methodtotheBE,wecomputedsparsefilterswithacardinalityof twofromanincreasingnumberofrecordingchannelsonsimulated data.Thetrainingwasperformedonaregulardesktopcomputer with4GBofRAMandequippedwithaCPUrunningat2.66GHz. Theelapsedtimeperfiltercomputationincreasedexponentially fortheBEmethodandlinearlyfortheSSPmethodasshownin Fig.4b.With128channels,theBEalgorithmcomputedasingle spatialfilterwithtwononzeroentriesin90s.FortheSSPmethod withthesamesetupabove,theelapsedtimewaslessthana sec-ond.Although,weusedtherelativechangeintheIRQtoidentify theoptimumsparsitylevel,onecanalsorunatypicalk-foldcross validationproceduretoidentifytheoptimumlevel.However,in suchacasetrainingthesystemwithBEmethodwilltakeseveral hourswhichmaynotbefeasibleforBMIapplications.Ontheother handwiththeSSPmethodtrainingthroughcrossvalidationcanbe executedinafewminutes.

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0 10 20 30 40 0 0.2 0.4 0.6 0.8 1 Cardinality 0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 Cardinality 1 10 20 30 40 All Channels 10 15 20 25 30 35 40 Cardinality Classification Error (%) BE SSP 20 40 60 80 All Channels 10 15 20 25 30 35 40 Cardinality Classification Error (%) BE SSP

Fig.2.TheaverageIRQofallsubjectsversuscardinality(a)ECoGand(b)EEGdata.Theredlineisthe10percentthresholdthatdeterminestheoptimumcardinalityto beusedinthetestdata.TheoptimumcardinalitylevelsforECoGandEEGarefiveand15respectively.TheclassificationerrorcurvesofSSPandBEmethodsversusthe cardinalityaregivenin(c)forECoGandin(d)forEEGThelastdatapointcorrespondstotheresultsobtainedfromstandardCSPwhichusesallchannels.(Forinterpretation ofthereferencestocolorinthisfigurelegend,thereaderisreferredtothewebversionofthearticle.)

Inordertoevaluatetheeffectofnoiseandintersession variabil-ityontheperformanceofourapproachwestudiedtwodifferent controlledexperiments:

i AddingGaussiannoisetoarandomlyselectedchannelinECoG andEEGdata.

Fig.3.TheCSPandSSPﬁltersforhandandfootmovementimagination.

iiSimulationofelectrodedisplacementinEEGdata.

Duringthefirstexperiment,weaddedzeromeanGaussiannoise tooneofthechannelsandcalculatedtheresultingclassification errortoobtainfinalclassificationaccuracyforthenoisydata.This experimentwasrepeatedforallchannelsandthentheresulting averageclassificationerrorwascomputed.Thevarianceofthe addi-tionalnoisewasincreasedinacontrolledmannerandproportional totheaveragevarianceofallchannels.Theratioofthenoiseto thevarianceoftheoriginaldatawasnamedasnoisetosignalplus noiseratio(NSNR)sincetheoriginalsignalalreadycontaminated fromdifferentnoisesources.InFig.5,wedepictedthe classifica-tionaccuracyvs.NSNRwhichwasexpressedinDecibelscale.Itwas notedthatanincreaseinNSNRcausederrortoincreaseforall meth-ods,howevertheincreaseinCSPmethodwasmorethanthesparse filters.WhileusingthesparsemethodsintheECoGdata,the classi-ficationerrorreachedaplateauafter5dBwhereasthestandardCSP errorincreasedmonotonically.Asimilarbehaviorwasobservedin theEEGdata.

Inthesecondexperiment,inordertostudyintersession vari-abilitywesimulatedelectrodedisplacementsbyinterpolatingthe EEGtestdataatdifferentpositions.Sinceelectrodelocationsfor theECoGdatawerenotavailable,weconductedthisexperiment withtheEEGdataonly.Werandomlydeterminedthedirectionand theamountofdisplacementofthenewelectrodelocations.Tobe morerealistic,weintroduceddisplacementwhichwasuniformly distributedover118electrodesvaryingbetween0and50%ofthe

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20 40 60 80 100 5 10 15 20 25 30 Number of Trials Minimum Error BE SSP CSP 20 40 60 80 100 120 0 20 40 60 80 100 Number of Channels Elapsed Time (s) BE SSP

Fig.4. (a)Theminimumerrorvs.thenumberoftrials.(b)Theaverageelapsedtimetoestimateaspatialﬁlterwithacardinalityoftwovs.thenumberoftotalrecording channels. −20 −10 0 10 20 10 20 30 40 50 60 70 80 90 100

Noise to Signal Plus Noise Ratio (dB)

Classification Error (%) BE SSP CSP −20 −10 0 10 20 10 20 30 40 50 60

Noise to Signal Plus Noise Ratio (dB)

Classification Error (%)

BE SSP CSP

Fig.5. Thenoisetosignalplusnoiseratio(NSNR)vs.classiﬁcationaccuracyfor(a)ECoGand(b)EEGsets.

distancetothenearestelectrodeasshowninFig.6a.Thetestingof algorithmswasconductedontheinterpolatedEEGdataonthe dis-placedelectrodelocations.Theclassiﬁcationresultsvs.cardinality isshowninFig.6b.ItwasnotedthattheerrorchangeforCSP,BE andSSPmethodsare13%,12%and10%,respectively.Theincrease inerrorratewasnotassevereasinthepreviousnoise contami-natedchannelexperiment.Thesparsemethodshadslightlylower errorincreasethanthestandardCSP.Initially,itwasexpectedthat thesparsemethodswouldnotbethisvulnerabletodisplacement in theelectrode locations.At this point, we speculate that the smoothnessoftheprojections have a certainadvantageonthe

displacedelectrodes.Forinstance,duetosparseness,itislikelythat theshiftedelectrodelocationsareweightedwithzero.Thismakes thesparseprojectionsdiscardtheshiftedchannelsthatfalloutside theprojectionzone. Incontrary,thestandard CSPis associated withsomesortofsmoothnessandmanycrucialchannels,although displaced,couldstillbeprojectedwithsimilarweightsduetothe correlateddistribution.Consequently,itshouldbenotedthatnot onlysparsenessbutalsosmoothnessof projectionscouldbean importantparametercontributingtothegeneralizationcapability ofthemethods.Nevertheless,onceagainweneedtohighlight that,theincreaseinerrorratewaslowerforthesparsemethods.

0 10 20 30 40 50 0 5 10 15 20 Displacement (%) Number of Electrodes 20 40 60 80 All Channels 10 20 30 40 50 60 Cardinality Classification Error (%) BE SSP

Fig.6. (a)Thehistogramofthenumberofelectrodeswithrespecttothedisplacementinducedonthetestdata.(b)Thecardinalityvs.classiﬁcationerroronthetestdata withdisplacedelectrodelocations.

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4. Conclusion

Theneedforthesparsefiltersisapparentwhenthereislarge numberofrecordingelectrodesandinsufficientamountoftraining data.Tominimizeoverfittingonthetrainingdataandeliminate noisychannels,weintroducedaspatiallysparseprojection tech-nique(SSP)basedonanovelobjectivefunction.Unlike theRQ, thisnewobjectivefunctionhasadependencyonthefilter magni-tude.Byusinganapproximated1norm,wecomputedthesparse spatialfiltersthroughanunconstrainedminimizationformulation withstandardoptimizationalgorithm.Weappliedourmethodto ECoGand EEGdatasetsandcompared itsefficiencytostandard CSP,and toa0 normbasedgreedytechnique.TheSSPmethod outperformed thestandard CSP onboth datasets and provided comparable resultsto0 norm based method,which is associ-atedwithhighercomputationalcomplexity.OntheECoGdata,the SSPmethodprovided44%decreaseintheerrorratecomparedto standardCSPmethodandusedonlyfivechannelsineachspatial projection.TheerrordifferencebetweenregularCSPandSSPisless apparentintheEEGdatasetasSSPmethodprovided26%decrease intheerrorrate.IncontrarytotheECoGdata,wealsoobserved thatmorechannelswereusedtoachieveminimumclassification accuracyintheEEGdataset.Thiscouldbeduethelowspatial res-olutionoriginatingfromthevolumeconductionandlowSNRof theEEG.Nevertheless,theSSPalgorithmwasabletoreacha mini-mumerrorratewithonly15channels.OurresultsindicatethatSSP methodcanbeeffectivelyusedtoextractfeaturesfrombothEEG andECoGdatasetswithlargenumberofrecordingchannels.We notethatthesparsemethodsprovidedsuperiorresultscompared tothestandardCSPwhenthereisanoisy/corruptedchannelinthe testdata.Inanothersetupwheredisplacedchannelswereusedto simulateintersessionvariability,wenotethatthesparsemethods hadslightlybetterrobustnessthanthestandardCSP.These obser-vationsindicatethatthesparsespatialprojectionframeworkcan beeffectivelyusedasarobustfeatureextractionengineoffuture BCIsystems.

Acknowledgments

Thisresearchwassupportedin partby theNationalScience Foundation,awardCBET-1067488,andbyagrantfromthe Univer-sityofMinnesotaInterdisciplinaryInformatics(UMII).Itwasalso

supportedinpartbytheTheScientiﬁcandTechnologicalResearch CouncilofTurkey(TUBITAK),2211PhDfellowshipforTurkish citi-zensandproject111E057.

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