119

119

3D Geological Modeling: Solving as a Classification Problem with the Support Vector Machine

ByAlexSmirnoff,EricBoisvert,andSergeJ.Paradis

GeologicalSurveyofCanadaGSC-Quebec

490delaCouronneQuebec,CanadaG1K9A9Telephone:(418)654-3716

Fax:(418)654-2615e-mail:alsmirno@nrcan.gc.ca

ABSTRACT

Theprocessofcreatingmulti-unit3Dgeologicalmodelsbysuccessiveunitinterpolationmaybetediousandtime-consuming.Here,weproposetoautomatethisprocedure through presenting the problem as a classifica-tiontaskandsolvingitsimultaneouslywiththeSupportVector Machine (SVM), a method known from the field of artificial intelligence. Experiments with various input data andkernelparametersdemonstratedthattheSVMhasgreatpotentialin3Dreconstructionsfromsparsegeologi-calinformation.Anextendedversionofthispaperhasbeenacceptedforpublicationin“ComputersandGeosci-ences”(Smirnoffetal.,2008).

INTRODUCTION

Often,geologistsarefacedwithavarietyofdiverseinformationthatrequiresgeneralizationandanalysis.3DmodelingsoftwarepackagessuchasGocadofEarthDecisionScienceshaveprovenanexcellentmeansfordatapresentationandinterpretation.Theprocedurenormallyrequiresreconstructionofindividualgeologicalunitsusingsurfacesinterpolatedfromcontrolpointswithsubsequentfusionoftheseunitsintoasinglemodel.Thepopularinterpolationtechniquesinclude�nverseDistanceWeighting(�DW),DiscreteSmooth�nterpolation(DS�),and various flavors of kriging preceded by semi-vario-gramanalysis.

Theaboveprocedurecaneasilybecomeatediousandtime-consumingtaskwhenacomplexgeomodelisconsidered.�naddition,thetraditionalinterpolationtechniquesassumereasonablearealcoverageoftheinputdata.Therefore,thereisastrongneedforanalgorithmthatwouldautomatetheprocessofmodelcreationevenincaseswhenonlyafewpiecesofinformationonregionalgeology,(e.g.,afewsparsecross-sections)areavailable.Findingsuchanalgorithmandtestingitsperformanceonavailabledatasetswastheobjectiveofthisstudy.

Here,weproposetheuseoftheSupportVectorMa-chine (SVM), a tool routinely applied in the field of image analysisandpatternrecognition.TheSVMisbecomingincreasinglypopularandhasbeensuccessfullyusedtosolve classification and regression problems in biol-ogy(e.g.,Nobleetal.,2005),hydrology(e.g.,Yuetal.,2004),medicine(e.g.,El-Naqaetal.,2002),andenviron-mentalscience(e.g.,Gilardietal.,1999).�nthisstudy,wedemonstratethattheapplicationofSVMingeologyallows sparse data to be efficiently combined in order to reconstructshape,area,andvolumeofmultiplegeologi-calunits.

METHODOLOGY

The SVM Algorithm

TheSVMalgorithmisbasedontheStatisticalLearn-ingTheorydevelopedbyV.Vapnik(Vapnik,1995).�tusesasetofexampleswithknownclassinformationtobuildahyperplanethatseparatessamplesofdiffer-entclasses.�nmachinelearningtheory,thisisknownassupervisedlearningasopposedtounsupervisedlearn-ingwhennoaprioriclassinformationisavailable.Thisinitialdatasetisknownasatrainingset,andeverysamplewithin it is characterized by features upon which classifi-cationisbased.Figure1Ademonstratesthisfortheone-dimensional(single-feature)case.Thesamplesclosesttothe hyperplane are termed support vectors (filled marks in Figure1).

�nmorecomplicated,non-linearcases,thetaskofdiscoveringtheseparatoristurnedintoalineartaskbytransferringinputdataintoahigher-dimensionalspaceknownasthefeaturespace.Figure1Bshowsanon-sepa-rableone-dimensionaldatasetintheinputspace.Theproblemiseasilysolvedthroughre-mappingdatatoahigher,two-dimensionalspacewherealinearsolutionisfound(Figure1C).Functionssatisfyingcertainconditionsandknownaskernelsarenormallyemployedforthis

120 D�G�TALMAPP�NGTECHN�QUES‘06

Figure 1.Buildingtheseparatinghyperplane,inseparableandnon-separableone-dimensionalcase.Filledmarksrepresentsupportvectors.(A)Linearlyseparablecaseanddecisionfunctionininputspace;(B)non-separablecaseininputspace.(C)trainingdatare-mappedintotwo-dimen-sionalfeaturespaceusingφ(x) = (x, x2) and linearsolutioninthisspace;(D)solutionre-plottedbackininputspace.

transfer(e.g.,Abe,2005).Thesolutionbecomesnon-lin-earwhenshownintheoriginaldataspace(Figure1D).

Once the equation for the optimal classifier is found, newdatawithunknownclassinformation(testsamples)can be classified based on the value of this decision func-tion.Unlikemostinterpolationmethodsbasedontheprin-ciplethatvaluesatpointscloserinspacearemoresimilar,the SVM is a boundary classification method where the boundaryisbuiltbasedontheinitialtrainingsetamongwhichonlyasmallnumberofsamples(supportvectors)are involved in the final decision making.

TheclassicalSVMtaskisabinary(two-class)clas-sification. However, a number of methods have been developed to support multi-class classification through variouscombinationsofbinarymethodssuchas“one-against-all”,“one-against-one”,etc.(seeHsuandLin,2002forreferences).Therefore,theSVMapproachisalsoapplicableformodelswithmorethantwoclasses.MoredetaileddescriptionsoftheSVMalgorithmareavailablefromanumberofsources(e.g.,CristianiniandShawe-Taylor,2000;Abe,2005).

SVM Application to 3D Modeling

ToapplytheSVMalgorithmtoourgeologicalrecon-structions, we defined the 3D space-partitioning task as a pure spatial classification problem. Three coordinates uniquelydescribeeverypointinthe3Dreconstructionspace.However,onlyalimitednumberofthosepointspossessdescriptionsorclassinformationthatcanbeiden-tified through well drilling, surficial geology mapping, and seismic profiling. The class information describes the geologicalunittowhicheachparticularpointbelongs.Therefore,thepointswithknownclasslabelsbecomesamplesintheSVMtrainingset,andpointcoordinatesinthethree-dimensionalspaceareusedassamplefeatures.Once a classification model based on this training set is built,therestofthepointsinthereconstructionspacecanbe classified based on their coordinates (features).

WeemployedoneofthemanySVMimplementationsfreelyavailableovertheinternet,namelyL�BSVMdevel-opedattheNationalTaiwanUniversity(seeChangandLin,2001fordetaileddescription).AsrecommendedinHsuetal.(2004),weusedL�BSVMwiththeradialbasisfunction(RBF)kernel,themostgeneralformofkernelresultinginapredictionmodelcontrolledbyonlytwohy-perparameters,Candγ.Asinglesolutionisobtainedforeverypairofparameters,anditissensitivetothechoiceoftheirvalues.However,selectingtheappropriatevaluesisadarkartnormallydoneonatry-and-seebasis.

For multi-class classification, LIBSVM uses “one-against-one”approach,whichwasshowntobeadvanta-geoustoothermethodsforpracticaluse(HsuandLin,2002).�naddition,asetofin-houseJavautilitieshasbeendevelopedforscaling,validation,andformatconversionpurposes.

1213DGEOLOG�CALMODEL�NG

Data

A3DgeomodelcreatedattheGeologicalSurveyofCanada,Quebec,inthecourseoftheEsker/Abitibiproject(Bolducetal.,2005)wasusedasthereferencedatasetinalloftheexperiments.Thismodelisbasedonsurfacegeology,well,andcross-sectiondata.SixgeologicalunitsweresequentiallyinterpolatedfromtheabovecontrolpointsusingtheDiscreteSmooth�nterpolation(DS�)technique(Mallet,1989)intheGocadG�S.

Experimental Work

Theexperimentalworkwasdesignedtoperformthefollowingtasks:(1)investigatewhethertheSVMcanbeefficiently used in binary geological reconstructions, (2) testtheSVMformulti-unitmodeling,and(3)examinehowtheresultingmodeldependsontheRBFkernelpa-rametersusedinthereconstruction.

General Approach

Thegeneralapproachtakeninalloftheexperimentswasasfollows:

Prediction

• �nGocad,createareconstructionspaceasasetofvolumeelements(voxet)oftheshaperepresenta-tiveofstudyareageometry.Thereconstructionspace was defined by a voxet with the following numberofvolumeelements(voxels)ineachdirec-tion:X-110,Y-240,Z-24.

• Addavailabledatatothereconstructionspace.Theunittypepropertyforeachofthesixgeologi-calunits(SVMclasses)wastransferredfromthestratigraphicgrid(SGrid)structurerepresentingthereferencemodel.

• �nGocad, using a DEM, define all voxet nodes abovethesurfaceasairorno-datapoints.

• Define a training set for the experiment. For the re-maininggroundpoints,settheunittypepropertytozero; these are the points that will be later classified bytrainedSVM.

• Scalecoordinatevaluesforthetrainingsetbetween0and1asrecommendedinHsuetal.(2004).

• UsingL�BSVM,buildapredictionmodelbasedonthetrainingset.Asinglereferencesetofkernelparameters(C=104andγ=102)previouslydeter-minedfrom2Dexperimentswasusedinallrecon-structionsexcepttheparametersensitivitytests.

• Scalecoordinatevaluesforthepointstobeclassi-fied and classify them using the prediction model createdinthepreviousstep.

• �mportthepointsetwithpredictedclassinformationbackintoGocad,forvisualizationandanalysis.

Validation

• Based on the available reference data, define the validationsetandextractitasasetofpointswithattachedclassproperty.

• Testpredictedclasslabelsagainstthevalidationsettodeterminehowmanyoriginalpointsineachclass and overall were adequately classified, a mea-surealsoknownastherecallrate.

Binary reconstruction

ThetrainingsetwascomposedoftheEsker/Abitibimodelpointslocatedon11arbitrarilychosenparallelsectionsorientedalongaxisX.PointsweregroupedintotwoclassesasshowninTable1.TheinputdatastatisticsaregiveninColumn4ofTable1.Asseenfromthetable,thetrainingsetwasdominatedbypointsrepresentativeofClass2,whichcombinedallmodelunitsexcepttheEskerUnit.Thevalidationsetwascomposedofalltheremain-ingmodelpoints(notincludedinthetrainingset).ThenumberofpointsusedforvalidationineachofthetwoclassesisshowninColumn1aofTable3.

Table 1.Geologicalunits,SVMclasses,andtrainingsetstatisticsforEsker/AbitibiBinaryModel.Totalnumberofpoints to be classified is 371783.

1. Geological Unit 2. SVM Class 3. All Points (#/%)b 4. Training Points (#/%)c

Esker 1 20300/5.22 995/0.26 Non-Eskera 2 368935/94.78 16457/4.23 AllUnits - 389235/100 17452/4.48

aNon-EskerunitincludedRock,Till,Clay,LittoralandOrganicunitsbNumberofallclasspointsandtheirpercentageofallmodelpointscNumberoftrainingpointsperclassandtheirpercentageofallmodelpoints

122 D�G�TALMAPP�NGTECHN�QUES‘06

Multi-Class reconstruction

Thesametrainingpoints,arrangedintosixclassescorrespondingtothesixgeologicalunitsfoundintheoriginalmodel,wereusedtotesttheSVMcapabilitiesin multi-class classification (Table 2). For training data statistics,seeColumn4ofTable2.Thistime,bedrock(Class6)entirelydominatedthetrainingset,withorgan-ics(Class1)beingtheleastrepresentative.ThevalidationsetalsocontainedinformationaboutallsixgeologicalunitsasshowninColumn2aofTable3.

Hyperparameters Sensitivity tests and Multiple Parameter-Set reconstructions

Toanalyzethesensitivityofpredictionresultstothevaluesofhyperparameters,Candγ,weusedasimplegridsearchprocedureasproposedinHsuetal.(2004).Thegrid search was run for the above training set configu-rations,andtherangeofparametersscannedbyevery

searchwasfrom2-8to215forCandfrom2-15to212forγincrementingparametervaluesbyapowerof2.Asinpreviousexperiments,thesuccessratewasdeterminedthroughdirectcomparisonwiththevalidationsetextract-edfromthereferencemodel.Wealsoexaminedthede-pendencyofsuccessrateonparametervaluesfortheclasswiththeleastnumberoftrainingpoints(Class1–Organ-ics).Finally,binarymodelswerebuiltwithcombinationsofparametersdrawnfromthemarginsofthereasonableworkingrange.TheseincludedlowC(2-2)–highγ(28),lowC(2-2)–lowγ(25),averageC(27)–averageγ(26),highC(214)–highγ(28)andhighC(214)–lowγ(25).

RESULTS AND DISCUSSION

Binary Reconstruction

Theoriginaleskerbody,trainingsections,andthere-sultsofbinaryreconstructionwiththereferenceparametersetareshowninFigure2.Column1bofTable3describes

Table 2. Geologicalunits,SVMclasses,andtrainingsetstatisticsforEsker/AbitibiMulti-ClassModel.Totalnumberofpoints to be classified is 371783.

1. Geological Unit 2. SVM Class 3. All Points (#/%)a 4. Training Points (#/%)b

Organics 1 1210/0.31 48/0.01 Littoral 2 3819/0.97 193/0.05 Clay 3 13295/3.42 628/0.16 Esker 4 20300/5.22 995/0.26 Till 5 15865/4.08 747/0.19 Bedrock 6 334746/86.00 14841/3.81 AllUnits -- 389235/100389235/100389235/100 17452/4.4817452/4.4817452/4.48

aNumberofclasspointsandtheirpercentageofallmodelpointsbNumberoftrainingpointsperclassandtheirpercentageofallmodelpoints

Table 3.ValidationdataandresultsforEsker/Abitibibinaryandmulti-classmodel.Numberofvalidationpointsinorigi-nal model and percentage of points properly classified by SVM.

SVM Class 1. Binary 2. Multi-Class

a. Validation Points (#/%)a b. Success (%) a. Validation Points (#/%)a b. Success (%)

1 19305/5.19 71.50 1162/0.31 18.76 2 352478/94.81 98.87 3626/0.98 37.20 3 - - 12667/3.41 57.10 4 - - 19305/5.19 67.65 5 - - 15118/4.07 45.72 6 - - 319905/86.05 95.45 All 371783/100 97.34 371783/100 89.87

aNumberofvalidationpointsperclassandtheirpercentageofallmodelpoints

1233DGEOLOG�CALMODEL�NG

Figure 2.Binaryeskerreconstruction.(A)OriginalEskerUnit;(B)trainingset;(C)reconstructedEskerUnit.

thevalidationresults.AsseenfromTable3,thesuccessrateofSVMpredictionisexceedinglyhigh.Especiallyremarkableresults,98.87%,areachievedinClass2.�npart,thiscanbeattributedtothefactthatpointsofthisclassentirelydominatethetrainingset.WhentheSVMcannot classify a point in binary classification, it tends toattributeittothepredominantclass.Consideringthat

bedrockpointsconstitute94.81%ofallpointsthatneedto be classified (352478 of 371783 as shown in Column 1ofTable3),itisnosurprisethattheoverallsuccessofpredictionachieves97.34%.

With the above explanation in mind, the classifica-tionsuccessinClass1,whichrepresentsonlyabout6%ofthetrainingset,isstillashighas71.50%.This,inouropinion,provesthattheSVMcanbeeffectivelyusedforbinary(single-unit)reconstructionsevenwithtrainingsetssubstantiallyskewedtowardoneoftheclasses.

WefurtheranalyzedsuccessrateinClass1onallmodelsectionswheretheEskerUnitwaspresent(234sections).TheresultsarepresentedinFigure3.Thefigure clearly demonstrates that the success of prediction decreasesasthedistancefromatrainingsectionincreases.Astrainingsection#1didnotintersecttheeskerbody,the success rate on the first 18 sections drops to 0%. Therefore,ascouldbeexpected,theoverallreliabilityofpredictionisdirectlyproportionaltothedensityofsec-tionswithtrainingdata.

Multi-Class Reconstruction

TheresultsofthisexperimentarefoundinFigure4andColumn2bofTable3.Theoverallsuccessscoreis89.87%,whichismainlycontrolledbythepredominantbedrockclass(Class6).Twootherclasses,eskerandmarineclay,demonstratesuccessratesover50%.TheseunitsaresomewhatbetterrepresentedintheSVMtrainingsetthantheremainingclasses.Figure5showsthatthesuccessofreconstructionforaparticularclassinthisex-perimentisalmostdirectlyproportionaltothenumberofthoseclasspointsinthetrainingset,exceeding75%whenthenumberoftrainingpointsexceeds1%ofthetotal.

Wealsocomparedareaandvolumeofgeologicalunitsintheoriginalmodelanditsreconstructedcounter-part.TheresultspresentedinTable4andFigure6showthat,forbothareaandvolumecalculations,therecon-structedandoriginalvaluesforanyunitarethesameorderofmagnitude.Thissuggeststhatalongwithsingleunit modeling, the SVM can efficiently be applied in multi-unitvolumetricreconstructions.

Hyperparameter Sensitivity Tests and Multiple Parameter-Set Reconstructions

Figure7summarizestheresultsofgridsearchforthebestpairofhyperparametersinbinaryandmulti-classreconstructions.Thebestoverallsuccessrates,97.79%and92.03%,wereachievedat[C=21,γ=26]and[C=2-1,γ=26],respectively.Theanalysisofsuccessratesintheclasswiththeleastnumberoftrainingpointsyielded77.38%and22.46%at[C=22,γ=26]and[C=29,γ=25],correspondingly.Asseenfromtheresults,parametersarefairlystable,andasinglerangeforC[2-3-215]andγ[24-

124 D�G�TALMAPP�NGTECHN�QUES‘06

Figure 3.Validationresultsforbinaryeskerreconstructionfrom11parallelsections.SuccessinClass1(Esker)againstsectionnumber.Verticallinesindicatetrainingsections.Totallengthofhorizontalaxisis24kmandsectionsarespacedat100m.

Figure 4.Multi-classeskerreconstructionfrom11parallelsections.(A)Originalmodel;(B)trainingset;(C)reconstructedmodel.

1253DGEOLOG�CALMODEL�NG

Figure 5.Validationresultsformulti-classeskerreconstructionfrom11parallelsections.Successperclassvs.numberoftrainingpointsperclass.

Figure 6.Surfaceareaandvolumecomparisonfororiginal(referencemodel)andreconstructedgeologicalunits.(A)Surfacearea;(B)volume.

126 D�G�TALMAPP�NGTECHN�QUES‘06

Table 4.ResultsofSVMreconstructionfrom11parallelsectionsforEsker/Abitibi.Unitareaandvolumecomparison(originalmodelvs.reconstructed).SeeTable2fortrainingsetstatistics.

Geological Unit 1. Area (m2) 2. Volume (m2. Volume (m3)

a. Original b. Reconstructed a. Original b. Reconstructed

Organics 2.55E+07 2.99E+07 8.06E+07 1.05E+08Littoral 7.56E+07 7.34E+07 2.54E+08 3.30E+08Clay 1.99E+08 1.97E+08 8.86E+08 1.01E+09Esker 1.48E+08 1.33E+08 1.35E+09 1.16E+09Till 2.15E+08 1.99E+08 1.06E+09 1.07E+09Bedrock 2.70E+08 3.06E+08 2.23E+10 2.23E+10

Figure 7.Summaryofbestresultsfromparametersensitivitytestsforbinaryandmulti-classreconstructionsandproposedrangeforRBFkernelparameters(C[2-3,215]andγ[24,29]).

29]canberecommended.Withinthisrange,higheroverallscoresandhigherscoresforover-representedclassesareachievedatsomewhatlowerCvalues.Ontheotherhand,proper classification of points in the least represented classesrequireshigherCvalues.VisualexaminationofbinarymodelsbuiltwithcombinationsofparametersdrawnfromdifferentcornersoftheaboverangealsoshowthatamoregeneralizedpicturecanbeachievedatlowerC’s(Figure8a,8b)whilehighervaluesofthisparameterpromotemoredetailedinterpretation(Figure8d,8e).AverageCvaluesresultinwell-balancedmodels(Figure 8c). The influence of the second parameter (γ)isnotasobvious.

CONCLUSIONS

OurexperimentsclearlyshowedthattheSVMwithRBF kernel can be efficiently used for both single- and multi-unit3Dreconstructions.Theprocedureisper-formedinasinglestep,whicheliminatestheneedforunit-by-unitinterpolation.Evenfromalimitedtrainingset(e.g.,severalcross-sectionssparselydistributedacrossthestudyarea)reasonablereconstructionresultscanbeachieved.

�tisimportant,however,thatallclassestoberecon-structedarereasonablyrepresentedinthetrainingset.�nthemulti-classcase,thereconstructionsuccesswas

1273DGEOLOG�CALMODEL�NG

showntobedirectlyproportionaltothenumberofunitsamplesinthetrainingdata.Thereliabilityofpredic-tionisgreaterinthevicinityofthetrainingdata,andtherefore,thedensityoftrainingsectionsandspatialcontinuityoflithologicalunitsmaydirectlyaffectthereconstructionresults.

Thekernelparametersshouldbechosenfromtherange2-3-215forCand24-29forγ.Whenmoremodeldetailsarerequiredorclasseswithasmallnumberoftrainingpointsareinvolved,higherCvaluesshouldbeconsidered.LowerCvaluesresultinmoregeneralizedmodelswithfewerdetails.Thisfavorsclassesthatdomi-natethetrainingset.

Finally,ourresultsindicatethatwhenappropri-ateparametersarechosen,notonlythegeneralshapeofageologicalbody,butalsosuchcharacteristicsasitssurfaceareaandvolumecanbereconstructedwithresultsclosetothoseobtainedfromtheapplicationofclassicalG�Smethods.

Figure 8.BinaryreconstructionswithparametersdrawnfromdifferentcornersoftherangepresentedinFigure7.(A)lowC(2-2)–highγ(28);(B)lowC(2-2)–lowγ(25);(C)averageC(27)–averageγ(26);(D)highC(214)–highγ(28);(E)highC(214)–lowγ(25).

ACKNOWLEDGEMENTS

TheresearchwasconductedattheQuebecDivi-sionoftheGeologicalSurveyofCanada(GSC),andwasfundedbytheGroundwaterProgramoftheEarthSci-encesSectorofNaturalResourcesCanada(contributionnumber20060335).WeareespeciallygratefultoAndréeBolduc(GSC)forherinvolvementinthisworkandforhavingmadetheGocad files with initial data and 3D Es-ker/Abitibimodelusedinexperimentsreadilyavailable.

REFERENCES

Abe, S., 2005, Support Vector Machines for Pattern Classifica-tion:Springer-Verlag,London,343p.

BolducA.M.,ParadisS.J.,RiverinM.N.,LefebvreR.,andMi-chaudY.,2005,A3Deskergeomodelforgroundwaterre-search:ThecaseoftheSaint-Mathieu–Berryesker,Abitibi,Quebec,Canada,inRussell,H.,Berg,R.C.,andThorleif-

128 D�G�TALMAPP�NGTECHN�QUES‘06

son,L.H.eds.,Three-DimensionalGeologicalMappingforGroundwaterApplications,GeologicalSurveyofCanada,Ottawa,Ontario,OpenFile5048,p.17-20.

Chang,C-C.,andLin,C-J.,2001,L�BSVM:aLibraryforSup-portVectorMachines,accessedathttp://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf.

Cristianini,N.,andShawe-Taylor,J.,2000,SupportVectorMachines:CambridgeUniversityPress,189p.

El-Naqa,�.,Yang,Y.,Wernik,M.N.,Galatsanos,N.P.,andNishikawa,R.,2002,Supportvectormachinelearningforthe detection of microcalcifications in mammograms: IEEE TransactionsonMedical�maging21,p.1552-1563.

Gilardi,N.,Kanevski,M.,Maignan,M.,andMayoraz,E.,1999,Environmental and Pollution Spatial Data Classification withSupportVectorMachinesandGeostatistics:WorkshopW07“�ntellegenttechniquesforSpatio-TemporalDataAnalysisinEnvironmentalApplications”,ACA�99,July,Greece,p.43-51.

Hsu,C-W.,Chang,C-C.,andLin,C-J.,2004,APracticalGuide

to Support Vector Classification, accessed athttp://www.csie.ntu.edu.tw/~cjlin/papers/guide/guide.pdf.

Hsu,C-W.,andLin,C-J.,2002,Acomparisonofmethodsformulti-classsupportvectormachines:�EEETransactionsonNeuralNetworks13(2),p.415-425.

Mallet,J-L.,1989,DiscreteSmooth�nterpolation:ACMTrans-actionsonGraphics8(2),p.121-144.

Noble,W.S,Kuehn,S.,Thurman,R.,Yu,M.,andStamatoyan-nopoulos,J.,2005,Predictingtheinvivosignatureofhumangeneregulatorysequences:Bioinformatics21(1),p.338-343.

Smirnoff,A.,Boisvert,E.,andParadis,S.J.,2008,SupportVectorMachinefor3Dmodellingfromsparsegeologicalinforma-tionofvariousorigins:ComputersandGeosciences34,p.127-143.

Vapnik,V.,1995,TheNatureofStatisticalLearningTheory:Springer-Verlag,NewYork,311p.

Yu,X.,Liong.,S-Y.,andBabovic,V.,2004,EC-SVMapproachforreal-timehydrologicforecasting:JournalofHydroinfor-matics06(3),p.209-223.