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A complete probabilistic framework for learning input models for power and crosstalk estimation in VLSI circuits

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Title:
A complete probabilistic framework for learning input models for power and crosstalk estimation in VLSI circuits
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Ramalingam, Nirmal Munuswamy
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Subjects / Keywords:
leakage
noise
sampling
learning Bayesian network
power estimation
Dissertations, Academic -- Electrical Engineering -- Masters -- USF
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bibliography   ( marcgt )
theses   ( marcgt )
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ABSTRACT: Power disspiation is a growing concern in VLSI circuits. In this work we model the data dependence of power dissipation by learning an input model which we use for estimation of both switching activity and crosstalk for every node in the circuit. We use Bayesian networks to effectively model the spatio-temporal dependence in the inputs and we use the probabilistic graphical model to learn the structure of the dependency in the inputs. The learned structure is representative of the input model. Since we learn a causal model, we can use a larger number of independencies which guarantees a minimal structure. The Bayesian network is converted into a moral graph, which is then triangulated. The junction tree is formed with its nodes representing the cliques. Then we use logic sampling on the junction tree and the sample required is really low.Experimental results with ISCAS '85 benchmark circuits show that we have achieved a very high compaction ratio with average error less than 2%. As HSPICE was used the results are the most accurate in terms of delay consideration. The results can further be used to predict the crosstalk between two neighboring nodes. This prediction helps in designing the circuit to avoid these problems.
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Thesis (M.S.E.E.)--University of South Florida, 2004.
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Includes bibliographical references.
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by Nirmal Munuswamy Ramalingam.
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A complete probabilistic framework for learning input models for power and crosstalk estimation in VLSI circuits
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ABSTRACT: Power disspiation is a growing concern in VLSI circuits. In this work we model the data dependence of power dissipation by learning an input model which we use for estimation of both switching activity and crosstalk for every node in the circuit. We use Bayesian networks to effectively model the spatio-temporal dependence in the inputs and we use the probabilistic graphical model to learn the structure of the dependency in the inputs. The learned structure is representative of the input model. Since we learn a causal model, we can use a larger number of independencies which guarantees a minimal structure. The Bayesian network is converted into a moral graph, which is then triangulated. The junction tree is formed with its nodes representing the cliques. Then we use logic sampling on the junction tree and the sample required is really low.Experimental results with ISCAS '85 benchmark circuits show that we have achieved a very high compaction ratio with average error less than 2%. As HSPICE was used the results are the most accurate in terms of delay consideration. The results can further be used to predict the crosstalk between two neighboring nodes. This prediction helps in designing the circuit to avoid these problems.
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ACompleteProbabilisticFrameworkforLearningInputModelsforPowerandCrosstalkEstimationinVLSICircuitsbyNirmalMunuswamyRamalingamAthesissubmittedinpartialfulllmentoftherequirementsforthedegreeofMasterofScienceinElectricalEngineeringDepartmentofElectricalEngineeringCollegeofEngineeringUniversityofSouthFloridaMajorProfessor:SanjuktaBhanja,Ph.D.Yun-LeeiChiou,Ph.D.SrinivasKatkoori,Ph.D.DateofApproval:October6,2004Keywords:PowerEstimation,LearningBayesianNetwork,Sampling,Noise,LeakagecCopyright2004,NirmalMunuswamyRamalingam

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DEDICATIONTomydad,mom,sisterandmybeautifulnieceShruti.

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ACKNOWLEDGEMENTSIliketorstthankmyadvisorDr.SanjuktaBhanja,whoseguidinglightIfollowedthroughouttheproject.ShehelpedmeinvariouswaysasIhitthewallmanyatimesthroughthecourseofthiswork.IwouldalsoliketothankDr.SrinivasKatkooriandDr.Yun-LeeiChiouforservinginmycommittee.Mycolleagues,themembersoftheVLSIDesignAutomationandTestLab,namelyThara,Karthik(Bheem),Shiva,Sathish(Ponraj),Vivek(awk),alsohelpedmewithmanyinsightfulideas,nottoforgettheircompanywhichIenjoyedwhileworkinginthelab.Also,IwouldliketothankBodka,Venky,Sathishforhelpingmewithcodingproblems.IwouldalsoliketothankMelodieandHenryformakingmyworkexperinceattheDivisionofResearchComplianceapleasantone.Butitwastheunconditionallovefrommyparentsandmysisterthatkeptmegoing.

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TABLEOFCONTENTSLISTOFTABLESiiiLISTOFFIGURESivABSTRACTvCHAPTER1INTRODUCTION11.1PowerEstimation31.2InputModel31.3CrosstalkEstimation91.4ContributionoftheThesis10CHAPTER2RELATEDWORK122.1VectorCompaction122.2CrosstalkEstimation15CHAPTER3LEARNINGBAYESIANNETWORKS173.1BayesianNetworks173.2Learning213.3AlgorithmforLearningBayesianNetworkGivenNodeOrdering233.3.1Step1:Drafting243.3.2Step2:Thickening273.3.3Step3:Thinning273.3.4FindingMinimumCut-Sets293.3.5ComplexityAnalysis30CHAPTER4SAMPLING324.1CausalNetworks334.2Inference374.2.1MoralGraph374.2.2Triangulation384.2.3JunctionTree394.2.4PropagationinJunctionTrees414.2.5ProbabilisticLogicSampling43CHAPTER5EXPERIMENTALRESULTS45CHAPTER6CONCLUSION55i

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REFERENCES57ii

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LISTOFTABLESTable5.1.PowerEstimatesofISCAS'85BenchmarkSuite.47Table5.2.ErrorEstimatesinPowerEstimation.47Table5.3.JointProbabilitySwitchingEstimateforNodes300and330ofC432for60K.51Table5.4.JointProbabilitySwitchingEstimateforNodes300and330ofC432forCR40.51Table5.5.JointProbabilitySwitchingEstimateforNodes557and558ofC499for60K.51Table5.6.JointProbabilitySwitchingEstimateforNodes557and558ofC499forCR40.52Table5.7.JointProbabilitySwitchingEstimateforNodes141and113ofC1355for60K.52Table5.8.JointProbabilitySwitchingEstimateforNodes141and113ofC1355forCR40.52Table5.9.JointProbabilitySwitchingEstimateforNodes2427and2340ofC1908for60K.53Table5.10.JointProbabilitySwitchingEstimateforNodes2427and2340ofC1908forCR40.53Table5.11.JointProbabilitySwitchingEstimateforNodes4899and4925ofC3540for60K.53Table5.12.JointProbabilitySwitchingEstimateforNodes4899and4925ofC3540forCR40.54iii

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LISTOFFIGURESFigure1.1.PowerDensityasPredictedin[26].2Figure1.2.PowerEstimationTechniques.4Figure1.3.SimpleModelofCoupledWires.10Figure2.1.DMCModeling.13Figure2.2.DataCompactionforPowerEstimation.14Figure3.1.ExampleforaDAG.18Figure3.2.Exampleford-separation.23Figure3.3.Step1:Drafting.25Figure3.4.WorkingMechanism.26Figure3.5.Step2:Thickening.28Figure4.1.CausalDiagram.34Figure4.2.ASmallCircuit.35Figure4.3.BNCorrespondingtotheCircuit.36Figure4.4.MoralGraph.38Figure4.5.TriangulatedGraph.39Figure4.6.JunctionTree.40Figure4.7.TwoCliqueswiththeSeparatorSet.42Figure5.1.ProcessFlow.46Figure5.2.LearnedBayesianNetworkoftheInputsofC432.48Figure5.3.LearnedBayesianNetworkoftheInputsofC3540.49Figure5.4.LearnedBayesianNetworkoftheInputsofC1908.49iv

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ACOMPLETEPROBABILISTICFRAMEWORKFORLEARNINGINPUTMODELSFORPOWERANDCROSSTALKESTIMATIONINVLSICIRCUITSNirmalMunuswamyRamalingamABSTRACTPowerdisspiationisagrowingconcerninVLSIcircuits.Inthisworkwemodelthedatade-pendenceofpowerdissipationbylearninganinputmodelwhichweuseforestimationofbothswitchingactivityandcrosstalkforeverynodeinthecircuit.WeuseBayesiannetworkstoeffec-tivelymodelthespatio-temporaldependenceintheinputsandweusetheprobabilisticgraphicalmodeltolearnthestructureofthedependencyintheinputs.Thelearnedstructureisrepresentativeoftheinputmodel.Sincewelearnacausalmodel,wecanusealargernumberofindependencieswhichguaranteesaminimalstructure.TheBayesiannetworkisconvertedintoamoralgraph,whichisthentriangulated.Thejunctiontreeisformedwithitsnodesrepresentingthecliques.Thenweuselogicsamplingonthejunctiontreeandthesamplerequiredisreallylow.ExperimentalresultswithISCAS'85benchmarkcircuitsshowthatwehaveacheivedaveryhighcompactionratiowithaverageerrorlessthan2%.AsHSPICEwasusedtheresultsarethemostaccurateintermsofdelayconsideration.Theresultscanfurtherbeusedtopredictthecrosstalkbetweentwoneighboringnodes.Thispredictionhelpsindesigningthecircuittoavoidtheseproblems.v

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CHAPTER1INTRODUCTIONMoore'sLawwhichstatesthatthenumberoftransistorsperintegratedcircuitwoulddoubleeverycoupleofyearshasbeenoneofthedrivingforcesforthedevelopmentteamstobreakdownthesebarriers.Butthepowerconsumedhasnotreducedandisnotpredictedtoreduceeveninthenano-domain.Powerdensityorpowerdissipatedperunitareaisincreasingduetothepackingofmillionsoftransistorsinasinglewafer(showninFig.1.1.).Now,becauseoftheincreaseinpowerdensity,costincreasedforhighercoolingandincreasedbatteryweight,nottoforgetthereductioninsystemreliability.Designershadtomakelowpowerdevicestoovercometheseproblems,andtheimpetusonlow-powerdesign,sawanincreasedattentiontopowerestimation.Thetotalpowerconsumedbyacircuitiscalledtheaveragepowerisgivenbytheformula,PavgPdynamicPshortPleakage(1.1)CMOScircuitsconsistofapull-upandpull-downnetwork,whichhaveaniteinputfall/risetimelargerthanzero.Duringthisshorttimeinterval,whenboththepull-upandpull-downnetworkareconducting,acurrenticcowsfromsupplytoground,calledtheshort-circuitcurrent,resultinginshortcircuitpower.Staticleakagepowerdissipiationcanbeattributedtoreversebiasdiodeleakage,sub-thresholdleakage,gateoxidetunneling,leakageduetohotcarrierinjection,Gate-InducedDrainLeakage(GIDL),andchannelpunchthrough.Notethatthistypeofpowerdissipationdependsonthelogicstatesofacircuitthanitsswitchingactivites.1

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Intel Pentium IIIntel Pentium ProIntel Pentium486Power(W/cm386 Time11010001002)Power DensityIntel Pentium III Figure1.1.PowerDensityasPredictedin[26].Themostimportantisthedynamiccomponentwhichconstitutesabout80%oftheaveragepower.ThecurrentidthatowsduetothecharginganddischargingoftheparisiticcapacitanceduringswitchingcausesthepowerdissipationPdynamic.Havingagatelevelimplementationofatargetcircuit,toestimatethetotalpowerdissipation,wecansumoverallthegatesinthecircuittheaveragepowerdissipationduetocapacitiveswitchingcurrents,thatis:Pavg05fclkV2ddnCnswn(1.2)wherefclkistheclockfrequency,Vddisthesupplyvoltage,Cnandswnarethecapacitanceandtheaverageswitchingactivityofthegaten,respectively.Fromthisequationwecanseethattheaverageswitchingactivityofeverygateinthecircuitisakeyparameterthatneedstoestimatedaccurately,particulalryifoneneedsthenode-by-nodepowerasthevoltageandclockfrequencyareknowntothedesigners.Switchingactivityisameasureforthenumberofgatesandtheiroutputsthatchangetheirbit-valueduringaclockcycle.Thetogglingbetweenlogiczeroandlogicone,capacitancesgetchargedanddischarged.2

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1.1PowerEstimationAveragepowerestimationtechniquescanbedividedintothreebroadcategories,namelyesti-mationbysimulation,statisticalsimulationandprobabilistictechniquesasshowninFig.1.2.Simulationbasedpoweranalysisrequiresasetofsimulationvectorsattheprimaryinputsofthecircuittotriggerthecircuitactivities.Toobtainthepowerdissipationofthecircuit,theswitchingactivityinformationiscollectedandappliedtopowermodelsaftersimulation.Thesevectorshavesubstantialimpactonthepowervaluesbecausethepowerdissipationreliesheavilyontheswitchingactivity.Eachvectorcausessomeenergytobedissipatedandthetotalpoweristhesummationoftheenergyofeachvectoranddividingoverthetotalsimulationtime.Usuallythesetechniquesprovidesufcientaccuracyattheexpenseoflargerunningtimesasthesemethodsuseaverylargesetofinputvectors.Howeverthemethodbecomesunrealistictorelyonwhendoneonlargecircuits.Statisticalmethodsareusedincombinationwithsimulationtechinquesinstatisticalsimula-tion[41],[42]todeterminethestoppingcriterion.Thoughthesetechniquesareefcientintermsofthetimerequired,onehastobecarefulinmodelingstatisticalpatternsattheinputsandcareshouldbetakentonotgettrappedinalocalminima.Inprobabilistictechniquestheinputstatistics(e.g.,switchingactivityoftheinputs,signalcor-relations,etc.)arerstgatheredintermsofprobabilitiesandthentheyarepropagated.Theyarefastandmoreadaptable,butinvolveanassumptionaboutjointcorrelations.Theyprovidesufcientaccuracywithalowcomputationaloverhead,butfactorssuchasslewrates,glitchgenerationandpropagationaredifculttocapture.Anotherchallengeinthemethodistheabilitytoaccountforinternaldependenciesduetoreconvergentfanoutofthetargetcircuit.1.2InputModelItiscrucialthatthevectorsusedinsimulationrepresentthetypicalconditionsatwhichthepowerestimateissought.Ifthesimulationvectorsdonotcontaintheproperinstructionmixthepoweranalysisresultwillbeskewed.Regardlessofhowthesimulationvectorsaregenerated,ifwesimulatethecircuitwithonlyseveralvectors,thepowerdissipationresultobtainedisnottruthfulbecausethevectorlengthistooshort.Mostpartofthecircuitisprobablynotexercisedenough3

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(Logic Level) Average Power Estimation for Combinational CicuitsSimulationStatistical SimulationProbabilistic Benini et al.'93 [27]Kang et al.'86 [28]Murugavel et al.'02 [29]Yacoub et al.'89 [30]Krodel '91 [31]Deng et al.'88 [32]Tjarnstrom '89 [33]Burch et al.'93[34]Najm et al.'98[35]Parker et al.'75 [36]Cirit et al.'87 [37]Bryant et al.'86 [38]Ercolini et al. [39]Chakravarti et al.'89 [40]Burch et al.'88 [41]Najm et al.'90 [42]Ghosh et al.'92 [43]Stamoulis et al.'93 [44]Najm et al.'93 [45]Kapoor et al.'94 [46]Schneider et al.'94 [47]Marculescu et al.'94 [48]Schneider et al.'96 [49]Ding et al.'98 [50]Marculescu et al.'98 [21]Bhanja et al.'02 [1]Figure1.2.PowerEstimationTechniques.4

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toobtainactualtogglingactivites.Ontheotherhandwecansimulatethecircuitforaverylargenumberofvectorstoobtainanaccuratemeasureofthepowerdissipation.Butisitnecessarywastingcomputerresourcestosimulatethatmanyvectors?Howmuchextraaccuracycanweacheivebysimulatingamillionvectorsversusonlyathousandvectors?Howdoweknowthatwehavesimulatedenoughvectorlength?Whatisthecostinvolved?Assaidin[5]thevectorcompactionproblemreducesthegapbetweensimulativeandnonsim-ulativeapproaches.Theinputstatisticsthatmustbecapturedandthelengthoftheinputsequenceswhichmustbeappliedaresomeoftheissuesthatmustbetakenintoconsideration.Generatingaminimallengthsequencethatstaisestheaboveconditionsisnotatrivialtask.Sovectorcom-pactionisthetechniquebywhichasmallersetofvectorsisderivedfromalargersetbyaseriesofstepssuchthatthesmallerderivedsetpreservestheoriginalstatisticalpropertiesofthelargerinitialset.Comingbacktothequestiononthelengthofthevectors,simulationmethodsareaccuratebutsuffercostandmemoryoverhead,whichlimitthesizeoftheinputvectorsettohundredsorthousandsofvectors.Butthisresultsininaccuracyinthepowerestimationprocessbecausethepowerconsumptionindigitalcircuitsisinputpatterndependant,thatisdependingontheinputvectorsappliedtothetargetcircuitverydifferentpowerestimatesmaybeobtained.Toobtainanaccuratepowerestimate,asetofinputsthatresemblethecharacteristicsofthedatafortypicalapplicationsisrequired.Sotoacheivethisgoalthevectorsethasasizeofmillionsofvectors.Vectorsofthesizeofhundredsorthousands,ifselectedrandomlyfromthelargevectorsetmaynotbeabletocapturethetypicalbehaviorandthusmayleadtoanunderestimationoroverestimationofthepowerconsumedbythecircuit.Soourmethodofvectorcompactionsolvestheproblembycompactingthemillionsofvectorsintoacharacteristicinputsetofamuchsmallerlength,yetstatisticallyequivalenttotheoriginallargervector,thusprovdingapowerestimateveryclosetothepowerestimatedbythelargersequence.SwitchingmodelofaVLSIcircuitisacomprehensiverepresentaionofswitchingbehaviorofallthesignalsinthecircuit.Ateachsignalwestorethestateattimetandthestateattimet1.ThedependencyamongalltheswitchingvariablecanbecapturedasajointProbability5

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distributionfunction.Onemightlookintoswitchingmodelasawaytoestablishtheroleofdataasinputstothecircuits.Inessence,theswitchingmodelcapturesthedata-drivenuncertaintyinVLSIcircuitsinacomprehensiveprobabilisticframework.Needlesstosaythat,modelinginputsbecomeanintegralpartoftheswitchingmodel,eventhoughahandfulofpriorworkinpowerorswitchinganalysis,hasmodeledinputsefciently.Evenanalysisthatarevectordriven(simulationandstatisticalsimulation)havemostlybecomeassumedinputpriorsasrandominputs.Inthiswork,wetakealookintomodelinginputsthroughacausalgraphicalprobabilisticapproachwhichmodelsinputspaceinacompactway.Thisinputmodelthencanbefusedwithavectorlessprobabilisticmodelorbeusedtogeneratesamplesforstatisticalorsimulativeapproachesbyarandomwalkintheprobabilsiticnetwork.NotethattheswitchingmodelisextremelyrelevantofbothstaticanddynamiccomponentofpowerasshowninEq.1.3.InEq.1.3,Pdgrepresentsthedynamiccomponentofpowerattheoutputofagateg.Theimpactofdataondynamiccomponentofpowerisencasulatedina,thesingletonswitchingactivity.ThestaticcomponentofpowerPsgisdominatedbyPleaki,leakagelossinaleakagemodei.Ithastobenotedthateachleakagemodeisdeterminedbythesteadystatesignalsthateachtransistorinthegatewouldbein.Forexample,inatwoinput(sayAandB)NANDgate,thegatewouldhavefourdominantleakagemode(i=4:A@0B@0A@0B@1A@1B@0andA@1B@1).bistheprobabilityofeachmodei.Probabilisticallybothaandbaresingletonprobabilityofswitchingandjointprobabilityofmultiplesignalsinagaterespectivelyandaredependentontheinputdataprole.Theswitchingmodelisaffectedbyvariousfactorssuchasthetopologyofthecircuit,theinputstatistics,thecorrelationbetweennodes,thegatetype,andthegatedelays,thusmakingtheestimationprocessacomplexprocedure.PtgPtgPdgPsg05afV2ddCloadwireiPleakibi(1.3)Inthiswork,wefocusonmodelingtheinputsandgenerateaprobabilisticstructureamongtheinputsthatcanbeusedtostudythebehaviorsofinternalnodes.Eventhoughestimationofsingletonswitchinghasbeendiscussedingreatdepthandmanyoftheproceduresareinputdriven(simulativeandstatisticalsimulative),inputsarestudiedinalimitedsetofworks[2,3,57].6

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Eq.1.4demonstratestheeffectofinputmodel.LetXibeanintermediatesignalwhichcouldbesingletonswitchingvariablerepresentingswitchingstates(0tn10t0tn11t1tn10tand1tn11t),ofasignalorXicouldbecompositesignals(A,B)inatransistorstackandcanhavecompositestatesnamelyA0tn10tB0tn10t.InestimatingXi,asshowninEq.1.4,wehavetwocomponents,namelythesetPXirIj,Ijdenotingtheprimaryinputsofthecircuits,wherejisthejthinputandPIjapriorprobabilityofthetheinputspace.Therstcomponentcanbeanalysedbyprob-abilisticmeasureconsideringajointpdfoftheentiresignalsthroughaprobabilisticframeworkasin[43,21,1]orcanbemeasured`[28,20]forspecicinputpatterns.ThesecondcomponentPIjhowever,isimportantforbothstimulus-sensitiveapproachandstimulus-freeapproachwhereweneedtomodelthedependencystructureoftheinputsandtheneitheruseitinthejointpdfofthesignalsforprobabilisticmethodsoruseittogeneraterepresentativespecicinputsamplesformeasurement-basedestimates.ThethemeofthisworkistoobtaincorrectpriorsintheinputspacePIjandthentogeneratesamplesthatcloselyemulatethebehavioroftheinputspaceortofusethepriorontoanexistingprobabilisticset-up.Notethat,wedonotconcentrateoncompaction,butourclaimisthatthesamplesaresoclosetotheactualdistributionthatthesamplerequirementsarereallylowandahighcompactionratioisautomaticallyachieved.PXiPXirIjPIj(1.4)Notethatinputmodelingisnotasimpletaskandcanbeusefulinotherareasliketestvectorandpatterngeneration.Thisworkisanefforttolearnacausalstructureintheinputdata.Causalstructuresarecommontoobtainforreal-lifedataandhavebeenprovensuccessfulinmodelingcomplexdata-setslikegene-matchingandspeechprocessing.Acausalmodelcanencapsulatetwoadditionalindependencies(induceddependencyamongthecauseofacommonchild,andweaklytransitivedependency)overandabovetheirundirectedMarkovmodelsthatmakesitspeciallyat-tractiveinreducingthestructure.Toourknowledge,thisistherstmodelwherecausalityinthedataisutilizedinlearningastructureforaninputmodelinVLSI.WeusedependencyanalysisininputstoarriveataBayesianNetworkwhichinvolvesthreesteps,drafting,thickeningandthinningasdonein[16].Intherststepmutualinformationbetween7

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pairsofnodesiscalculatedandanapproximategraphbasedontheinformationiscreated.Inthenextsteparcsareaddedwhenthepairsofnodesarenotconditionallyindependentonaceratinconditionset.IfthemodelisDAGfaithful,thegraphobtainedhereisanI-mapofthemodel.Inthelaststepeacharcaddedischeckedusingconditionalindependencetestsandwillberemovedifthenodesareconditionallyindependent.ThenalresultisaP-mapoftheunderlyingmodel.Bayesiannetworksaredirectedacyclicgraph(DAG)representationswhosenodesrepresentrandomvariablesandthelinksdenotedirectdependencies,whicharequantiedbyconditionalprobabilitiesofanodegiventhestatesofitsparents.ThisDAGstructureessentiallymodelsthejointprobabilityfunctionoverthesetofrandomvariablesunderconsiderationinacompactmanner.Theattractivefeatureofthisgraphicalrepresentationofthejointprobabilityfunctionisthatnotonlydoesitmakeconditionaldependencyrelationshipsamongthenodesexplicitbutalsoservesasacomputationalmechanismforefcientprobabilisticwalkgeneratingsamples.ProbabilisticLogicSamplingisdoneinsidethecliquesofthejunctiontreetoobtaintheneces-sarysamples.Thestepsinvolvedinformationthejuctiontreeisthecreationofamoralgraph,andtheprocessiscalledcompilation,thenthemoralgraphistriangulated.Thecliquesetisidentiedandthejunctiontreeofcliquesisformed.GivenaBayesiannetwork,amoralgraphisobtainedby'marryingparents',thatisaddingundirectededgesbetweentheparentsofacommonchildnode.Beforethisstep,allthedirectionsintheDAGareremoved.Themoralgraphissaidtobetrian-gulatedifitischordal.TheundirectedgraphGiscalledchordalortriangulatedifeveryonofitscyclesoflengthgreaterthanorequalto4posessesachord[9],thatisweaddadditionallinkstothemoralgraph,sothatcycleslongerthan3nodesarebrokenintocyclesofthreenodes.Thejuntiontreeisdenedasatreewithnodesrepresentingcliques(collectionofcompletelyconnectednodes)andbetweentwocliquesinthetreeTthereisauniquepath.ProbabilisticLogicSamplingisamethodproposedbyHenrion[56]whichemploysastochasticsimulationapproachtomakeprobabilisticinferencesinlargemultiplyconnectednetworks.IfwerepresentaBayesiannetworkbyasampleofmdeterministicscenarioss=1,2,.....mandLsxisthetruthofeventxinscenarios,thenuncertaintyaboutxcanberepresentedbyalogicsample.8

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1.3CrosstalkEstimationAnevaluationnodeisacircuitnodethatformsaconnectionbetweenchannelconnectedcompo-nentsinthedesign.Noisecanbedenedasanythingthatcausesthevoltageofanevaluationnodetodeviatefromthenominalsupplyorgroundrailswhenitshouldnormallyhaveastablehighorlowasdeterminedbythelogicofthecircuitinconsideration.Thenoisesourcesthatareofinteresttodigitaldesignareleakagenoise,charge-sharingnoise,powersupplynoiseandcrosstalknoise.LeakageisduetotheoffcurrentofFETs,andislargelyduetothesubthresholdcurrentandisdirectlydeterminedbythethresholdvoltageandtemperature.WhereaspowersupplynoiseisthenoiseappearingonthesupplyandgroundnetsandcoupledontoevaluationnodesthroughaFETconductionpath.Chargesharingnoiseiscausedduetothechargeredistributionbetweenthedynamicevaluationnodeandinternalnodesofthecircuit.Crosstalknoiseisthevoltageinducedonanodeduetocapacitivecouplingtoaswitchingnodeofanothernet.Itcanalsobesaidasthecapacitiveandinductiveinterferencecausedbythenodevoltagedevelopedonsignallineswhenthenearbylineschangestate.Itisafunctionoftheseparationbetweensignallines,thelineardistancethatsignallinesrunparallelwitheachother.Intoday'slogicdevicesthefasteredgerategreatlyincreasesthepossibilityofcouplingorcrosstalkbetweenthesignals.Crosstalkisoneoftheissuesthathamperdesignerstoacheivehigherspeeds,andsoitmustbereducedtolevelswherenoextratimeisrequiredforthesignaltostabilize.Theimportancetoestimatecrosstalkcanbefurtherunderstoodaswediscussthefaultsthatitmaybeinducedbetweentwocoupledinterconnects,namelytheaggressorandthevictim.1.DelayFaultItoccurswhensignalsofthetwocoupledinterconnectionsundergooppositeswings.Thisfaultaffectsthegatedelaywhichinturncanchangethecriticalpathdelayandcauseglitches.2.LogicFaultHerealogicerroroccurswhenthevoltageinducedinthevictiminterconnectbytheaggressorinterconnectisgreaterthanathreshold.Thiscausesthecircuitmalfunctionwhentherisktoleranceboundisexceeded.9

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RRCCimCiLiLLimaggressorvictimaggressorvictimtimetimecoupled noiseVVFigure1.3.SimpleModelofCoupledWires.3.Noise-inducedRaceFailuresTheracefailuresareaconsequenceofthedelayfault.Thechangeinthedelaycausesahold-timeviolation,commonlynoticedinpipelinedcircuits.InasimpliedmodelasshowninFig.1.3.,couplingcanbeconsideredbetweenthetwolinesasacapacitivevoltagedivider.HigherintegrationcauseslargermutualcapacitanceCmandsmallerintrinsiccapacitanceCiofaline,bothworseningcrosstalk.ThesameisvalidfortheinductivecouplingasthemutualinductanceLmgrowssignicantlyindenserstructures.TakingtheoutputresistancesRandtheinputcapacitancesofthereceivingcircuitsintoaccountthemodelshowsacrosstalkvoltagepropotionaltothecouplingwirelength.1.4ContributionoftheThesisThecontributionofthisworkistwo-fold.First,wearriveataprobabilisticgraphicalmodelintheinputsthatis(i)edge-minimal(ii)exactintermsofdependenceand(iii)easytolearn.Second,itiselegantasamodelandalsoasasourceofgeneratingsamplesfromthisgraphicalprobabilisticstructurethatcloselyresemblethedependencyintheinputsandafewsamplesconvergestothemeanoftheunderlyingdistribution.Thus,thisinput-modelcanbeofdualpurpose:(i)Itcanbe10

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fusedwithagraph-basedprobabilisticsetupforthecircuitdependency[1]andmakethewholeestimationprocessstimulus-freeand(ii)canbeusedtogeneratesamplesthatcloselymatchtheoriginaldependencymodelininputsforstatisticalsimulationandsimulationbasedestimation.ThesalientfeaturesoftheproposedBayesianNetwork(BN)learningmodelforinputsareasfollows.1.Itgeneratesanedge-minimalstructurethatmodelsdependencyexactlyunderacausaldataenvironment2.ThecomputationsareeasyandlearningalgorithmsareO(N2toO(N4)intermsofnumberofinputs3.Thedependencymodeloftheinputscanbefusedwithgrphicalstructureoftheinternalcircuitmakingtheestimationstimulus-freeandinsensitivetomeasurements4.Thedependencymodelcanbeprobabilisticallyefcientlysampledsuchthatsamplescloselyemulatethedependenciesintheinputsforstatisticalsimulationandsimulationbasedesti-mationprocess.5.TheperformancethatisseentheISCAScircuitsgeneratesamaximumerrorof1.8%withacompressionratioupto300.11

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CHAPTER2RELATEDWORK2.1VectorCompactionThepioneeringworkinthiselddonebyMarculescuet.al,[2]-[7],haveapproachedtheproblemrstusingaStochasticSequencialMachineandlaterbyusingMarkovNetworks.In[3]theapproachtotacklethisproblemwasbasedonthestochasticsequentialmachines(SSMs)theoryandemphasiswasonthoseaspectsrelatedtoMoore-typemachines.FiniteAutomataaremathematicalmodelsforvarioussystemswithanitenumberofstates.Thesemodelsacceptpar-ticularinputs,atdiscretetimeintervalsandemitoutputsaccordingly.Theautomataisconsideredtoshowstochaticbehaviorasthecurrentstateofthemachineandtheinputgivendetermineaccord-inglythenextstateandtheoutputoftheautomaton.Heretheauthorssynthesizerstastochasticmachinethatisprobabilisticallyequivalenttotheinitialsequence.Then,byapplyingrandomlygeneratedinputstothestochasticmachine,arandomwalkthroughthestatesofthetransitiongraphisdoneandashortersequnceisgenerated.Theideaproposedinthispaperisthattheautomatonwheninstatexiandreceivinginputn,canmoveintoanewstatexjwithapositiveprobabilitypxin.ThebasictaskisofsynthesizinganSSMwhichiscapableofgeneratingconstrainedinputsequencesanduseitinpowerestimation.ThemethodproposedforpowerestimationofatargetcircuitforagiveninputvectorsequenceoflengthL0istoderiveaprobabilisticmodelbasedonSSMsandgenerateashortersequenceL,whichisusedtocalculatethepowerofthetargetcircuit.Themethodusedin[3]hassomelimitationsastheauthorsuseprobabilisticautomatatheorytosynthesisestochasticmachines,wherethecompactiontechniquebecomesamulti-stepprocess.Aninitialpassthroughthesequenceisperformedrsttoextractthestatisticsneededandthenthestochasticmachineissynthesisedtogeneratethenewcompactedsequence.Thismethodbearsa12

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Intial SequenceL0DMC ModellingGenerate Compacted Sequence LCompacted Sequence L
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inouttargetcircuitL1 DMCmodelingsequencegenerationintargetcircuitoutrandom input sequenceL1L
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intoseveralgroupsaccordingtotheirpowercharacteristics.Thentheshortersequenceisobtainedbyconsecutivesampling.Inthemethodproposedin[14],asetoftestvectorsisgeneratedusinganytestgenerationalgorithm.Thenthecompactionprocedure,COMPACTisused,andthetestsobtainedhere,arecomparedtotheothertestswhicharegeneratedpreviously.IfthesetgeneratedbyCOMPACTiscompatablewiththeotherteststhenthelattersetisreplacedbythecommonvectors.Threedynamictestvectorcompactionmethodsarespeciedherein[15]forareducedtestsequencegeneration.InallthemethodsthesequencegeneratedfromanAutomaticTestPatternGenerator(ATPG)forasinglefault,isassignedspecicvaluesforunspeciedprimaryinputs.IntherstmethodcalledtheBestRandomFill,alltheunspeciedprimaryinputsaregivennitenumberofrandomnumbers,andeachonesimulatedtodeterminewhichdetectsthemostnumberoffaults.ThenextmethodcalledtheBestRandomFill2hasslightmodicationsofthepreviousalgorithm.Hereeachfaultisgivenaweight,theharderonegettingthehighest,andthellwhichgetsthehighestnumberofharder-to-ndfaultsisselected.Thisalgorithmismademoreeffectivebynotproducingsimilarrandomlls.ThenextmethodknownastheWeightedDeterministicFill,givesdeterministicvaluestotheunspeciedinputs.Firstthenumberofstuck-at-0andstuck-at-1faultspropogatedisfound.Thenthegatewithhighestnumberoffaultspassedtoitisfoundanditsinputsareassignedbybacktrackingtoprimaryinputs.2.2CrosstalkEstimationIn[24]theauthorsaimtondthesetofinterconnectionpairswhichwillnotposecrosstalkproblemsduringnormaloperationofthecircuit.Theystatethattheeffectoftheenvironmentuponanytargetcircuit,morespecicallyonitslayoutthatitisdesignedtowork,andunderconditonswhichareassumedtobeknowntothedesignercanbemodeledbycapturingtheinputpatterndependencies.Takingintoaccountthistypeofmodelingcanresultinamuchsimplercircuitthanwhenitdesigneddiscardingtheinputpatterncorrelations.In[25]thenecessitytoconsideron-chipsimultaneousswitchingnoise(SSN)isdealtwith,particularlywhendeterminingthepropagationdelayofaCMOSlogicgateinahighspeedsynchronousCMOSICs.Thepaperpresentsanalyticalexpressionswhichcharacterizeon-chipSSNandalsotheeffectofSSNonthepropagationdelay15

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andoutputvoltageofalogicgate.In[51]theauthorsextracttheelectricalparametersforabasesetofadjacentwireconguration.Thisstepisdoneonlyoncedependentonthethetechnology,thenthegeometryofthecompletechiproutingspacesisextracted.Theadjacencylengthsbetweenthechosenvictimanditsagressorsconsideringthewirewidthsandspacingsiscalculated.Thentheeffectiveouputresistancesinuencingthesignalslopesandtheamountofcouplednoiseisdetermined.Finallytheswitchingtimesofvictimandagressorlinesfromthechip'stimingreportisreadandalltiming-uncriticalwireadjacenciesaredisregarded.Acomprehensivemethodolgyforunderstandingandanalyzingthenoiseimmunityofdigitalintegratedcircuitsispresentedin[52].Anoiseclassicationbasedonnoiselevelrelativetothesupplyandgroundrailsisintroduced.Anoisestabilitymetricasapracticalformalbasisforensuringnoiseimmunityisdescribed.Astaticnoiseanlaysisapproachisalsoexplained,whichcanbeusedasatechniqueforidentifyingallpossibleon-chipfunctionalfailureswithoutfullpatterndependentdynamicalsimulation.TheHarmonyisthenusedtocombinestatictiminganalysisandreducedordermodellingwithtransistor-levelanalysis.16

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CHAPTER3LEARNINGBAYESIANNETWORKS3.1BayesianNetworksBayesianNetworkisagraphicalprobabilisticmodelbasedontheminimalgraphicalrepre-sentationoftheunderlyingjointprobabilityfunction.Itcanbedenedasadirectedacyclicgraph(DAG)withaprobabilitytableforeachnodeandthenodesinthenetworkrepresentthepropositional(random)variablesinthedomainandthearcsbetweenthenodesrepresentthedepen-dencyrelationshipamongthevariables.Beforedelvingdeepintothedenitionletusdiscusssomebasicconcepts.Denition:LetEandFbeeventssuchthatPF0.ThentheconditionalprobabilityofEgivenF,isgivenbyPEFPEF PF(3.1)Bayes'Theorem:TheBayestheoremdevolepedbyThomasBayesin1763formsthebasisofBayesianNetworks.Denition:GiventwoeventsEanfFsuchthatPE0andPF0,thenPEFPEFPE PF(3.2)Proof:weuseEq.3.1toderivetheBayestheorem.PEFPEF PFandsimilarlywecanwrite,PFEPFE PE.Multiplyingtheseequalitiesbythedenominator,weget,PEFPFPFEPEasPEFPFE,so17

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Figure3.1.ExampleforaDAG.PEFPEFPE PF.SomebasicdenitionsrelatedtoBayesiannetworksareasgivennext:Denition1:AdirectedgraphGdenotedas(V,E),whereVisanite,non-emptysetwhoseelementsarecalledverticesornodes,andEisasetoforderedpairsofdistinctelementsofV.ElementsofEarealsocalledtheedgesorarcs.If(x,y)E,wecansaythatthereisadirectededgefromxtoyandthatxandyareincidenttotheedge.Itisdenotedbyanarrowfromxtoy,andwecansaythatxandyareincidenttotheedge.xandyaresaidtobeadjacentorneighborsifthereisanedgefromxtoyorfromytox.Ifthestartofthearrowisatxandtheendpointaty,thenxiscalledtheparentofyandyiscalledthechildofx.Similarlyxiscalledtheancestorofyandythedescendentofx.Thesetofedgesconnectingthenodesxandyiscalledthepathfromxtoy.Adirectedcycleisapathfromanodetoitself.Asimplepathisonewithnosubpathswhicharedirectedcycles.Denition2:Adirectedgraphthatcontainsnodirectedloopsorcyclesiscalledadirectedacyclicgraph(DAG).AnexampleofaDirectedAcyclicGraph(DAG)isshowninFig.3.1.Denition3:LetUbeanitesetofdiscretevaluevariables.LetPbeajointprobabilityfunctionoverthevariablesinU,andletX,Y,ZbeanythreesubsetsofvariablesinU.XandY18

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aresaidtobeconditionallyindependentgivenZifPxyzPxzifPyz0.ConditionalindependencemeansthatknowledgeofZmakesXandYindependentofeachother.Denition4:ADAGGisadependencymap(D-map)ofadependencymodelM,ifeveryindependencerelationshipderivedfromMcanbeexpressedinG.Denition5:ADAGGisanindependencemap(I-map)ofadependencymodelM,ifeveryindependencerelationshipderivedfromGcorrespondstoavalidconditionalindependencerela-tionshipinM.Denition6:ADAGGisaminimalI-mapofadependencymodelMifitisanI-mapofManddeletionofanyofitsedgesfromGdestroystheindependencerelationofM.Denition7:ADAGGisbothaD-mapandanI-mapofthedependencymodelM,ifitisaperfectmap(P-map)ofM.Denition8:ADAGGiscalledaBayesianNetworkofaprobabilityfunctionPonasetofvariablesU,ifGisaminimalI-mapofP.AlsoaBayesiannetworkisadirectedacyclicgraph(DAG)representationoftheconditionalfactoringofajointprobabilitydistribution.AnyprobabilityfunctionPx1xncanbewrittenas1Px1xNPxnxnn1xnn2x1Pxnn1xnn2xnn3x1Px1(3.3)Thisexpressionholdsforanyorderingoftherandomvariables.Therepresentationandinfernceusingthisequationbecomesverytediouswhennbecomesverylarge.Inmostapplications,avariableisusuallynotdependentonallothervariables.Therearelotsofconditionalindependenciesembeddedamongtherandomvariables.Theaboveequationdoesnotconsidertheseindependecies,butthesedependenciescanbeusedtoreordertherandomvariablesandtosimplifytheconditionalprobabilities.Px1xNPvPxvPaXv(3.4)wherePaXvaretheparentsofthevariablexv,representingitsdirectcauses.Thisfactoringofthejointprobabilityfunctioncanberepresentedasadirectedacyclicgraph(DAG),withnodes(V) 1ProbabilityoftheeventXixiwillbedenotedsimplybyPxi orbyPXixi .19

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representingtherandomvariablesanddirectedlinks(E)fromtheparentstothechildren,denotingdirectdependencies.Denition9:ABayesiannetworkGofadistributionPdeterminesasetofindependencerela-tions.Theseindependencerelationsinturnmayentailothers,inthesensethateveryprobabilitydis-tributionhavingindependencerelationswillalsohavefurtherindependencerelationsasexplainedin[18].InotherwordsGofthedistributionPmaynotrepresentalltheconditionalindependencerelationsofP.But,whenGcanactuallyrepresentalltheconditionalindependencerelationsofP,thenPandGarefaithfultoeachother.In[10],GiscalledaperfectmapofPandPiscalledtheDAG-IsomorphofG.Denition:AdependencymodelMissaidtobecausaloraDAGisomorphifthereisaDAGDthatisaperfectmapofMrelativetod-separation,i.e.,IXZYM!#"%$XZYD(3.5)Theorem:AnecessaryconditionforadepndencymodelMtobeaDAGisomorphisthatIXZYMsatisesthefollowingindependentaxioms.Symmetry:IXZY!#"IYZX(3.6)Composition/Decomposition:IXZY&W!#"IXZY&IXZW(3.7)Intersection:IXZ&WY&IXZ&YW'"IXZY&W(3.8)Weakunion:IXZY&W'"IXZ&WY(3.9)Contraction:IXZ&YW&IXZY'"IXZY&W(3.10)20

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WeakTransitivity:IXZY&IXZ&gY'"IXZgorIgZY(3.11)Chordality:Iag&db&Iga&bd'"IagborIadb(3.12)WhyBayesianNetworks?1.BayesNetworksprovideanaturalrepresentationforconditionalindependence.2.TopologyandtheConditionalprobabilitytablesprovideacompactrepresentationofthejointdistribution.3.Bayesiannetworksaregenerallyeasytoconstruct.AnequivalentrepresentationnamelyMarkovnetworksaretedioustoconstructandmoreovertheyareundirectionalandinferenc-ingbecomesmoretimeandmemoryconsuming.4.InferenceinBayesnetworksiseasier,forexamplepolytreeinferencingisanNPhardproblemongeneralgraphs.5.LearningofBayesiannetworksprovideacompactrepresentationofthedata.Havingthisrepresentationofthedata,inferencingiseasierandfaster.3.2LearningInaBayesiannetwork,thegraph,calledtheDirectedAcyclicGraph(DAG),isthestructureoftheBayesianNetwork,andtheconditionalprobabilitydistributioniscalledtheparameter.Boththeparameterandthestructurecanbeseperatelylearnedfromthedata.Whilelearningtheparameter,weassumethatweknowthestructureoftheDAG,butinstructurelearningwestartwithonlyasetofrandomvariableswithunknownrelativefrequencydistribution.Learningstructurecanalsobedonewithmissingdataitems,andhiddenvariablesandalsointhecaseofcontinuousvariablesasdiscussedin[11].TherearetwomajormethodstoBayesiannetworklearning,namelysearchand21

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scoringmethodanddependencyanalysismethods.Inthisexperimentweareusingthedependencyanalysismethodtolearnthebayesiannetworkstructurefromdata.TheinputtolearntheBayesiannetworkisadatabasetable.Eachnodeinthenetworkisarepresentativeoftheeldsinthedatabase.Eachrecordinthegivendatabaseisacompleteinstantiationoftherandomvariables.Weassumethatthedatabasetablehasdiscretevaluesandthedatasetiscompletewithnomissingvalues.Also,thevolumeofthedatasetshouldbelargeenoughsothatreliableCItestscouldbeperformed.Independencyanalysismethod,conditionalindependenceplayanimportantrole,andbyusingtheconceptofdirectiondependentseparationord-separation(Pearl,1988),alltheindependencerelationsoftheBayesiannetworkcanbecomputed.Denition:ForaDAGGVE,whereX,YVandXY,andC(V)XY,wesaythatXandYared-seperatedgivenCinGifandonlyifthereexistsnoadjacencypathPbetweenXandY,suchthat:(i)everyconvergingarrowonPisinCorhasadescendentinCand(ii)noothernodesonpathPisinC.Ciscalledthecut-set.IfXandYarenotd-seperatedgivenC,wesaythatXandYared-connectedgivenC.Inmoresimpleterms,wecanexplaind-separationwiththeanexampleasshowninFig.3.2..Asimpleexplanationisgivenas,twonodesared-separatedgiventhethirdifallactivepathsfromtherstnodetothesecondnodeareblockedgiventhethird.Itcanalsobeexplainedas,ifknowledgeabouttherstnodegivesnoextrainformationaboutthesecondoncethethirdnodeisknown.Thatis,onceweknowaboutthethirdnode,therstnodeaddsnothingtowhatwealreadyknowaboutthesecondnode.Intheexampe,Fig.3.2.Zisd-separatedfromB,givenYasallpathsfromXtoYareblockedgivenZ.AlsoZisnotd-separatedoritisd-connectedfromBgivenCasallpathsbetweenZandBarenotblockedasthereexistsapaththroughY.Iftwonodesaredependent,thentheknowledgeofthevalueofonenodewillgiveussomeinformationofthevalueoftheothernode.Thisinformationgaincanbemeasuredbyusingmutualinformation.Thereforetheknowledgeofmutualinformationcantellusaboutthedependencyrelationbetweentwonodes.ThemutualinformationoftwonodesXi,Xjisexpressedas:22

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XYZABCFigure3.2.Exampleford-separation.IXiXjxixjPxixjlogPxixj PxiPxj(3.13)andtheconditionalmutualinformationisdenedasIXiXjCxixjcPxixjclogPxixjc PxicPxjc(3.14)whereCisasetofnodes.WhenIXiXjissmallerthanacertainthresholde,wesaythatXiXjaremarginallyindependent.WhenIXiXjCissmallerthane,wesaythatXiXjareconditionallyindependentgivenC.3.3AlgorithmforLearningBayesianNetworkGivenNodeOrderingThealgorithmistheworkofChenget.al[16]whichconstructsaBayesiannetworkbasedondependencyanalysisfromadatabasetableasinput.AsimilarworkistheChow-Liualgorithmasexplainedin[10].23

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Therearethreesteps,thatthealgorithmperforms,namelydrafting,thickeningandthinning.Intherststepadraftiscreatedbasedonthemututalinformationofeachpairofnodes.Inthesecondstepknownasthickening,arcsareaddedwhenthepairofnodesarenotconditionallyindependentbasedonacertainconditionset.WegetanI-mapattheendofthisstepwhichisoftheunderlyingdependencymodelgivenitisDAGfaithful.Inthelaststepthearcsaddedintheprevioussteparecheckedusingconditionalindependencetestsandthearcwillberemovedifanytwonodesarecondtionallyindependent.SoattheendofthethirdstepwegetaperfectmapofthemodelwhenitisDAGfaithful.3.3.1Step1:DraftingWeinitiateagraphGVE,whereVhasalltheattributesofadatasetandEisanemptyset.WealsoinititateanemptylistL.ForvivjVandij,foreachpairofnodesvivj,wecomputemutualinformationusingequation3.13.ThenwesortthepairsofnodesinascendingorderthathavemutualinformationgreaterthanacertainsmallvalueeandareputintothelistL.ThepointerppointstotherstpairofnodesinL.ThentherstpairofnodesisremovedfromLandthearcsareaddedtoE,andthedirectionofthearcdependsonthenodeordering.Thepointerismovedtothenextpairofnodes.WegetthepairofnodespointerbyL,andifthereisnoopenpath,thecorrespondingarcisaddedtoE,andthepairofnodesisremovedfromL.Thesestepsarerepeatedtillthewecoverallpairsofnodes.Thereasontosortthemutualinformationinascendingorderisheuristic,whichstatesthathighermutualinformationrepresentsadirectconnectionthanalowerone.Ithasbeenprovencorrectiftheunderlyinggraphisasinglyconnectedgraph.Aowchartrepresentationofthestepsisshownbelow.Thissteptriestondadraftwhichismorelikethenalmodelasmuchaspossiblebyonlyusingpairwisemutualinformationtestswithoutinvolvingconditionalindependencetests.Theresultobtainedattheendofthestepcanbeanythingfromanemptygraphtoacompletegraphwithoutaffectingthenalgraphattheendofallthestepsofthealgorithm.Thedraftlearningprocedurecomestoastopwheneverypairwisedependencyisexpressedbyanopenpathinthedraft.24

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is it the end ofstart yesnonoyesfor each pair of nodesG(V,E), and empty create pointer p. get first 2 nodes. add arcs stopto E.move p to next pair.order, for values greatersort them in ascendingthan a particular value.compute mutual information Initiate a graph list L.get next pairis there an open path betweenthem?L.remove the pair from add arc to E and the list? Figure3.3.Step1:Drafting.25

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(a)(b)(c)(d)ABCDEABCDEABCDEABCDEFigure3.4.WorkingMechanism.Theworkingmechanismcanebegraphicallyillustratedbythefollowingsimplemulti-connectednetworkborrowedfrom[18].Foreachpairofnodeswegetthemutualinformation,asexplainedbefore.Intheexample,supposewehaveIBD+*ICE,*IBE+*IAB-*IBC,*ICD-*IDE+*IAD+*IAE-*IAC,andthesepairshavemutualinformationgreaterthane.NowthelistLcontains[B,D,C,E,B,E,A,B,B,C,C,D,D,E,A,D,A,E,A,C].NextwegetapairofnodesiterativelyfromthelistL,andconnectthetwonodesbyanarc,andremovethepairfromthelistL,ifthereisnopreviouslyexistingopenpathbetweenthem.Inthisexample,thelistLequals[C,D,D,E,A,D,A,E,A,C]attheendofthephaseasthesepairsofnodesarenotdirectlyconnectedintherstphasebuttheydohavethemutualinformationgreaterthane.ThegraphgeneratedattheendofthisstepisshowninFig.3.4.(b).Thisstepoftheprocesstriestomodelthegraphasclosetothenalmodelaspossibleusingonlypair-wisemutualinformationtests,thatisnoconditionalindependencetestsareinvolved.Thoughthegraphobtainedfromherecanbeanywherefromanemptygraphtotheexactnalresult,theefciencyisimprovedifthegraphisclosetothenalmodel.Whensearchingthedraft,wetryto26

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minimizethenumberofmissingorwronglyaddedarcs.Astopconditionisusedtominimisetheseerrors.Thisdraftingstependswheneverypair-wisedependencyisexpressedbyanopenpathinthedraft.ThisstepisessentiallytheChow-Liualgorithmanditguarenteestheconstructednetworkisthesameasthenalone.ThereforethisalgorithmcanbeviewdasaspecialcasefortheBayesiannetworksthathavetreestructures.3.3.2Step2:ThickeningThestepshereareagaineasilyexplainedwiththehelpofanotherowchart.InthisstepCItestsandd-separationanalysisareusedtondoutifthepairofnodesthatwereleftoutinstep1becausetheyalreadyhadsomeopenpaths,shouldbeconnectedornot.Wecannotbesureiftheconnectionisreallyneeded,butwecanknowthatnoarcswillbemissedattheendofthisstepasweuseonlyoneCItesttomakeadesicion.Theprocedurend-cut-settriestogetacut-setthatcand-seperatethetwonodes,andthisisdoneforeverypairofnodes.ThenitusesaCItesttoseeifthetwonodesareindependentconditionalonthecut-set.Thearcisaddedifthenodesarenotconditionallyindependent.Somearcscanbewronglyadded,becausesomeneededarcsmaybemissingandtheyhinderndingapropercut-set.ThegraphafterthesecondstepisshowninFig.3.4.(c).Arc(D,E)isaddedbecauseDandEarenotindependentgivenB,whichisthesmallestcut-setbetweenDandE.Also,arc(A,C)isnotaddedbecausetheconditionalindependencetestsshowthatAandCareindependentgivenB.Similarly,forthesamereasonswedonotaddarcs(A,D),(C,D),(A,E).FromthegrpahwecanalsoseethatafterthisstpethegraphobtainedisanI-mapoftheunderlyingmodel.3.3.3Step3:ThinningForthetwonodesunderconsideration,ifthereareotherpaths,otherthanthearc,thenthearcistemporarilyremovedfromE,andtheprocedureiscalledagaintondacut-setthatcand-seperatethetwonodes.Nowgiventhacut-settheCItestisdonetondoutwhetherthetwonodesareconditionallyindependent.Ifso,thearcispermanentlyremovedorelseitisaddedback.Thisstepisbasicallytondthearcswhichmaybebeenwronglyaddedintheprevioussteps.Herewe27

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are theyconnect pair of nodes is it the end ofthe list?stopyesnoyesnouse CI testuse pointer tomove p to nextget pair of nodesadd arc to Epairstartcall the procedureto findcut-setconditionallyindependent? Figure3.5.Step2:Thickening.28

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alsomakesurealltheaddesarcsareneededandtherearenoextraones,byusingonlyoneCItest.SincethecurrentgraphhadbecomeanI-mapattheendofthepreviousstep,wecanbesurethatthedesciontoremovearcsiscorrect.Inthisstep,wetrytondthatanarcconnectingapairofnodesarealsoconnectedbyotherpaths.Thisisdonebecauseitispossiblethatthedependencybetweenpairofnodescouldnotbeduetothedirectarc.Thenthealgorithmremovesthisarcandusestheproceduretondthecut-set.Thentheconditionalindependencetestisusedtocheckifthetwonodesareindependentcondtionalonthecut-set.Ifso,thearcispermanentlyremovedasthetwonodesareindependent,orelsethearcisplcedbacksincetheyareconditonallydependent.Thisisrepeatedtillallthearcsareexamined.ThisstepisshowninFig.3.4.(d)fortheexample,andithasthenalcompletegraph.Herewecanseeedge(B,E)isremovedpermanentlybecauseBandEareIndependentgivenC,D.SoattheendofthisthirdstepthecorrectBayesiannetworkisobtained.3.3.4FindingMinimumCut-SetsInthesteps1and2wecalltheprocedurend-cut-settogetacut-setbetweenthetwonodesandthenusethiscut-setintheconditionalindependencetest.[19]alsoproposesanalgorithmtondminimumcut-sets,whichprovesthataminimumcut-setbetweentwonodescanbealwaysfound.Begininputnode1,node2;Explore(listOfPath,node1,node2);Store(ListOfPath,OpenPath,Closepath);DoWhilethereareopenpathswithonlyonenodeDoputthesenodesofeachsuchpathincut-set;takeoffalltheblockedpathsbythesenodesfromtheopenandclosedpathset;Findpaths,fromtheclosedpathset,openedbythenodesintheblocksetandmovethemtotheopenpathset;Removethenodesthatarealsointhecut-settoshortensuchpaths;29

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EndWhileIfthereareopenpathsDoInthecut-setfindandputthenodethatcanblockthemaxnumberofremainingpaths;Removealltheblockedpathsbythenodefromboththeopenandclosedpathset;Findpaths,fromtheclosedpathset,openedbythisnodeandmovethemtotheopenpathset;Removethenodesthatarealsointhecut-settoshortensuchpaths;EndIfUntilthereisnoopenpathEndWehavetwosetsnamelytheopenandclosedpathset.Wendrstalltheadjacencypathsbetweenthetwonodesandandputthepathsintothetwosets.Inthecut-setweputthenon-convergingpathsthatareconnectedtoboththenodes.Thisisdonebecausethesehavetobeineveryvalidcut-set.Thisisrepeatedtondanodethatcanblockthemaximumnumberofpaths.Putthisinthecut-setandrepeattillalltheopenpathsareblocked.3.3.5ComplexityAnalysisIngeneral,mostoftherunningtimeoflearningalorithmsisconsumedbydataretrievalfromdatabases.SupposeadatasethasNattributes,themaximumnumberofpossiblevaluesofanyattributeisr,andanattributehasamaximumofkparents.Thecomputationofmutualinforma-tionrequiresatthemostOr2basicoperationsandcomputaionofconditionalmutualinformationrequiresatthemostOrk2basicoperations,sincetheconditionsethasknodesatthemost.So,theconditionalmutualinformationtesthasacomplexityofOrN.TherststephasacomplexityofONlogNsteps,forsortingthemutualinformationofpairs.TheprocedurehasONbasicoperations.SoasawholewithrespecttotheCItests,Step1needsON2mutualinformationcomputationsandStep2needsatmostON2numberofCItests.SimilarlyStep3requiresatmost30

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ON2numberofCItests,becauseadescisionrequiresoneCItest.SoasawholethealgorithmrequiresON2CItestsintheworstcase.31

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CHAPTER4SAMPLINGBayesiannetworksarethebestsuitedtorepresentjointprobabilitydistributions,astheyincludetheconditionalindependencerelationshipsamongthevariablesinthenetwork.AnattractivefeatureofBayesiannetworkisthatitsamenabilitytorecursiveandincrementalcomputationofschemes.LetHdenoteanhypothesis,ene1e2..endenoteasequenceofdataobservedinthepast,andedenoteanewfact.TheoriginalwaytocalculatethebeliefinH,PHenewouldbetoaddthenewdataetothepastdataenandperformaoverallcomputationoftheeffectonHoftheentiredataseten1ene.ThiscomputationistimeandmemoryintensivebecausethecomputationofPHenebecomesmoreandmorecomplexasthesizeofthesetincreases.ThiscalculationcanbesimpliedbydiscardingthepastdataoncewehavecomputedPHen.NowwecancalculatethevalueofthenewdatabyEq.4.1PHenePHenPeenH Peen(4.1)Also,weknowfromEq.3.2PHePeHPH Pe(4.2)ComparingEq.4.1andEq.4.2wecanseethattheoldbeliefPHenassumestheroleofpriorprobabilityinthecomputationofthenewimpact.Itcanalsobeseenthatitcompletelysum-marizesthepastexperiencesandifweneedupdating,weneedtomultiplyonlybythatlikelihoodfunctionPeenH,whichaswecanseegiventhehypothesisandpastobservationsitmeasurestheprobabilityofthenewdatae.Likelihoodisthehypotheticalprobabilitythataneventthathasalreadyoccuredwouldyieldaspecicoutcome.Theconceptdiffersfromthatofaprobabilityin32

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thataprobabilityreferstotheoccurenceoffutureevents,whilealikelihoodreferstopasteventswithknownoutcomes.ThisrecursiveformulationstillwouldbecumbersomebutforthefactthatthelikelihoodfunctionisoftenindependentofthepastdataandinvolvesonlyeandH.Usingthecondtionalindependenceconditionwecanwrite,PeenHPeHandPeen0/HPe1/H(4.3)anddividingEq.4.1bythecomplementaryequationfor/H,wecanwriteOHen1OHenLeH(4.4)IfwemultipythecurrentposterioroddsOHenbythelikehoodratioofe,uponthearrivalofeachnewdatae,asshowninEq.4.4,showsasimplerecursiveprocedureforupdatingtheposteriorodds.Oddsistheprobabilitythattheevenwilloccurdividedbytheprobabilitythattheeventwillnotoccur.OHenistheprioroddsrelativetothenextobservation,whileOHistheposterioroddsthathasevolvedfromthepreviousobservationnotincludedinen.TakingthelogarithmofEq.4.4,logOHenlogOHenlogLeH(4.5)Thesimplicityofthelog-likelihoodcalculationhadledtoavarietyofapplications,especiallyinintelligencegatheringtasks.Foreachnewreport,wecanestimatethelikelihoodratioL,whichcanbeeasilyincorporatedinthealreadyaccumulatedoverallbeliefinH.4.1CausalNetworksTheimportantrequirementofBayesiannetworkisthatitincorporatesd-separationpropertiesofthedomainmodeledanditneednotreectcause-effectrelations.Butthereis,however,agoodreasontostriveforcausalnetworks.Causationisoneofthebasicprimitivesofprobabilitybecauseitisanindispensibletoolforstructuringandspecifyingprobabilisticknowledge.Sincetheseman-ticsofcausalrelationshipsarepreservedbythesyntaxofprobabilisticmanipulationsnoauxillary33

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DAB DABDAB(a)(b)(c)Figure4.1.CausalDiagram.devicesareneededtoforceconclusions.ForexampleifIguredoutthatthecauseofmyslippingwasduetoawetpavement,Icouldnolongerconsiderothereventsasthewetnessofthepavementisconrmed.Thefactsthatitrainedthatdayorthesprinklerwasonormyfriendalsoslippedandbrokealegshouldnolongerbeconsideredoncethewetnessofthepavementisidentiedasthedirectcauseoftheaccident.Theasymmetryconveyedbythecausaldirectionalitycanbeusedforencodingmorecomplexandintricatepatternsofrelationships.Itcannowbeunderstoodthatonceaconsequenceisobserveditscausescannolongerremainindependentbecauseconrmingonecauselowersthelikelihoodoftheother.Causaldirectionalityconveysthemessagethattwoeventsdonotbecomereleventtoeachothermerelybyvirtueofpredictingacommonconsequence,buttheydobecomerelevantwhentheconsequenceisactuallyobserved.Theoppositeistruefortwoconsequencesofacommoncause,typicallythetwobecomeindependentuponlearningthecause,asdiscussedwhenlearningthestructurefromdata.ConsidertheFig4.1.,whichrepresentsadiseaseDandtwotestsAandB.Itisusuallyrepre-sentedbythephysicianasshowninFig4.1.(b),generallyasopposedtothemodelFig4.1.(a).ButitisnotcorrecttorepresentthesituationasdoneinFig4.1.(b)becausethetwotestsareindepen-dent,andtocorrectthemodel,wemustintroduceanextralinkasshowninFig4.1.(c).Nowthe34

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7 96548321 Figure4.2.ASmallCircuit.structureisnotminimalandtogetacorrectmodel,itisnotsufcienttoacquirePDABtogetherwithP(A)andP(B).TheconditionalprobabilitiesforFig4.1.(a)reectthegeneralpropertiesoftherelationbetweendiseasesandtestsandtheyaretheonesthatamanufactureroftestscanpublish,whereastheconditionalprobabilitiesforFig4.1.(b)areamixtureofdisease-testrelationsandpriorfrequenciesofthedisease.Thepresenceofinterventionsprovideanothergoodreasonfortheuseofcausalnetworks.Aninterventionisanactionthathasanimpactonthestateofparticularvariables.Soincausalmodelstheimpactoftheinterventionwillspreadinthecausaldirectionandnotintheoppositeway.Ifthemodeldoesnotreectcausaldirectionsitcannotbeusedtosimulatetheimpactofinterventions.Asdenedbefore,Bayesiannetworksaredirectedacyclicgraphsinwhicheachnoderepresentsauncertainquantityorrandomvariable,whichcantaketwoormorevalues.Thearcswhichconnectthenodessignifytheexistenceofdirectcausalinuencesbetweenthelinkednodesorvariablesandthestrengthsoftheseinuencesarequantiedbytheconditionalprobabilities.Theadvantageofnetworkrepresentationisthatitiseasytoexpressdirectlythefundamentalqualitativerepresenta-tionofdirectdependencies.Also,itcandisplayaconsistentsetofadditionaldirectandindirectdependenciesandpreservesitasastablepartofthemodel,independentofthenumericalestimates.Thedirectionalityofthearrowsisessentialtodisplaynontransitivedependenciesthatisiftwocausesareindependenttoeachother,theyarestillrelevanttotheeffectandinduceddependencies,thatisonceaconsequenceisobserved,itscausescannolongerremainindependent,becauseonecauselowersthelikelihoodoftheother.Ifthearcsarestrippedoftheirdirectionssomeofthe35

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X1XXXXXXXX76984325Figure4.3.BNCorrespondingtotheCircuit.relationshipswouldbemisinterpreted.Thusbydiplayingirrelevancesinthedomainmodelled,thecausalschemataminimizesthenumberofrelationshipsthatneedtobeconsideredwhileamodelisconstructedandthisineffectlegitimizesmanyfuturelocalreferences.AMarkovnetworkisanundirectedgraphwhoselinksrepresentsymmetricalprobabilistic,whileabayesiannetworkisadirectedacyclicgraphwhosearrowsrepresentcausalinuences.ThemainweaknessofMarkovnetworksistheirinabilitytomodelinthestructuretheinducedandnon-transitivedependencies.Inthisrepresentationanytwoindependentvariableswillbedirectlyconnectedbyanedge,merelybecausesomeothervariabledependsonboth.Duetothismethodofmodelling,manyusefulindependenciesgounrepresentedinthenetwork,thusmakingitlongandcomplex.Bayesiannetworksefcientlyovercomesthisdeciencybyusingamuchricherlanguageofdirectedgraphs.Sothesedirectionsofarrowshelpustodistinguishgenuinedependenciesfromspuriousdependenciesinducedbyhypotheticalobservations.Theconceptofd-separationasex-plainedinSection3.2,isefcentlyusedtoaddressthesedependencies.Anevenbiggeradvantageofdirectedgraphicalrepresentaionsisthat,theymakeiteasytoquantifylinkswithlocal,concep-tuallymeaningfulparametersthatturnthenetworkasawholeintogloballyconsistentknowledgebase.36

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Fromthesediscussionsitisclearthatcausalrepresentationisbestsuitedforrepresentationofvariousphenomenons.Thusitisnoproblemtocongureagraphwhichrepresentsphenomenawithexplicitnotionsofneighborhoodoradjacencylikeelectroniccircuitsandcommunicationnetworks,asdiscussedin[10].Thedeterminationofdirectionofthenetworktobemodelledbecomestheimporatanttaskincausalnetworks.Inmodellinglogiccircuits,thedirectionofowisknownanditbecomesmucheasiertoconstructacausalnetwork.SoBayesiannetworksarethebestsuitedfortherepresentationofVLSIcircuits,thanundirectedgraphslikeMarkovnetworks.4.2InferenceTheBayesiannetworknotonlymodelscausalitybutalsomakesinferencemucheasier.Theinferencedonehereissplitintomanysteps.Firstconvertthenetworkintoajunctiontreeofcliquesandthenuseprobabilisticlogicsamplingtoacheiveourpurpose.Thestepsinvolvetheformationofamoralgraph,andtheprocessiscalledcompilation,thenthemoralgraphistriangulated.Thecliquesetisidentiedandthejunctiontreeofcliquesisformed.4.2.1MoralGraphGivenaBayesiannetwork,amoralgraphisobtainedby'marryingparents',thatisaddingundirectededgesbetweentheparentsofacommonchildnode.Beforethisstep,allthedirectionsintheDAGareremoved.Thisstepensuresthateveryparentchildsetisacompletesubgraph.Theseundirectedlinksarecalledthemorallinks.ThemoralgraphrepresentstheMarkovstructureoftheunderlyingjointfunction.SomeoftheindependenciesdisplayedintheoriginalDAGstructurewillnotbegraphicallyvisibleinthemoralgraphasitisundirectedandhasadditionallinksaddedtoit.Someoftheindependenciesthatarelostinthetransformationcontributestotheincreasedcomputationalefciency,butdoesnotaffecttheaccuracy.ThemoralgraphoftheFig.4.3.isshowninFig.4.4.Inthegureitseenthatthedirectionoftheedgeshavebeenremovedandtheparentsofacommonchildhavebeen'married',thatisalinkisaddedbetweenthem.Theaddedlinksareshown37

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X1XXXXXXXX76984325Figure4.4.MoralGraph.asdottedlinesandwecanseethattheyconnectX1andX2,X3andX4,X5andX6andbetweenX7andX8.IftheunderlyingDAGstructureisdenotedasD,thenthemoralgraphisdenotedasDm.4.2.2TriangulationThemoralgraphissaidtobetriangulatedifitischordal.TheundirectedgraphGiscalledchordalortriangulatedifeveryonofitscyclesoflengthgreaterthanorequalto4posessesachord[9],thatisweaddadditionallinkstothemoralgraph,sothatcycleslongerthan3nodesarebrokenintocyclesofthreenodes.IntheFig.4.5.wehaveaddedadash-dottedlinebetweenX4andX7totriangulatethemoralgraph.Theeasiestwaymaybetoaddallpossiblelinksbetweennodestomakethegraphchordal,alsotransformingitintoacompletegraph.Butthechallengeistouseminimalpossibleadditionallinks,whichisingeneralaNP-hardproblemandvariousheuristicsareused.InHUGIN,aminimumll-inedgesheuristicsisusedasexplainedin[9].38

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X1XXXXX79825XX43X6Figure4.5.TriangulatedGraph.4.2.3JunctionTreeThejuntiontreeisdenedasatreewithnodesrepresentingcliques(collectionofcompletelyconnectednodes)andbetweenCiandCjinthetreeTthereisauniquepath.Therunningintersec-tionpropertystatesthatintheintersectionsetCiCjarepresentinallthecliquesinthatuniquepathbetweenCiandCj.Thispropertyofthejunctiontreeisusedinthelocalmessagepassing.Theconstructionofthejunctiontreebreaksdowntotheformationofchordalgraphfromthemoralgraph,wherethesetofpotentialcliquesiscreated,referredtoastheeliminationset.Thentheelim-inationsetofcliquesisreducedandtreelinksbetweenthecliquesareaddedformingthejunctiontree.Inconstructingthechordalgraphandeliminationsetfromthemoralgraph,rstalltheverticesofthemoralgraphareunnumbered.Thevertexthatneedstheminimumnumberofadditionaledgesbetweenitsneighborsischosenandisgiventhehighesavailablenodenumber,sayi,startingfromthenumberequaltothetotalnumberofnodes.ThenthesetCiisformedoftheselectedvertexanditsstillunnumberedneighbors.Edgesarethenlledinbetweenanytwounlinkednodesinthissetandthenodenumberisdecrementedby1.Thisprocessisrepeatedtilltherearenounnumbered39

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46 C = {X X X } 71 C = {X X X }6 C = {X X X } 7C = {X X X } 7C = {X X X } 7C = {X X X } 7534 62748 354125 6789 Figure4.6.JunctionTree.nodes.TheresultantgraphischordalandthecliquesCi'saretermedtheeliminationsetofthegraph,subsetofwhichisusedtoconstructthejunctiontree.Inourexamplemoralgraph,nodeX9isrstselectedsincenoll-inedgeisneededbecausealltheneighbors(rememberthemoralgraphisundirected)arealreadylinked.ThisnodeX9isassignedthenumber9-thetotalnumberofnodesinthegraph.ThesetC9isthenformedbynodesX9X8X7.ThenodesX8andX7arenotyetnumbered.Forthesecondcycle,thenodesX8X7X6andX4cannotbeselectedastheyeachwouldrequireonell-inedgesamongstitsneigh-bors,whereastheneighborsofX3doesnotrequireanyll-inedges.HenceX3isnumbered8inourexampleandC8isformedbyX3X4X6.Forthethirdcycle,wethenselectX2,numberingitas7andformingC72X2X1X5.Inthefourthcycle,nodeX1isassigned6andC62X1isformed.WethenselectX5,assignanumber5,andformC53X5X6X7.NodeX8isassignednumber4,andC44X8X7X4isformed.Inthisstep,all-inedgebetweenX4andX7isadded.Wethenassignthenumber3toX7,thenumber2toX6,andthenumber1toX4.40

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TheresultanteliminationsetCi(whichisthesupersetoftheclique-set)obtainedfromtherunningexampleareC9C15X9X8X767X3X4X667X2X1X567X167X5X6X76X8X7X467X7X6X467X667X45ForeachcliqueCioftheCliquesetorderedtohaverunningintersectionproperty,wehavetoaddalinktocliqueCjwherejisanyoneoftheset128i1suchthatCiC1&C2&&Cin1:9CjFigure4.6.showsthejunctiontreefortheexampleconsidered.ItcanbeobservedthatthecliquesinourexampleareC1;X4X6X7,C2;X7X4X8,C3;X5X6X7,C42X1X2X5,C5X3X4X6,andC6
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A A,f f B,Bf S,SFigure4.7.TwoCliqueswiththeSeparatorSet.tomoralgraphs,arealsopreservedinthetriangulatedmoralgraph.Also,thejointprobabilityfunctionthatfactorizesonthemoralgraphwillalsodosoonthetriangulatedonsinceeachcliqueinthemoralgraphiseitheracliqueinthisgraphorasubsetofaclique.Letxcbethesetofnodesincliquecinthejunctiontree.Thejointprobabilityfunctionoverthesevariablesisdenotedbypxc.Letxsbethesetofnodesinaseparatorsetsbetweentwocliquesinthejunctiontree.Thejointprobabilityfunctionoverthesevariablesisdenotedbypxs.LetCSdenotethesetofallcliquesandSSdenotethesetofseparatorsbetweentheadjacentcliquesinthejunctiontree.Thejointprobabilityfunctionfactorizesoverthejunctiontreeinthefollowingform[9]:px1xN'c=CSpxc>'s=SSpxs(4.6)Aseparatorsetsiscontainedintwoneighboringcliques,c1andc2.Ifweassociateeachoftheproducttermsovertheseparatorsinthedenominatorwithoneitstwoneighboringcliques,sayc1,thenwecanwritethejointprobabilityfunctioninapureproductformasfollows.Letfc1xc1pxc1>pxsandfc2xc2pxc2,thenthejointprobabilityfunctionasexpressedas:Px1xN'c=CSfcxc(4.7)wherethefactorsfcxcarealsocommonlyreferredtoasthepotentialfunctionoverthenodesxcincliquec,andCSisthesetofcliques.Thesefunctions,fcxcs,canbeformedbymultiplyingtheconditionalprobabilities,fromtheinputBayesiannetworkspecication,ofnodesinthecliquec.LettwocliquesAandBhaveprobabilitypotentialsfAandfB,respectivelyandSbethesetofnodesthatseparatesAandBasshowninFig4.7.Whenthereisnewevidenceforsomenode,itwillchangetheprobabilitiesofalltheothernodessuchthattheneigboringcliquesagreeonthe42

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probabilitiesofS,theseparatorset.ToachievethiswerstcomputethemarginalprobabilityofSfromprobabilitypotentialofcliqueAandthenusethattoscaletheprobabilitypotentialofBascapturedbyEq.4.9.Toachievethisweneedtotransmitthescalingfactoralongthelinkandthisprocessisreferredtoasmessagepassing.WehavetorepeatthisprocessinthereversedirectionbycomputingthemarginalprobabilityofSfromprobabilitypotentialofcliqueBandthenusethattoscaletheprobabilitypotentialofA.Thiswillensurethatevidenceatboththecliquesaretakenintoaccount.Newevidenceisabsorbedintothenetworkbypassingsuchlocalmessages.Thepatternofthemessageissuchthattheprocessismulti-threadableandpartiallyparallelizable.Becausethejunctiontreehasnocycles,messagesalongeachbranchcanbetreatedindependentlyoftheothersandhenceparallelmessagepassingispossible.f?SX=AX@=SfA(4.8)f?BfBf?S fS(4.9)Howeverweneednotinitiatethemessagepassingoverthewholenetworkforeverynewevi-denceasthereisatwophasemessagepassingschemethatcanintegrateallnewevidenceintwopasses.Acliqueisselectedtobetherootnode,thenalltheleafnodessendmessagestowardstherootnode,whicharere-computedusingEqs.4.8and4.9ateachnode.Thenasthemessagesfromalltheleafnodesarereceived,theyarepassedbackfromtherootcliquetowardstheleaves.Notethatthisensuresthatalonganyonelink,wehavemessagesalongbothdirections,thus,ensuringallnodeshavebeenupdatedbasedoninformationfromallthenewevidence.4.2.5ProbabilisticLogicSamplingProbabilisticLogicSamplingisamethodproposedbyHenrion[56]whichemploysastochasticsimulationapproachtomakeprobabilisticinferencesinlargemultiplyconnectednetworks.IfwerepresentaBayesiannetworkbyasampleofmdeterministicscenarioss=1,2,.....mandLsxisthe43

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truthofeventxinscenarios,thenuncertaintyaboutxcanberepresentedbyalogicsample,thatisthethevectoroftruthvaluesforthesampleofscenarios:LxA4BL1xL2x..LmxDC(4.10)Ifwehavethepriorprobabilitypx,wecanusearandomnumbergeneratortoproducealogicsampleforx.Thismethodofsamplingproceedsasexplainedbelow,givenaBayesiannetworkwithpriorsspeciedforallsourcevariablesandconditionaldistributionsforallothers:1.Usearandomnumbergeneratortoproduceasamplevalueforeachrootnodeinthenetworkandasampleimplicationruleforeachparameterofeachconditionaldistribution,usingthecorrespondingprobabilities.2.Proceeddownthroughthenetworkfollowingthedirectionofthearrowsfromtherootnodesusingsimplelogicaloperationstoobtainthetruthofeachvariablefromitsparentsandtheimplicationrules.3.Repeatsteps2and3mtimestoobtainalogicsampleforeachvariable.4.Estimatepriormarginalprobabilityofanyeventbycomputingthenumberoftimesinwhichtheyaretrue.5.Estimateposteriorprobabilityonanyofthealreadycomputedvariableasthefractionofthenumberoftimestheeventoccursoutofthepossiblescenariosinwhichthegivenconditionistrue.Westopwithstep3asweneedonlythelogicsampleandwedonotcomputetheposteriorprobabilites.Weusethislogicsamplingmethodinsideeachcliqueofthejunctiontree.Tostartwith,therstcliqueisconsideredandthesamplingisdoneinsidetoarriveatthesamplesforthenodesinsidetheclique.Oncevaluesarexedforthesenodes,theyarepropagatedtothenextcliquewiththehelpoftheseparatorset.Asthenodesoftheseparatorsetareinboththecliques,oneormoreofthenodesareinstantiated.Withthesevaluestherestofthenodesaresampledwithlogicsamplingandthisgoesontilltheendofthejunctiontree.44

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CHAPTER5EXPERIMENTALRESULTSThismodelprovestobeanefcienttechniqueforpowerestimationbyprovidingaminimumnumberofsamples.Assaidbeforethemethodbridgesthegapbetweensimulativeandprobabilistictechniques.ThisisacheivedbyaneffecientmodelofBayesiannetwork,bylearningthestructurefromtheinputdata.ThelearnedstructureoftheBayesiannetworkrepresentsthegraphicalstructureoftheinputdata.Thelearnedstructurecannowbeusedtogenerateanynumberofvectors.ThegenerationofthevectorsetisdoneefcientlybyexploitingthesamplingtechniquesappliedonBayesiannetworks.Samplingisdonetoderivevectorsinanyneededcompactionratio.ThereducedvectorsetisthenfedintoHSPICEtogetanaccurateestimateoftheaveragepower.Thepowerestimateisthencomparedwiththevaluegotfromsimulatingthelargevectorset.Ifaistheoriginallengthofthevectorandbisthecompactedlength,thena>biscalledthecompactionratio(CR).Wehaveacheivedveryhighcompactionratiosofupto300whichhaveneverbeenacheivedbefore.Wehavexedthelargervectorsettobe60,000highly-correlatedinputpatternsgeneratedbysimulatingthevariousbenchmarkcircuits.Thevectorswerethentakenfromrandomoutputnodesofthesecircuits.Thisisdonetoinduceahighcorrelationbetweenthevectorssothattheyareascloseaspossibletotherealworlddata.Alsodifferentbenchmarkcircuitswereusedforgenerat-ingthesevectors,forexampletoobtainvectorsforC499thecircuitC3540wassimulated.ThecorrelationbetweenthevectorscanbeseenintheconstructedBayesiannetwork.TheoverallprocessowisasshowninFig.5.1..Asshown,werstgeneratethethelargevectorset.Thenthepowerconsumedbyeachbenchmarkcircuitiscalculatedusingthelargevectorsetastheinputstimuli.Thisvalueisusedasthebasisforcomparisonwiththepowerestimategeneratedfromthecompactedvectorset.45

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Generate the larger sequenceLearn Bayesian NetworkSample the network andgenerate required lengthhspice power estimatehspice power estimate compareFigure5.1.ProcessFlow.ThenthesecondstepistousethisvectorsettolearntheBayesiannetwork.WeusePowerConstructor[16]toacheiveourpurpose.ThepowerconstructorisanefcienttoolwhichprovidesanaccurateBayesiannetworkintheHUGIN[23]readableformat.Thealgorithmusedheretakesadatabaseasinputandconstructsthebeliefnetworkstructureasoutput.ThisalgorithmextendstheChowandLiu'streeconstructionalgorithmtobeliefnetworkconstructionbyusingathreephasemechanism.Thethreephasesarenamelydrafting,thickeningandthinning.Intherstphase,thisalgorithmcomputesthemutualinformationofeachpairofnodesasameasureofclosenessandcreatesadraftbasedonthisinformation.Inthesecondphase,thealgorithmaddsedgeswhenthepairofnodesthatcannotbed-separated.Thethirdphase,eachedgeofthethecurrentgraphisexaminedusingCItestsandwillberemovedifthetwonodesoftheedgecanbed-separated.Thealgorithmguarenteesthattheperfectmapoftheunderlyingmodelisgenerated.Theprocessismountedona32-bitwindowssystemsonPCandcanberunonWindows9xandXPversions.SincetheconstructionengineisanActiveXDLL,itcanbeintegratedintootherbeliefnetwork,dataminingorknowledgebasesystems.Also,modiedconditionalindependencetestmethodisusedtomaketheresultsmorereliablewhenthedatasetisnotlargeenough.AttheendoftheconstructionstagethelearnedBayesiannetworkfortheinputsofC432isasshowninFig.5.2.46

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Table5.1.PowerEstimatesofISCAS'85BenchmarkSuite. Circuit NumberofInputs 60K CompactionRatio 40 80 200 300 C17 5 2.9973 2.9826 2.9889 2.9829 2.9799 C432 36 4.1304 4.1669 4.1980 4.1474 4.1545 C499 41 3.0005 2.9972 3.0177 3.0927 3.1116 C1355 41 3.2257 3.2348 3.2478 3.3319 3.3245 C1908 33 5.3925 5.3634 5.4259 5.4095 5.3511 C3540 50 1.5969 1.5875 1.5866 1.5499 1.5769 C6288 32 2.2521 2.3112 2.3116 2.3142 2.3197 Table5.2.ErrorEstimatesinPowerEstimation. Circuit NumberofInputs PercentageErrorforCR 40 80 200 300 C17 5 0.49 0.44 0.48 0.58 C432 36 0.88 1.6 0.41 0.58 C499 41 0.10 0.57 3.07 3.7 C1355 41 0.28 0.68 3.2 3.0 C1908 33 0.50 0.61 0.31 0.76 C3540 50 0.58 0.64 2.9 1.25 C6288 32 2.6 2.6 2.7 3.0 Average%Error 0.77 1.02 1.83 1.86 47

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Figure5.2.LearnedBayesianNetworkoftheInputsofC432.SimilarlyfortheinputsofC3540andC1908,theBayesiannetworklooksliketheoneshownFig.5.3.andFig.5.4.respectively.AfterconstructingtheBayesiannetwork,thenextstepasshownintheprocessowgraphisthegenerationofacompactedvectorsequence.ItisdonebysamplingthebeliefnetworkusingacodewritteninHUGINAPI.ThefunctionsinHuginaregivenintermsofISO/ANSICfunc-tionprototypes.ItcontainsahighperformanceinferenceenginethatcanbeusedasthecoreofknowledgebasedsystemsbuiltusingBayesianbeliefnetworksorinuencediagrams.Withtheprobabilisticdescriptionsofcausalrelationshipsinthedomain,onecanbuildknowledgebasesthatmodeltheapplicationdomain.Giventhisdescription,fastandaccuratereasoningcanbedonewiththeinferenceengine.Beforethenetworkcanbeusedforinference,itmustbecompiled.Duringcompilation,rstthelinkstodecisionnodesareremoved.Thenthenetworkisconvertedintoitsmoralgraph,thatistheparentsofeachnodeare”married”andthedirectionsoftheedgesaredroppedandtheutilitynodesareremoved.Thenthemoralgraphistriangulated,thatisll-inlinksareaddedsothateach48

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Figure5.3.LearnedBayesianNetworkoftheInputsofC3540. Figure5.4.LearnedBayesianNetworkoftheInputsofC1908.49

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cycleoflengthgreaterthanthreehasachord(anedgeconnectingtwonon-consecutivenodes).Thenthecliques(maximalcompletesets)ofthetriangulatedgraphareidentied,andthecollectionofcliquesisorganisedasatree,withthecliquesformingtheverticesofthetree.Suchatreeiscalledajunctiontree.Iftheoriginaltreeisdisconnected,therewillbeatreeforeachconnectedcomponent.Lastlypotentialsareassociatedwiththecliquesandthelinksofeachjunctiontree.Thesepotentialsareinitializedfromtheconditionalprobabilitytables,usingasumpropagation.Vectorsofvaluesoverthesetofvariablesinthenetwork(sampling)canbegeneratedwithrespecttotheconditionaldistribution.Ifthedomainiscompiledsamplingthecongurationwithrespecttothecurrentdistributiononthejunctiontreecanbedone.Sincethedistributionisaninuencediagram,allthedecisionsmustbeinstantiatedandpropagated.Thenetworkmustbeanacyclicdirectedgraphandistrueinourcase.Also,allchancenodesmusthaveavalidconditionalprobabilitydistributions,whichisalsotrueinourcase,andthenthedecisionsareinstantiated.Table5.1.showsthepowerestimatesoftheISCAS'85benchmarksuite.ThesecondcolumnshowsthenumberofinputsineacheachcircuitandtheBayesiannetworkmodelhasthenumberofnodesequaltothenumberofinputs.SothelargestBayesiannetworkusedwas50forC3540andthesmallestisof5nodesforC17.Table5.2.showsthepercentageerrorobtainedforeachcircuitandforeachcompactionratio.Itisseenthatasthecompactionratioincreasestheerror%increases,butitisnotthatsignicant.Also,theerrorwasonly0.77%forCR40withamaximumerrorof1.86%forCR300,withthehighesterrorof3.7%forC499forCR300.CircuitC1908isseentohavethelowesterror%forallcompactionratiosandC6288havinghigherrorsformostCR's.Thefollowingtablesshowthecrosstalkestimationvaluesforthevariousbenchmarkcircuitsforboththebiggervectorsetandthecompactedonewiththecompactionratioof40.Thesevaluesdenotethejointprobabilityofthetwonodesinconsideration.HSPICEisusedtoprintthevoltagesvaluesfortheparticularinputvectorsofthetwonodes,whicharethenchangedtotheirswitchingvalues.Fromthesevoltagevaluesthejointprobabilityiscalculatedbyndingthenumberofoc-curences.Themainideaofthispartoftheresultistoestimatetheworst-casecrosstalk,butthetablealsogivesanestimateoftheprobabilityofleakageoccuringinthenodes.Wehavelimitedto50

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Table5.3.JointProbabilitySwitchingEstimateforNodes300and330ofC432for60K. Node300 Node330 State 00 01 10 11 00 0.06 0.19 0.19 0.54 01 0 0 0 0 10 0 0 0 0 11 0 0 0 0 Table5.4.JointProbabilitySwitchingEstimateforNodes300and330ofC432forCR40. Node300 Node330 State 00 01 10 11 00 0.053 0.19 0.19 0.56 01 0 0 0 0 10 0 0 0 0 11 0 0 0 0 onlyafewnodesduetohighrunningtimetakenbyHSPICE,otherwiseanynumberorallofthenodescanbeestimated.Thetables5.3.and5.4.wereplottedforthenodes300and330whicharetheinputstoagateselectedatrandom.Thetable5.4.showsthatthecompactedvectorsetgavethesimilarcrosstalkestimationresultsastheoriginalvectorsetusedin5.3.,provingthatthesmallervectorsetisequallyefcientinmodellingcrosstalk.Thesimultaneousswitchingnoiseofonenodeswitchingfrom0to1andtheotherswitching1to0atthesametimeyieldsthemaximumcrosstalk.Herewecanseethattheparticularsetofinputusedhaszeroorverylowworstcaseswitchingnoise.Alsowecanseethattheleakageisprettyhigh,andsothedesignersshouldpaymoreattentiontoreducingtheleakage.Table5.5.JointProbabilitySwitchingEstimateforNodes557and558ofC499for60K. Node557 Node558 State 00 01 10 11 00 0.054 0.058 0.059 0.057 01 0.066 0.06 0.07 0.066 10 0.06 0.071 0.063 0.065 11 0.065 0.062 0.068 0.07 51

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Table5.6.JointProbabilitySwitchingEstimateforNodes557and558ofC499forCR40. Node557 Node558 State 00 01 10 11 00 0.056 0.057 0.059 0.057 01 0.064 0.06 0.07 0.064 10 0.06 0.07 0.063 0.067 11 0.066 0.06 0.068 0.07 Table5.7.JointProbabilitySwitchingEstimateforNodes141and113ofC1355for60K. Node141 Node113 State 00 01 10 11 00 0.25 0 0 0 01 0 0.24 0 0 10 0 0 0.25 0 11 0 0 0 0.24 Thetwotables5.5.and5.6.areforthecircuitC499,consideringthenodes557and558.Wecanseeheretoothatthecompactedvectorsetyieldedsimilarvalueswithrespecttotheresultfromthelargervectorset.Forthissetofinputsthecouplingnoiseisassignicantastheleakage.Theworst-casecrosstalkprobabilityis.07fornode557switchingfrom0to1andnode558switchingfrom1to0and.07for557switchingfrom1to0and558switchingfrom0to1.Thetables5.7.and5.8.representsthejointprobablitydistributionofnodes141and113whicharethetwoinputsofaNANDgate.Theparticularinputsetshowsthatthereisahighprobabilityofleakagewhenboththenodesareswitchingfrom0to0andfrom1to1.Table5.8.JointProbabilitySwitchingEstimateforNodes141and113ofC1355forCR40. Node141 Node113 State 00 01 10 11 00 0.25 0 0 0 01 0 0.24 0 0 10 0 0 0.25 0 11 0 0 0 0.24 52

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Table5.9.JointProbabilitySwitchingEstimateforNodes2427and2340ofC1908for60K. Node2427 Node2340 State 00 01 10 11 00 0.19 0.02 0.02 0 01 0.05 0.17 0 0.02 10 0.05 0 0.16 0.02 11 0.01 0.04 0.04 0.14 Table5.10.JointProbabilitySwitchingEstimateforNodes2427and2340ofC1908forCR40. Node2427 Node2340 State 00 01 10 11 00 0.19 0.02 0.02 0 01 0.06 0.2 0 0.02 10 0.06 0 0.16 0.02 11 0.01 0.04 0.04 0.15 Thetables5.9.and5.10.capturesthenodes2427and2340whichareinputsofaNORgate.Herewecanseeahighprobabilityofleakagethancrosstalknoiseasbothnodesswitchingfrom0to0and1to1haveahighjointprobability.Thenodesconsideredare4899and4925inthetables5.11.and5.12..Herewecanseethattheswitchingfrom1to1inboththenodeshasthehighestprobability,showinghighprobabilityofleakage.Table5.11.JointProbabilitySwitchingEstimateforNodes4899and4925ofC3540for60K. Node4899 Node4925 State 00 01 10 11 00 0 0 0 0.09 01 0 0 0 0.21 10 0 0 0 0.21 11 0 0 0 0.48 53

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Table5.12.JointProbabilitySwitchingEstimateforNodes4899and4925ofC3540forCR40. Node4899 Node4925 State 00 01 10 11 00 0 0 0 0.09 01 0 0 0 0.21 10 0 0 0 0.21 11 0 0 0 0.48 54

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CHAPTER6CONCLUSIONHereweareabletomodelaframeworkforlearninginputmodelsandusethemtoestimatepowerandcrosstalk.Thoughlotofresearchworkhasbeendoneinbothsimulativenadprobabilisticmethods,theyremainastwoseperateentities.Thisworkservesasabridgeeffectivelyconnectingthetwomethods.Thetechniqueproposedisuniqueandeffectivelymodelsthespatialcorrelationandrstordertemporalcorrelationbetweentheinputs.AlsousingBayesiannetworkswearealsoabletoincludethecausalitybetweenthenodeswhichcannotbemodelledusingundirectedgrpahslikeMarkovnetworks.Amongthemajoradvantagesofthismethodaretheexcellentscalabilitytolargecircuitsandtheabilitytomodelinputdependenciesusingtheconceptoftree-dependentBayesianNetworks.1.Wepresentauniedgraphbasedprobabilisticframeworkforpowerestimationthattakesintoaccounthighorderspatialandrstordertemporaldependenciesthatarepresentintheinputs.2.ModelingtemporalcorrelationattheinputnodesisautomaticinwaywesetupthestatesoftherandomvariablesoftheBN.Forspatialcorrelationamongtheinputnodes,welearnanapproximatetree-dependentdistributionsintheprimaryinputsbasedontheinputstatistics.3.TheeasewithwhichtheBayesiannetworkisbuiltconsideringthetimetakenmakestheapproachhighlyfavorableoverbuildingMarkovnetworks,asconstructingthelatterisbothmemoryandtimeintensive.4.WealsotriedothersamplingmethodsusingMatlabfromBayesNetToolboxformatlab(BNT),anopensourcesoftwarebyKevinMurphy[54]andalsotheopensourceProbabilisticNet-workLibrary(PNL)fromIntel[55],whichisaC++versionoftheBayesNettoolboxfor55

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matlabbyKevinMurphy.ButneitherofthemsuitedourpurposeandsoHugin[23]wasused.5.SamplingtheBayesiannetworkcanalsobedoneusingGeniewithanyoneoftheapproximateinferencemethods.TheresultsobtainedweresimilaranditcanbeusedtosamplealargeBayesiannetwork.6.Experimentalresultsshowthatthemethodproposedbythisthesisishighlyefcientandeffectiveinvariouswaysnamelytheverylesstimetakentoconstructtheinputmodelandalsosamplingit,nottoforgettheverylowerrorpercentageandhighcompactionratios.Ourfuturedirectionwouldbeintegartingthismethodwithavectorlessapproach.Weintendtousethisuniedframeworkalsowithsequentialcircuits.Also,thepresentworkmodelsonlyrstordertemporalcorrelation.DynamicBayesiannetworksmustbeusedtoacheivehigerordertemporalcorrelation.56

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