系统辨识未来发展

SystemIdentification:AnInverseProbleminControlPotentialsandPossibilitiesofRegularizationLennartLjungLinköpingUniversity,SwedenInverseProblems,MAI,Linköping,April4,2013SystemIdentificationSystemIdentification(since1956)TheWorldAroundSystemIdentificationStatisticalLearningtheoryMachineLearningSparsityCompressedsensingParticlefiltersManifoldlearningNetworkedsystemsStatisticsAbstractl SystemIdentificationisawellestablishedareainAutomaticControll Tofindasystemthat(may)havegeneratedobservedinput-outputsignalsl Aninverseproblem!l Theroleofregularization.–Recentlyre-discoveredinthefield.Outlinel Preamble:Aquickprimeronestimationandsystemidentificationl Thestandardapproachtobuildamodell Anewalgorithml …..APrimeronEstimationSoneedtomeetthedatawithaprejudice!Alldatacontaininformationandmisinformation(“Signalandnoise”)informationindataSqueezeouttherelevantButNOTMORE!Primer:EstimationPrejudicesl NatureisSimple!l Occam'srazorPrincipleofparsimonyl Godissubtle,butHeisnotmalicious(Einstein)l So,conceptually,whenyoubuildamodel:Primer:BiasandVarianceMSE=BIAS(B)+VARIANCE(V)Error=Systematic+RandomThisbias/variancetradeoffisattheheartofestimation!ThebestMSEtrade-offtypicallyhasnon-zerobias!TakeHomeMessagesfromthePreambleSeekparsimoniousmodelsThebias/variancetrade-offisattheheartofestimationMatureareawithtracestooldhistory….butstillopenfornewencountersAnEyeopeningEncounterIwasgiveninput-outputdatathatmimicstheC-peptidedynamicsinhumans:Findagoodestimateoftheimpulseresponse(transferfunction)ofthesystemTheTraditionalApproach(My)traditionalapproach(MaximumLikelihood,ML):Buildstate-spacemodelsofcertainordern(n:thorderDifferenceequations)bypem(data,n)!Usecrossvalidationtofindn:!ze=z(1:125);zv=z(126:end);m5=pem(ze,5);m10=pem(ze,10);m15=pem(ze,15);!ModelOrderChoiceOrder10isbest,reestimate:m10=pem(z,10);!Comparemodeloutputwithmodels’simulatedoutputscompare(zv,m5,m10,m15)!ComparewithTrueImpulseResponseHappentofindthetrueimpulseresponse.Howgoodwasmyestimate?compare(trueimp,m10,'ini','z')!81,1%Anotherproposedmethod:XXXHereisanewapproachtosystemidentification:mfilexxxhelpxxx!Thisisamagicalgorithmforsystemidentification.!Tryme!!JustdoModel=xxx(Data)!So,let’sdothat!EstimateoftheImpulseResponsemx=xxx(z);compare(trueimp,mx,m10,’ini’,’z’)!Fitmx:88.6%Fitm10:81,1%Surprise!Thetheoryofestimatinglinearsystemsisnotdeadyet!WhatistheKeyIdea?Regularization:l Useflexiblemodelstructureswith(too)manyparametersl Whichonesarenotquitenecessary?l Puttheparametersonleashesandcheckwhichonesaremosteagerinthepursuitforagoodfit!l Pullparameterstowardszero()l Pullparameterstozero()OutlineforRemainderofTalk• Regularizationforbias/variancetradeoff• Regularizationformanifoldlearning• RegularizationforsparsityandparsimonyRegularization:Curbthefreedominflexiblemodels.Outline• Regularizationforbias/variancetradeoff• Regularizationformanifoldlearning• RegularizationforsparsityandparsimonyRegularizationE.g.LinearRegression:Recall:(Too)manyparameters?Putthemonleashes!AFrequentistPerspectiveFrequentist(classical)perspectiveTrueparameternoisevarianceThechoiceminimizestheMSEto…BayesianInterpretationisarandomvariablethatbeforeobserving(apriori)isi.e.thenegativelogofitspdfisanditspdfafter(aposteriori)isThisistheRegularizedLScriterion!So,thereg.LSestimategivesthemaximumofthispdf(MAP),(theBayesianposteriorestimate)CluetothechoiceofP!pdf:probabilitydensityfunctionEstimationofImpulseResponseAgoodpriorfordescribesthebehaviourofthetypicalimpulseresponseg(k):l Exponentiallydecaying,sizeC,ratel Smoothasafunctionofk,correlationl Estimate(thehyperparameters)fromdataEstimationofHyperparametersl ”EmpiricalBayes”(EB)l xxx:estimatebyEBanduseinregularizedLS!(=RFIR)l OriginalresearchandresultsbyPillonetto,DeNicolaoandChiusoInaBayesianframework,Yisarandomvariablewithadistributionthatdependsonthehyperparameters.EstimatethosebyML!ALinktoMachineLearning”GaussianProcesses(GP)”TheIRestimationalgorithmisacaseofGPfunctionestimation,frequentlyusedinMachineLearning.(PillonettoetalusedthisframeworktodevicetheXXXalgorithm)GP:EstimateaFunctionf(x)ThesearethesameasthepreviousBayesiancalculations!AssumeaGaussianpriorforfComputetheposteriorestimategiventheobservationsMachineLearningofDynamicSystemsCarlRasmussen(MachineLearningGroup,Cambridge)hasperformedquitespectacularexperimentsbyswingingupaninvertedpendulumusingMPCandamodelestimatedbyGP.ThefunctionestimatedisthestatetransitionfunctionGP:DualitywithRKHSThisisamuchstudiedprobleminstatisticsandmachinelearning(Wahba,Schölkopf,…)ComparewiththefinitedimensionalFIRcase:LetthepriorpdfofthefunctionfhaveacovariancefunctionKassociatedwithaReproducingKernelHilbertSpace.ThentheBayesianposteriorestimateoffisgivenas1. SymbiosiswithBayesiancalculationsinGaussianframeworks.2. Welltunedregularizationnorm(e.g.byEB)cangivesignificantimprovementinmodelquality(MSE)RegularizationnormPriormodelknowledgeSummary:QuadraticNormRegularization• Regularizationforbias/variancetradeoffOutline• Regularizationforbias/variancetradeoff• Regularizationformanifoldlearning• R