11_Response Surface Methods_248001066

DR.KEDENGMATHEMATICALSCIENCESCENTERTSINGHUAUNIVERSITY,BEIJING邓柯清华⼤大学数学科学中⼼心[e-mail]&DataAnalysisLecture11ResponseSurfaceMethods12ResponseSurface3ResponseSurfaceMethods(RSM)v GoalsØ DiscoveraproperregiontocarryoutexperimentØ FindtheoptimalcombinationoffactorsØ Useasmallnumberofexperimentsv ChallengesØ TheresponsesurfacecanbehighdimensionalØ TheshapeofthesurfaceisunknownØ Thetargetregionoffactorsisunknown4ThreeBasicstepsv FactorscreeningØ StartwithalargenumberoffactorsØ Selectafew(≤5)importantfactorsforresponsesurfacev Aseriesof1storderexperimentsØ StartfromaninitialconfigurationofafewselectedfactorsØ Movetotheneighborhoodoftheoptimalconfigurationv A2storderexperimentØ AnadditionalexperimentinthetotheneighborhoodoftheoptimalconfigurationØ Helptofindtheoptimalconfiguration5SequentialExplorationofResponseSurface61stOrderApproximation&SteepestAscentv The1storderTaylorexpansionv Steepestascent2ndOrderApproximationv The2storderTaylorexpansionv Totalcurvature:v Matrixform72ndOrderApproximation8v The2storderTaylorexpansionv SVDofBv Amoreconvenientform92ndOrderApproximation:4PossibleScenariosv The2storderapproximationv PossiblescenariosØ Ellipticsystem:λj0or0foralljØ Hyperbolicsystem:someλj0,someλj0Ø Stationaryridgesystem:someλj≈0,andtheexperimentregionisclosetothecenterØ Rising/fallingridgesystem:someλj≈0,andtheexperimentregionisfarawayfromthecenter10AGraphicalIllustrationofRidgeSystemsStationaryridgesystemFallingridgesystemCentralCompositeDesignsfork=2&311v Cube/cornerpoints(2kdesign)v Axial/starpointsv CenterpointsRanitidineExperiment:AnExample12RanitidineExperiment:DesignMatrix1323designat8cornerpoints6runsalongthe3axes6replicatesatthecenter1stOrderExperiments:Outline14v Model:v Design:2k+additionalexperimentsatcentralpointsØ 2kfullfactorialdesignwithoutreplicatesatcornerpointsØ rreplicatesatthecenterpointv FunctionsofreplicatesatthecenterpointØ Helptoestimatethenoisetermσ2Ø HelptoestimateandtestthetotalcurvatureCv GoalsØ Suggestcenterpointsoflater1storderexperimentsifCisnotsignificantØ Or,launch2ndorderexperimentifCissignificant1stOrderExperiments:Estimation15v Model:v Design:2k+additionalexperimentsatcentralpointsØ 2kfullfactorialdesignwithoutreplicatesatcornerpoints(nf=2kruns)Ø ncreplicatesatthecenterpoint(ncruns)v ThreetypesofquantitiestobeestimatedØ Ø Ø v DeepestascentisgivenbyTestTotalCurvature:1stOrderModelvs2ndOrderModel16v H0:H1:v Estimateoftotalcurvaturev TestsignificanceoftotalcurvatureØ Ø AlargeCmeanslinearapproximationisnotsufficient,andthus,weareclosetotheneighborhoodofoptimalconfigurationDeepestAscentMethodtoSearchfortheOptimal17v Carryoutaseriesof1storderexperimentsalongthedeepestascentv Selectthecentralpointwithlargest/smallestaverageresponsev AmoreconvenientformDeepestAscentMethod:aCaseStudy18v Response:Processyieldv TwoFactors:Reactiontime&Reactiontemperaturev First-rounddesign:22factorialdesign+5centerpointsAnalysisoftheCaseData19v Conclusions:alinearmodelissufficientv Refinethemodelandfinallyget:v Thedeepestascent:(0.775,0.325)=(1.00,0.42)InteractionCurvatureMaineffectsTheConclusionisFurtherSupportedbyANOVA20v Conclusions:alinearmodelissufficientv Refinethemodelandfinallyget:v Thedeepestascent:(0.775,0.325)=(1.00,0.42)SearchalongtheDeepestAscent21v Thestepsizeis5minutesofreactiontimeand2degreesFv Wefinallyreachedtheoptimalpointin12steps2stOrderExperiments:Outline22v Model:v Design:CCDorBBDØ CCD:centralcompositedesignØ BBD:Box-Benhkendesignv Estimateβ0,{βi},{βij}byregressionv FindtheoptimalconfigurationoffactorsbasedonestimatedparametersCCDBBDCentralCompositeDesignsfork=2&323v DesignparametersØ Distance(α):defaultsettingα=k1/2(allpointsfallintoahyper-sphere)Ø Numberofreplicatesatcenterpoint:2≤nc≤6APracticalCCDwithk=224v ThreeResponses:yield,viscosity&molecularweightv TwoFactors:x1&x2APracticalCCDwithk=2:DataAnalysis25APracticalCCD:EstimatedResponseSurface26OptimizeMultipleResponses27v Multipleresponsesarecommoninpracticev Typically,wewanttosimultaneouslyoptimizeallresponses,orfindasetofconditionswherecertainproductpropertiesareachievedv Approaches:Ø 1.Modelallresponses&overlaythecontourplotsØ 2.OptimizedthemostimportfactorswhilesettingconstraintstootherfactorsØ 3.Combinemultipleresponsesintoa“integratedresponse”OverlayingtwoContourPlots28Aquestiontothinkabout29Whydoweneedthe“starpoints”inthecentralcompositedesign?Aquestiontothinkabout30Whydoweneedthe“starpoints”inthecentralcompositedesign?Answerv Cornerpointshelptoestimatemain&interactioneffectsv Centerpointshelptoestimatenoisetermσ2v Starpointshelptoestimatequadraticeffects{βii}Ø Withoutstarpoints,thequadraticeffectsareconfoundedtogether31Referencev Experiments:Planning,AnalysisandParameterDesignOptimization(《试验设计与分析及参数优化》)Chapter9.v Additionalreadingmaterial:“Lecture10-S1.pdf”