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含有定量和定性因子计算机试验的Kriging模型.docx

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含有定量和定性因子计算机试验的Kriging模型 Introduction Krigingisamethodofinterpolationusedtopredictthevalueofanunknownvariableataspecificlocation.Itisbasedontheassumptionthatvaluesoftheunknownvariablearespatiallycorrelated.ThemethodwasfirstdevelopedbyD.G.Krigein1951andhassincebeenwidelyusedinavarietyoffieldsincludinggeostatistics,engineering,andenvironmentalscience. Inthispaper,wewilldiscusstheKrigingmodelthatincludesbothquantitativeandqualitativefactorsinacomputerexperiment. KrigingModel TheKrigingmodelisbasedontheassumptionthatthespatiallycorrelatedvaluesofthevariableofinterestcanbemodeledasaprocesswithameanandaspatiallycorrelatedresidual.Themodelcanbewrittenas: Z(s)=μ+ε(s) WhereZ(s)isthevalueofthevariableatlocations,μisthemeanofthevariableovertheentiredomain,andε(s)isthespatiallycorrelatedresidual. InaKrigingmodelthatincludesbothquantitativeandqualitativefactors,thespatiallycorrelatedresidualisassumedtobeafunctionofboththequantitativeandqualitativefactors.TheKrigingmodelcanthenbewrittenas: Z(s)=μ+ε(s,X,G) WhereXisavectorofthequantitativefactorsandGisavectorofthequalitativefactors.ThespatiallycorrelatedresidualisassumedtobeafunctionofbothXandG. TheKrigingmodelthatincludesbothquantitativeandqualitativefactorscanbeestimatedusingavarietyofmethodsincludingmaximumlikelihoodestimation,cross-validation,andBayesianmethods. MaximumLikelihoodEstimation Maximumlikelihoodestimationisamethodofestimatingtheparametersofastatisticalmodelbasedonthelikelihoodfunction.InthecaseoftheKrigingmodelthatincludesbothquantitativeandqualitativefactors,thelikelihoodfunctionis: L(θ|Y,X,G)=f(Y|X,G,θ) Whereθisthevectorofparameters,Yisthevectorofobservations,Xisthevectorofquantitativefactors,Gisthevectorofqualitativefactors,andf(Y|X,G,θ)isthedensityfunctionfortheobservationsgiventheparametersandthefactors. Themaximumlikelihoodestimateoftheparametersisthevalueofθthatmaximizesthelikelihoodfunction. Cross-Validation Cross-validationisamethodofevaluatingtheperformanceofastatisticalmodel.InthecaseoftheKrigingmodelthat