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18-660:NumericalMethodsforEngineeringDesignandOptimizationXinLiDepartmentofECECarnegieMellonUniversityPittsburgh,PA15213Slide1OverviewLecture6:LinearEquationSolver^LUdecomposition^CholeskydecompositionLecture7:LinearRegression^Ordinaryleast-squaresregression^Minimaxoptimization^DesignofexperimentsSlide2LinearRegressionLinearregression(alsoreferredtoasresponsesurfacemodeling)iswidelyusedformanyengineeringproblems^Wedonotknowtheanalyticalformoff(x)^Butwecangenerateasetofsamplingpointsforf(x)^Fitanapproximatefunctionforf(x)fromthesesamplingpointsf(x)xSlide3LinearRegressionMajorstepsoflinearregression^Selectamodeltemplate(e.g.,polynomialfunction)^Generateanumberofsamplingpoints^Computeperformancevaluesatthesesamplingpoints^CreateasetoflinearequationstosolvemodelcoefficientsAsimpleexample^f(x)=exp(x),x∈[-11]1,1]^WewillusethissimpleexampletoshowhowwecangenerallybuildaregressionmodelfromsamplingdataSlide4LinearRegressionExampleStep1:selectamodeltemplatef()x≈bx+cStep2:generateanumberofsamplingpointsSlSamples12345x-1-0.500.51Step3:computeperformancevaluesatthesesamplingpointsSamples12345f(x)0.36790.60651.00001.64872.7183Slide5LinearRegressionExampleStep4:createlinearequationsformodelcoefficientsf(x)≈bx+cSamples12345x-1-0.500.51f(x)0.36790.60651.00001.64872.7183⎡−11⎤⎡0.3679⎤⎢−0.51⎥⎢0.6065⎥⎢⎥⎡b⎤⎢⎥⎢01⎥⋅⎢⎥=⎢1.0000⎥i-thsamplingpoint⎢⎥⎣c⎦⎢⎥⎢0.51⎥⎢1.6487⎥⎣⎢11⎦⎥⎣⎢2.7183⎦⎥xvaluesf(x)valuesSlide6LinearRegressionExampleStep5:solveover-determinedlinearequations^#ofequationsisgreaterthan#ofcoefficients–over-determined^Noexactsolutionexiststosatisfyallequations,butwecanfindtheleast-squaressolution:A⋅α=BSlide7LinearRegressionExampleA⋅α=BminA⋅α−B2α2Slide8LinearRegressionExample⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥Msamples⎢A⎥⋅α=⎢B⎥(M>N)⎢⎥⎢⎥⎢⎥⎢⎥⎣⎢⎦⎥⎣⎢⎦⎥NcoefficientsThereareseveralpossiblewaystosolveover-determinedlinearequationsforlinearregression^Wewillexplainthesealgorith
