Tutorial on Nonparametric Inference With R Chad Schafer and Larry Wasserman

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1、TutorialonNonparametricInferenceWithRChadSchaferandLarryWassermancschafer@stat.cmu.edularry@stat.cmu.eduCarnegieMellonUniversityTutorialonNonparametricInference–p.1/202OutlineGeneralConceptsofSmoothing,Bias-VarianceTradeoffLinearSmoothersCrossValidationLocalPolynomialRegressionCon

2、fidenceBandsBasisMethods:SplinesandWaveletsMultipleRegressionDensityEstimationMeasurementErrorInverseProbemsClassificationNonparametricBayesTutorialonNonparametricInference–p.2/202BasicConceptsinSmoothingProblemI:Regression.Observe(X1;Y1);:::;(Xn;Yn).Estimatef(x)=E(YjX=x).Equivalent

3、ly:Yi=f(Xi)+iwhereE(i)=0.Simpleestimator:fb(x)=meanfYi:jXixjhg:ProblemII:DensityEstimation.ObserveX1;:::;Xnf.Estimatef.Simpleestimator:fb(x)=histogram.TutorialonNonparametricInference–p.3/202Nonparametricregression:DarkEnergyYi=f(zi)+i;i=1;:::;nY=luminosityofithsupernovaiz=r

4、edshiftofithsupernovaiWanttoestimateequationofstatew(z):w=T(f;f0;f00)where003H2(1+z)2+2f(z)1+z0M(f0(z))3w(z)=:3H2(1+z)310M(f0(z))2TutorialonNonparametricInference–p.4/202Nonparametricregression:DarkEnergyGoldSNeSampleDistanceModulus32343638404244460.00.51.01.5RedshiftzTutorialonN

5、onparametricInference–p.5/202Nonparametricregression:DarkEnergyr:SecondDerivativeCo−MovingDistancer(Mpcperkm/s)0.000−0.005−0.010−0.015−0.0200.0150.0100.0050.0001.51.00.50.01.51.00.50.0RedshiftzRedshiftzDarkEnergyEOSParameterw(pressure/density)r:FirstDerivative420−2−40.0140.0100.00

6、61.51.00.50.01.51.00.50.0RedshiftzRedshiftzTutorialonNonparametricInference–p.6/202DensityEstimationpencilbeamredshiftEARTHTutorialonNonparametricInference–p.7/202DensityEstimation:HistogramExample:Redshifts(pencilbeam).80804040000.00.10.20.00.10.2UndersmoothedJustRight804000.00.1

7、0.215001000OversmoothedNumberofBinsTutorialonNonparametricInference–p.8/202DensityEstimation:KernelSmoother0.00.10.20.00.10.20.00.10.20.0000.008TutorialonNonparametricInference–p.9/202TheBias–VarianceTradeoffEverysmootherrequireschoosingasmoothingparameterh.Forahistogram,h=binwidt

8、h.Considertheregressionestimatorb