xmeanxavg ymeanysxxsumx0avg x2sxysumx0avg x3y0avg yb1isxysxxb0iavg y0b1i3avg xendDrawhistogramsofthecoe14cientsb0andb1histb0histb1 Figure1 ID: 846798
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1 y=Y(indx);avg x=mean(x);avg y=mean(y);sx
y=Y(indx);avg x=mean(x);avg y=mean(y);sxx=sum((x�avg x):2);sxy=sum((x�avg x):(y�avg y));b1(i)=sxy=sxx;b0(i)=avg y�b1(i)avg x;end;Drawhistogramsofthecoecientsb0andb1hist(b0)hist(b1) Figure1:Histogramofb0 Figure2:Histogramofb12 [n;xout]=hist(b0);n=6n=200;bar(xout,n)holdon;plot(a,pdfNormal)holdo xlabel('b0')ylabel('6*Frequency') Figure3:Histogramandthepdfcurveofb0onthesameplotb=1:0:1:2mu=1:5;sigma=sd B1;pdfNormal=normpdf(b,mu,sigma);[n;xout]=hist(b1);n=
2 40n=200;bar(xout,n)holdon;plot(b,pdf
40n=200;bar(xout,n)holdon;plot(b,pdfNormal)holdo xlabel('b1')ylabel('40*Frequency')4 Figure4:Histogramandpdfcurveofb1onthesameplotAswecanseefromFigure3andFigure4,theshapeofthehistogramofthecoecientsobtainedfromthe200timessimulationsissimilartothatofthecurveoftheestimateddistubtioinofthecoecients.2.(20points)Usethesamedatasetinthelastproblem,wewillestimate0and1usingNewton-Raphsonmethod.a.Drawa3dplotusingMATLAB(check"surf"commandforexample)toillustratehowth
3 eSSEvariesaccordingtodi erentcombina
eSSEvariesaccordingtodi erentcombinationsofestimatesof0and1.Sotospeak,drawa3dplotwherexandyaxesrepresentdi erentvaluesofslopeandinterceptoftheregressionlinerespectively,whilezaxisistheSSE.b.UseNewton-RaphsonmethodtominimizetheSSEandgiveestimatesoftheparam-eters(slopeandintercept)oftheregressionline.Giveageometricalinterpretationofthemethodandexplainhowitworks.Answer:a.Usethe"surf"commandinMatlabtodrawthe3Dplotz=zeros(61;61);x=[0:0:1:6];y=[�1:5:0:1:4:5];i=0;j=0;f
4 ori=1:61forj=1:615 z(i;j)=sum((Y�x(j)
ori=1:61forj=1:615 z(i;j)=sum((Y�x(j)�y(i)X):2);endendmeshgrid(x,y,z)surf(x,y,z) Figure5:3DplotofSSEversustheslopeandtheinterceptoftheregressionlineb.UseNewton-RaphsonmethodtominimizethefunctionF()=SSEandgettheesti-matesoftheparameters.HereweusetheiterationsHF(n)(n+1�n)=rF(n)whereHF(n)istheHessianmatrix(second-orderpartialderivativesofthefunctionSSE)andrF(n)isgradient.rF="�2Pi(Yi�0�1Xi)�2Pi((Yi�0�1
5 Xi)Xi)#HF="2n2PiXi2PiXi2Pi(X2i)#="&
Xi)Xi)#HF="2n2PiXi2PiXi2Pi(X2i)#="01#TheMatlabcodeis:function[beta,SSE]=NR linear(data,beta start)x=data(:;1);y=data(:;2);n=length(x);di =1;beta=beta start;whiledi �0:0001beta old=beta;J=[-2*sum(y-beta(1)-beta(2)*x);-2*sum((y-beta(1)-beta(2)*x).*x)]6 H=[2n;2sum(x);2sum(x);2sum(x:2)]H 1=inv(H);SSE=sum((y�beta(1)�beta(2)x:2)beta=beta old-H 1*Jdi =sum(abs(beta-beta old));endhw1=[X,Y]beta0=[0;0][betaml;sse]=NR linear(hw1;beta0)
6 UsingNewtonRalphsonmethod,wegotthesamere
UsingNewtonRalphsonmethod,wegotthesameresultwiththeleastsquaremethod,b0=2:7725b1=1:5297:ThegeometricinterpretationofNewton'smethodisthatateachiterationoneapproximatesbyaquadraticfunctionaroundF(x),andthentakesasteptowardsthemaximum/minimumofthatquadraticfunction.3.(10points)a.Insimplelinearregressionsettingy=0+1x+,writeouttheexplicitformtheerrorfunction.b.Provethisfunctionisconvexwithrespecttoitsvariables(0and1).Answer:TheerrorfunctionE=Pni=1(Yi�0�
7 ;1Xi)2Toprovetheerrorfunctionis
;1Xi)2Toprovetheerrorfunctionisconvexwithrespectto0and1,weneedtoshowthattheHessianmatrixoftheerrorfunctionispostive-semidenite.SupposewehaveanonzerovectorZ="z1z2#ZTHZ=hz1z2i"2n2PiXi2PiXi2Pi(X2i)#"z1z2#=h2nz1+2z2PiXi2z1PiXi+2z2PiX2ii"z1z2#=2nz21+4z1z2PiXi+2z22PiX2i=2[nz21+2z1z2PiXi+z22PiX2i]=2[(z1+Xiz2)2]0foranynonzerovectorZ2RnTheHessianMatrixoftheerrorfunctionwithrespectto0and1ispostive-semideniteandthereforetheerrorfunctionisaconvexfunct