实用多元统计分析答案08.pdf

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1、164Chapter88.1Eigenvaluesof*areÀ1=6.À2=1.TheprincipalcampenentsareYl=..894X1+.447X2Y2=.~47X1-.894XZ,.Vareyi)=À1'=6.Therefore.proporti()noftatalpopulationvarianceexplainedby~lis6/(15+1)=.86.8.2£=(1..6325J.63251(a)Y,=,.707L,+.7!J7L¿Var(Yi)=À,=1.6325Proportienoftotalpopulation'2=.7n7Z,~.707Z2

2、varianceexplainedbyYlis1.6325/(1+1.=..816(b)No.Thetwo(standardized)variablescontribute~quallytotheprincipalcomponentsin8.2(a).Thetwovariablescontributeunequallytotheprincipalcomponentsin8.1because'Oftheirunequalvarian~es.(c)PyL=.903;PYlZz=.903;.PyZ=.42911.2,8.3Ei~envaluesoftare2.'4.4.E;gen

3、vect~rsassaciatewiththeei~en-values4.4arenotunique.Onechoi~~is=i=iO1O)çnd:~=(001).Withtheseassignments.the'principalcomponentsarey1=Xl'Y2=X2andY3=X3.8.4figenvaluesof*areselutions.of1;-À11=(a2-Àp-2t~2_ÀH.a2,p)2=0Thus~0'2-À)H.a2_À)2_2cr4p2J=0S'OÀ=0'2'OrÀ='"2~lt~hl,).ForÀ1=(12,~i=(l/ff,.o,-l/

4、I2J.'ForÀ2=O'2(l+i'Ii;~fj/Z.)1NEiZ).ferÀ~'=a2'tl-pI2).~~=0/".-1/12;1/121.65Princi'fa1PropertionofTotalComponentVarianceVarianceExplained11021/3Y1=aXl-12X3111.a2(l+pm1(1+1'12)y2="2Xl+12X2+2"X31110'2(1-.p12)1(l-pl2)y3='2Xl-/žX2+2"X38.5(a)Eigenva.luesof2satisfyIE-ul=(l-À)3+2.p3-3(1:'À)p1=0or(

5、l+29-À)(1-p-À)2=O.HenceÀ1=1+2.p;À2=À3=1-?andresultsareconsistentwith(8-16)forp=3.(b)Bydirectmultiplication11.iy.p-y~-.ø(c1)=(1+(P-1)9Hc1)thusvaridyingthefir~teig~nvalu~-eigenv~ct~~pair.further~:i=(l-p)~i';=2.3....,p.1668.6(a)Yi=.999xl+.041x2SamplevarianceofYi=-l=7488.8varianceofY2=t=13.8Y2

6、=-.041xl+.999x2Sample(b)ProportionoftotalsamplevarianceexplainedbyYiis-l/(-l+t)=.9982(c)Centerofconstantdensityellpseis(155.60,14.70).Halflengthofmajoraxisis102.4indirectionofyi'Halflengthofperpndicularminoraxisis4.4indirectionofY2'(d)r)~x=1.000,ry"x=.687Thefirstcomponentisalmostcompletely

7、deterined191.'2byXi=salessinceitsvarianceisapproximately285timesthatofX2=profits.Thisisconfirmedbythecorrelationcoefficientry'x=1.000.¡.i8.7(a)SamplevaranceofYi=-l=1.6861Yi=.707z1+.707z2Y2=.707z1-.707z2SamplevarianceofY2=t=.3139(b)Proportionoftotalsamplevarian