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1X2XpX1Y2YqY=X(1X2X)′pX=Y(1Y2YqY)′p≤qXY()XE1µCov()X∑11()YE2µCov()Y∑22Cov()YX,∑12∑21'Z⎥⎦⎤⎢⎣⎡YXµE()ZE()()⎥⎦⎤⎢⎣⎡YEXE⎥⎦⎤⎢⎣⎡21µµ()()∑+×+qpqpE()µ−Z()′−µZ()()()()()()()()⎥⎥⎦⎤⎢⎢⎣⎡′−−′−−′−−′−−22122111µµµµµµµµYYEXYEYXEXXE()()()()⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡∑∑∑∑××××qqpqqppp222112111X2XpX1Y2YqYppXaXaXaU+++=L2211Xa′V=qqYbYbYb+++L2211Yb′()′=paaaa,,,21L()′=qbbbb,,,21LVar()Ua′Cov()aXa′∑11aVar()Vb′Cov()bYb′∑22bCov()VU,a′Cov()YX,ba′∑12bUVρρCorr()VU,bbaaba∑∑∑′′′221112UVUV)1(a)1(b1UXa)1(′1VYb)1(′UV)2(a)2(b2UXa)2(′2VYb)2(′1U1VUV121,−pUUUL121,−pVVVLpUpVpXY1U1V2U2VpUpVabρVar()UVar()VVar()UVar()VabCorr()VU,a′∑12b()=ba,φa′∑12b()−−′−∑1211aaλ()1211−′∑bbννλ,φab⎪⎪⎩⎪⎪⎨⎧=−=∂∂=−=∂∂∑∑∑∑0022211112bababaνφλφa′b′a′b∑12λλ=′=∑aa11b′a∑21νν=′=∑bb22a′()12′∑b=b′a∑12νλ=a′b∑12ρλUVa∑−11λb∑+120=a∑21b∑−22λ0=∑12∑−122(∑12∑−122∑21)∑−112λ0=a∑21∑−111(∑21∑−111∑12)0222=−∑bλ∑−111∑−122(∑∑−12111∑−122∑21)2λ−0=a(∑∑−21122∑−111∑12)02=−bλ=A∑−111∑12∑−122∑21=B∑−122∑21∑−111∑12⎪⎩⎪⎨⎧==bBbaAa22λλ2λABab2λABABpABABνλ=a′b∑12ρλUVAB21λ1λ21λ)1(a)1(b1UXa)1(′1VYb)1(′UV1λABpp∑12qp022221≥≥≥≥pλλλL()()()kaaaL21,()()()kbbbL21,k()()()()()()()()()⎪⎪⎩⎪⎪⎨⎧+++=+++=+++=pkpkkpppppXaXaXaUXaXaXaUXaXaXaULMLL22112222121212121111()()()()()()()()()⎪⎪⎩⎪⎪⎨⎧+++=+++=+++=qkqkkpqqqqYbYbYbVYbYbYbVYbYbYbVLMLL22112222121212121111pλλλ≥≥≥L21pλλλ≥≥≥L211U1V2U2VpUpV)(ia)(ibiXiYiUiVi1X2XPX1U2UpU1Y2YqY1V2VpVVar()kUVar()kVCov()lkUU,Corr()lkUU,lk=Cov()lkUU,Corr()lkUU,lk≠=lk,pCov()lkVV,Corr()lkVV,lk=Cov()lkVV,Corr()lkVV,lk≠=lk,qXYiUiViλiUjVji≠Cov()jiVU,Corr()jiUU,⎪⎩⎪⎨⎧≠==≠pijipijii002,1,0LλUVUGVGUGCov()UX,Cov()XaX′,aXXEaXXEUXE)()()(′=′=′∑11aVGCov()VY,Cov()XbY′,∑22bUGVGiXUiUGiXUUGVGCov()VX,Cov()YbX′,∑12bCov()UY,Cov()XaY′,∑21aUpGGUU′VqGGVV′pGGiUUλ′()iduR()pGGRiUUiduλ′=()qGGRiVVidvλ′=XYqp+n()qpn+×⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡nqnnnpnnqpqpyyyxxxyyyxxxyyyxxxLLMMLLLL21212222122211121111211S⎥⎦⎤⎢⎣⎡22211211SSSS11S∑=′−−−niiiXXXXn1))((1122S∑=′−−−niiiYYYYn1))((1112S211))((11SYYXXnniii′=′−−−∑=SΣΣˆΣˆΣ111−S12S122−S21S∑−111∑12∑−122∑21122−S21S111−S12S∑−122∑21∑−111∑120ˆˆˆ22221≥≥≥≥pλλλL()()()kaaaˆˆ,ˆ21L()()()kbbbˆˆ,ˆ21Lpλλλˆˆˆ21≥≥≥L1ˆU1ˆV2ˆU2ˆVpUˆpVˆ0ˆˆˆ22221≥≥≥≥pλλλL()()()kaaaˆˆ,ˆ21L()()()pbbbˆˆ,ˆ21L022221≥≥≥≥pλλλL()()()kaaaL21,()()()pbbbL21,XYCov()YX,0=2χXY∑120H∑120=Cov()YX,0=1H∑12≠0Ap0ˆˆˆ22221≥≥≥≥pλλλL1Λ()∏=−pii12ˆ1λ()()()22221ˆ1ˆ1ˆ1pλλλ−−−L2iλA∑−122∑21∑−111∑12n0H1Q()⎥⎦⎤⎢⎣⎡++−−1211qpn1Λqp×2χα1Q≥2χ()qp×0H1λ1λ1λ2Λ()∏=−pii22ˆ1λ2Q()⎥⎦⎤⎢⎣⎡++−−1212qpn2lnΛ()()11−−qp2χα1Q≥2χ()()11−−qp2λkkλkrrQ()⎥⎦⎤⎢⎣⎡++−−121qprnrΛrΛ()∏=−prii2ˆ1λrQ2χ()()11+−+−rqrp⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡nqnnnpnnqpqpyyyxxxyyyxxxyyyxxxLLMMLLLL21212222122211121111211R⎥⎦⎤⎢⎣⎡22211211RRRR11R22R12R21R′=A111−R12R122−R21R=B122−R21R111−R12RSPSSX1X2X3X4X5X1X2X3X4X510.9489.8964.51357773.0822.5892.3255.41298168.65313.4690.7138.2114819.08412.4690.0445.12112427.6858.9490.4641.83108036.1262.890.1750.64201150.8678.9191.4346.32138342.6588.8290.7847.33162847.1790.891.4762.36482266.23105.9490.3140.85169621.24112.692.4235.14171732.81127.0787.9729.5193317.91314.4488.7129.04131321.361415.2489.4331.0594320.4153.1691.2137.85137227.34169.0488.7639.7188015.521712.0287.2838.76124828.911811.1589.1336.3397618.231922.4687.7238.38184536.772024.3484.8631.0779815.12133.2183.7939.44119324.05224.7890.5731.2690320.252321.568622.3865418.932414.0980.9621.4995614.722532.3187.67.786512.592611.1889.7141.0193021.492713.886.3329.6993822.042825.3481.5631.3110027.352920.8481.4534.59102425.723039.664.938.47137431.91CANCORR1FilesNewSyntax2Canonicalcorrelation.sps12s1_cv001s2_cv001s1_cv002s2_cv002