EOF分析

A.7EOF(empiricalorthogonalfunction,EOF)(eigenvectoranalysis)(principalcomponentanalysis,PCA)Lorenz1950EOF²Xm£n²XXTCm£m=1nX£XTXCX(C01)C²C(¸1;:::;m)Vm£mCm£m£Vm£m=Vm£m£^m£m^m£m^=266664¸10:::00¸2:::0::::::::::::00:::¸m377775¸¸1¸2:::¸mX¸0042EOF¸1EOFVEOF1=V(:;1)¸kVkEOFk=V(:;k)²EOFX()PCm£n=VTm£m£Xm£nPCPC(1,:)EOFXEOF(PC)EOFPCXX=EOF£PCEOFXEOFPC1nPC£PCT=^PC0E£ET=II100EOFCMatlabEOF1[EOF,E]=eig(C)EOFEPC=EOFT£X(1kmNDVI)mC2XX=UXVTP(P)43²C=1nXXTC¸P=pn¸²C=XXTC¸P=p¸X1mXi=1X2i=mXk=1¸k=mXk=1PC2kX¸k¸kPmi=1¸i£100%EOFNorth(1982)95%¢¸=¸r2N¤¸N¤(4)¸¸A.161949¡20021EOF44[E,V]=eig(C)X%X=[26152;94054];X(1,:)=X(1,:)-mean(X(1,:));X(2,:)=X(2,:)-mean(X(2,:));XX=-1.202.80-2.201.80-1.204.60-0.40-4.400.60-0.40%%%co-variancematrixC=X*X'/5;C=3.760.920.928.24[EOF,E]=eig(C);%V:eigenvectors;E:eigenvaluesPC=EOF'*X;%%reversetheorderE=fliplr(flipud(E))lambda=diag(E);%retaineigenvaluesonlyEOF=fliplr(EOF)PC=flipud(PC)EOF=0.19-0.980.980.19E=8.42003.58PC=4.280.15-4.740.94-0.622.07-2.821.31-1.651.10%%checkEOF*EOF'%=IEOF1.00001.00PC*PC'/5%=lambdaPC8.420.000.003.58EOF*PC%=XX-1.202.80-2.201.80-1.204.60-0.40-4.400.60-0.4045[U,S,V]=svd(X)XX=[26152;94054];X(1,:)=X(1,:)-mean(X(1,:));X(2,:)=X(2,:)-mean(X(2,:));X-1.202.80-2.201.80-1.204.60-0.40-4.400.60-0.40[U,S,V]=svd(X);U=0.190.980.98-0.19S=6.49000004.23000V=0.66-0.490.560.09-0.060.020.670.63-0.320.22-0.73-0.310.530.25-0.160.140.390.030.910.06-0.10-0.26-0.020.060.96EOF=U;PC=S*V';PC=4.280.15-4.740.94-0.62-2.072.82-1.311.65-1.10E=S.^2/5;%=lambdaEE=8.42000003.58000EOF*PC%=XX-1.202.80-2.201.80-1.204.60-0.40-4.400.60-0.4046024681001000200030004000EigenvalueNumber−0.08−0.06−0.04−0.0200.020.04EOF126.1%195019601970198019902000−100−50050100YearPC#1−6−4−202hPaA.16:1EOF.(a)95%(b)(c)(d)+¾(hPa).1949¡2002NCEP/NCAR47XEOFPC¤EOF¤PC¤(k)=PC(k)p¸kEOF¤(k)=EOF(k)p¸kPC01XPCXA.16(d)North(1)(2)160EOF(A.17)(3)1SLPmnm£mA=1nXX0B=XX048(a)EOF190.7%0.020.040.060.080.10.120.140.16(b)EOF115.8%−0.2−0.15−0.1−0.0500.050.10.15(c)EOF113.4%−0.1−0.0500.050.10.15(d)EOF111.3%−0.1−0.0500.050.1A.17:EOF(a)(b)(c)(d).1951¡2002.49C=XX0C¤=X0X,X0XXX0C¤£V¤=V¤£^V¤C¤^V¤CVa=X£V¤VaCnVkVk=1p¸kVa(:;k)VPC=VT£XEOFm£mEOFm£nnX5£2XX052X=[-1.204.602.80-0.40-2.20-4.401.800.60-1.20-0.40][V1,E1]=eig(X*X');%%V1=fliplr(V1);%%E1=fliplr(flipud(E1));%%V1=-0.660.49-0.45-0.15-0.32-0.02-0.67-0.14-0.15-0.720.730.31-0.56-0.15-0.1750-0.14-0.39-0.42-0.580.570.100.260.53-0.77-0.19E1=42.110000017.89000000000000000000[V2,E2]=eig(X'*X);%%V2=fliplr(V2);%%E2=fliplr(flipud(E2));%%V2=0.19-0.980.980.19E2=42.110017.89E1E2XX'Va=X*V2;%%V_k1=Va(:,1)/sqrt(E2(1,1));0.660.02-0.730.14-0.10XX'V_k2=Va(:,2)/sqrt(E2(2,2));0.4951-0.670.31-0.390.262XX01nXX012.5±1000hPa(©)EOFm=10512n=54C=©©T10512£10512C¤=©¤©54£5454NDVI20EOFA.7.1REOF:varimax.mA.8SVDEOFXEOF(combinedempiricalorthogonalfunctionCEOF)(singularvaluedecompositionSVD)(canonicalcorrelationanalysisCCA)SVDSVDSVDSVD52²XYm£np£nC=1nXYTCm£p²SVDC=UXVTUXVYP°Matlab[U,S,V]=svd(C)²XA=UTXYB=VTY²°XY°2P°2£100%1982¡2000NDVINDVI1982¡2000SVDNDVIA.18NDVI42.619.510.37.75.04.22.3%791.6%NDVINDVINDVINDVINDVI53EUNDVIEU0.720.78WPPNANDVINAO0.520.5895%NDVIWA¡0:52¡0:56PNASOEANP(:,.NDVI.2002,57(5),505-514)5430N40N50N60N70N(A)SVD1150W100W50W050E100E150E30N40N50N60N70N(B)SVD2150W100W50W050E100E150E30N40N50N60N70N(C)SVD3150W100W50W050E100E150E30N40N50N60N70N(D)SVD4150W100W50W050E100E150E30N40N50N60N70N(E)SVD5150W100W50W050E100E150E30N40N50N60N70N(F)SVD6150W100W50W050E100E150E30N40N50N60N70N(G)SVD7150W100W50W050E100E150E-0.1-0.04-0.03-0.02-0.0100.010.020.030.040.1NDVIA.18:NDVISVD.1982¡2000.55