-1-5. EOF EOF EOF EOF (REOF) EOF Matlab EOF EOF

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1 -- 5. EOF EOF EOF EOF (REOF) EOF Matlab EOF EOF Dual foralis EOF (CEOF,EEOF,REOF,RCEOF,SSA,MSSA) EOF References EOF Epirical orthogonal functions (EOFs).,., (Principal coponent analysis). EOF SLP SS EOF EOF(cobined EOF) (Canonical Correlation Analsysis) (Singular Value Decoposition Analysis) EOF 5.. (NP) Z N, P, Z ( ) EOF EEOF

2 -2- Z M = tx = = X x EOF N t P (principal coponent (PC)) (load) (score) M N, P (5.) Z= X (5.2) (NM) X (PM) EOF (5.)(5.2) EOF t t 2 ( 2) ( ) λ = = 0, 2 EOF (5.) (5.3) 5.2. EOF ( p p) (covariance atrix), V, N V = Z Z v( p, p ) = z( n, p ) z( n, p ) (5.4) 2 2 n= (p, p2) N- N- (5.)(5.2) V M M M M = Z Z= t x t x 2 = 2 xt t x 2 2 i= j= 2 = 2= (5.5) (5.3)

3 -3- M V = λ x x, λ = t t = t = 2 (5.6) P N ( ) z n, p (total variance) p= n= 2 (trace ) N P N ( ) ( ) ( ) ( ) ( ) 2 = =,, 2 =, trace V trace Z Z trace z n p z n p z n p n= (5.7) p= n= (5.6) trace v i i x i x i x i P P M M P 2 ( V) = (, ) = λ ( ) ( ) = λ ( ) i= i= = = i= M 2 = λ x = M trace( V ) = λ (5.9) = λ λ x 2 (explained variance) (5.8) λ M = λ (5.0) 5.3. EOF EOF EOF Vx = λx (5.) Λ VX = XΛ (5.2) (5.2)(5.2)

4 -4- = X X= ZX (5.3) M 2 Vx = λx x x = λx x x = x λx (5.4) i= Vx = λx (5.5) V XVX= Λ (5.6) SLP EOF

5 -5- Fig. 5. Spatial patterns of the first (a) and second (b) EOF odes and their teporal coefficients (c and d) for the SLP in winter season (Dec. Feb.) over the North Pacific (20 E 90 W) (after Minobe and Mantua 999).

6 EOF EOF EOF EOF SLP EOF e p Ze ( Ze ) ( Ze ) = e Z Ze ZZ V e ' e +δ e e δ e e ' e' Ve' = ( e + δe ) V( e+ δe) (5.7) = e Ve+ 2( δe) Ve+ δe Vδe V a Vb = b Va e e e Ve = e ' Ve ' (5.7) δ e ( δ e) Ve = 0 e ' e e (5.8) δ ( δ e) e = 0 (5.9) λ (5.8) ( δe) { Ve λe} = 0 (5.9) Ve = λe (5.20) Proisendorfer p EOF (REOF) EOF ((5.2)(5.3)) EOF

7 -7- EOF EOF EOF EOF EOF(Rotated EOF, REOF) EOF EOF EOF 5.6. EOF Matlab %

9 EOF EOF EOF EOF EOF REOF EOF REOF Doenget and Latif (2002)EOF REOF 5. EOF EOF2 REOF REOF2 SLP SS EOF2 ono pole ono pole SS ono-pole ono-pole EOF REOF EOF EOF REOF REOF EOF EOF REOF REOF EOF

10 -0- EOF EOF x ' = x λ, t ' = t / λ (5.2) i i i i i i x =, = 2 2 i λi ti i ( ) EOF i PC (5.22) Dual foralis EOF 7236= P>>N Z(NP) V( PP ) Z L(NN)=ZZ EOF dual foralis parallel algorith VX=XΛ., Z. ZVX=ZZ ZX=LZX=ZXΛ B=ZX LB=BΛ L B D=Z B D D=Z B=Z ZX=VX=XΛ X x ij, x ij = d ij λ j

11 -- X EOF N znp (, ) z( np, 2) N n = n n0 n n/n0 (p, p 2 ) EOF EOF PC EOF PC EOF open open write(, 5.9. EOF (CEOF,EEOF,REOF,RCEOF,SSA,MSSA) EOF : Epirical Orthogonal Function EOF EOF EOF (CEOF, REOF ) EOF : coplex EOFEOF : Extended EOF ~ SSA/MSSA : EOF ( )Variax rotation

12 -2- EOF EOF REOF EOF : Rotated Coplex Epirical Orthogonal FunctionREOF : Singular Spectru Analysis (Ghil and Vautard, 99) : Multi-Channel Singular Spectru AnalysisEEOF 20~40 SSA EOF EOF, CEOF REOF, RCEOF EOF REOF EOF, REOF 90 CEOF, EEOF, MSSA EEOF,MSSA 5.0. EOF EOF (nonlinear Principal Coponent Analysis, nonlinear PCA) EOF REOF EOF North Rule of hub (North et al. 982)

13 -3- (Doenget and Latif 2002 ) 5.. References Doenget, D. and M. Latif, 2002: A Cautionary Note on the Interpretation of EOFs. J. Cliate, 5(2), , Preisendorfer, R. W. 988: Principal coponent analysis in Meteorology and oceanography. Elsevier, pp (EOF ) North, G. R.,. L. Bell, R. F. Cahalan, and F. J. Moeing, 982: Saping errors in the estiation of epirical orthogonal functions. Mon. Wea. Rev., 0, Richan, M. B., 986: Review article: Rotation of principal coponents. J. Cliatol., 6,