-- o C inter Arctic Oscillation Index Sapporo Air-Temp Matlab randn (red n

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1 (autoregressive process) (MA) (ARMA) (ARIMA) (persistency) (serial correlation) m amias (988) 970 SST (re-emergence) Alexander (999)

2 -- o C inter Arctic Oscillation Index Sapporo Air-Temp Matlab randn (red noise model) yt () = ryt ( ) + σε() t (.) y(t) t r ε σ (white noise) r=0 r -.

3 (.) r σ σ R σ yt () = r yt ( ) + rσ yt ( ) ε() t R = r σ + σ R + σ ε() t σ = σ σ = σ (.) ( r ) R R / ( r ) <> σ σ R (.) Allen and Smith (996) (first order Markov process)

4 tlng, nsmpl, r rtsrs=randn(tlng,nsmpl); % tlngnsmpl rtsrs(,:)=rtsrs(,:)/sqrt(-r^); % for t=:tlng rtsrs(t,:)=rtsrs(t-,:)*r+rtsrs(t,:); end - r=0.5, nsmpl=0000, tlng=0 m n (/ n) y ( t) i= m

5 (autoregressive process) (autoregressive model) p yt ( ) + φ() yt ( ) φ( pyt ) ( p) + σε( t) = 0 (.3) φ() i y y t AR(q) φ AR AR Matlab yt () = yt () yt ( ) + φ() yt ( ) φ( pyt ) ( p) = 0

6 -6- t p yt () = λ yt ( p) (.4) yt () = λ yt ( ) p p λ + φ() λ... + φ( p) = 0 λ> y λ= λ< AR AR φ () <..4. Maximum Entropy Method (MEM) Spectrum φ τ ρτ ( ) yt ( ) yt ( + τ) ρτ ( ) yt ( ) yt ( τ) { ()... ( p) p σ ε } = φ yt ( ) + + φ yt ( ) + ( t) yt ( τ) = φ() yt ( ) yt ( τ) φ( p) yt ( pyt ) ( τ) + σ ε( t) yt ( τ) (.5) ρτ ( ) + φ() yt ( ) yt ( τ) φ( p) yt ( pyt ) ( τ) + σ ε( t) yt ( τ) = 0 ρτ ( ) + φ() ρτ ( ) φ( p) ρ( t p) = 0 σ = ρ(0) r r() τ + φ()( r τ ) + φ()( r τ ) φ( p)( r t p) = 0 (.6) AR r() = φ() AR 0 (.5) yt ( ), t=,..., n yt ( + τ )

7 -7- ρ(0) + φ() yt ( ) yt ( τ) φ( p) yt ( pyt ) ( τ) + σ ε( t) yt ( ) = 0 φ() r() φ() r()... φ( p) r( p) σ / σ = (.7) (.6) (.7) r() r() r( p) σ / σ () r() r( p ) φ r() φ() r() r() r( p ) = r() φ( p) r( p ) r( p ) r( p) (.8) Toeplitz( ) r() r() r( p) σ / σ r() r() r( p ) φ() 0 r() r() r( p ) φ() = 0 r( p) r( p ) r( p ) φ ( p) 0 (.9) (Yule-alker equation) Matlab lpc, pyulear p p p p (Akaike s Information Criterion, AIC) p AIC (996) AIC AIC = + + p + (.0) ln πσ ( ) AIC = + p + (.) ln σ ( ) AIC AIC AR ) AR

8 -8- ) AR AIC 3) AIC AR φ( p) (Burg) Matlab pburg p p (.6)..5. (MA) q (moving-average process, MA process) yt ( ) + ε( t) + θ() ε( t ) + θ() ε( t ) θ( q) ε( t q), t=,..., n y y t MA(q) MA AR() MA..6. (ARMA) p,q (autoregressive-moving average process, ARMA process) y t ARMA(p,q) (,985, (.5)) yt ( ) + φ() yt ( ) φ( pyt ) ( p) + ε( t) + θ() ε( t ) θ( q) ε( t q) = 0, t =,..., T p,q ARMA AR ARMA..7. (ARIMA) (autoregressive-integrated-moving moving average) ARIMA ARIMA y(t)-y(t-)arma ARIMA..8. (correlogram).3.

9 -9- (time between effectively independent samples, Trenberth (984) Metz (99) ) (effective decorrelation time) T e (effective number of degrees of freedom) τ τ Te = rxx( τ) = + rxx( τ) τ= τ= (.) τ τ Te = rxx ( τ) = + rxx ( τ) τ= τ= (.3) τ Te = rxx( ) ryy( ) rxy( ) ryx( ) τ τ + τ τ τ = τ = + rxy (0) ryx(0) + rxx ( τ) ryy ( τ) + rxy ( τ) ryx ( τ) τ = (.4) r τ τ xtxt ( ) ( + τ) yt ( ) yt ( + τ) τ t= τ t= rx ( τ), ryy( τ), x xt () yt () t= t= τ xt () yt ( + τ ) τ t= rxy ( τ ) xt () yt () t= t= Lieth (98) Bayley and Hammersley (946) Lieth (98)Trenberth (984) Bartlett 930 Bartlett(955) Davis (976) Lieth 98 (Trenberth 984 reference ) Katz (98)Trenberth (984) AR AR

10 -0- AR Trenberth(984) (/ ) x( t) x( t+ τ ) AR AR AIC τ t=.3.. {} x x ({} ) {} {} {} v x x = x x x + x = x x (.5) () (.6) t = v= x t x x x= x ' + µ, µ= x v= x'( t) + µ x t= = = ( x'( t) + µ )( x'( s) + µ ) µ s= t= ( x'( t) x'( s) + ( x'( s) + x'( t) ) µ + µ ) µ s= t= = x'( t) x'( s) + x'( s) + x'( t) + µ s= t= µ x'( t) x'( s) = s = t = { } τ= j i σ τ v= = r τ= τ= (.7) ( τ ) ρxx ( τ) xx ( τ) (.8)

11 -- σ e / = σ T / (.9) e e Te Te = / e (.8)(.9) T e τ = rxx( τ ) τ = (.0) (.) (.7) v= x'( t) + x x t= s= t= s= t= s= t= = ( x'( t) + x)( x'( s) + x) x = ( x'( t) x'( s) + ( x'( s) + x'( t) ) x + x ) x { '( ) '( ) '( ) '( ) } = x t x s x s x t x x x (.).3.. (.4) <> {} (sample mean) q(i), i=,, {} q = (/ ) q () i x, y < xy >= V sample variance { xy} = V + v (sample variance) v= { xy} < xy > v = { xy} { xy} < xy >+< xy> v n= v = { xy} { xy} xy + xy = { xy} xy {} q = q

12 -- = x y x y xy v () t () t () s () s t= s= = xt () ytxs () () ys ( ) t= s= = xt () yt () xs ( ) ys () t= s= xy xy <> ( p.33) x, x, x, x m n p q xmxnxpxq xmxn xpxq + xmxp xnxq + xmxq xnxp v = () ( ) () () () () () ( ) () () ( ) ( ) x t x s y t y s x t y s y t x s x t x t y s y s xy + + t= s= = xt () xs ( ) yt () ys () xt () ys () yt () xs ( ) + t= s= τ= j i v τ = ρxx ( τ) ρyy ( τ) + ρxy ( τ) ρyx ( τ) τ = (.) ρxx( τ) = xtxt ( ) ( + τ), ρyy ( τ) = yt ( ) yt ( + τ), ρ ( τ) = xt ( ) yt ( + τ), ρ ( τ) = ytxt ( ) ( + τ) xy yx x, y Q = x, Q = y r ( τ) = ρ ( τ)/ Q, r ( τ) = ρ ( τ)/ Q, r ( τ) = ρ ( τ)/ Q Q xx xx x yy yy y xy xy x y (.) QQ τ ( τ) ( τ) ( τ) ( τ) (.3) x y v = τ = rxx ryy + rxy ryx x y

13 -3- xt ()', yt ()', t=,..., t= xt () yt () = QQ x y v ( the effective number of degrees of freedom (effective sample size), Metz (99)) QQ τ QQ QQ x y x y x y v = rxx ( ) ryy ( ) rxy ( ) ryx ( ) τ τ + τ τ = = τ = e t/ Te T e τ Te = t rxx( τ) ryy( τ) + rxy( τ) ryx( τ) τ = (.4) DOF = t / Te [] (.4) Te = t rxx( τ) ryy( τ) + rxy( τ) ryx( τ) τ = (, ) e.4. Finite Impulse Reponse (FIR) Infinite Impulse Response (IIR) FIRIIR Press (993) FIR IIR IIR

14 -4- Matlab fft ifft filter filtfilt butter Allen, Myles R., and Leonard A. Smith, 996: Monte Carlo SSA: Detecting irregular oscillations in the Presence of Colored oise. J. Climate, 9 (), Alexander, M. A., C. Deser, and M. S. Timlin, 999: The Reemergence of SST anomalies in the orth Pacific Ocean. J. Climate, (8), Bartlett, M. S., 955: An introduction to stochastic processes with special reference to methods and applications. Cambridge university press, pp (third edition 978) Bayley G. V. and J. M. Hammersley, 946: The 'effective' number of independent observations in autocorrelated time series. J. Roy. Statist. Soc. Suppl., 8(), Davis, R. E., 976: Predictability of sea surface temperature and sea level pressure anomalies over the orth Pacific Ocean. J. Phys. Oceanogr. 6 (3), Hanan, E. J. 970: Multiple Time Seies. iley. Katz, R.., 98: Statistical evaluation of climate experiments with genal circulation models: A parametric time series modeling approach. J. Atmos. Sci., 39, pp. 37, 989. Lieth, C. E.,98: Statistical methods for the verification of long and short range forecasts. ECMF seminar on Problems and prospects in long and medium range weather forecasting [available from ECMF] Metz,., 99 : Optimal relationship of large-scale flow patterns and the barotropic feedback due to high-frequency eddies, J. Atmos. Sci., 48, amias, J., X.,Yuan, and D. R. Cayan, 988: Persistence of orth Pacific Sea Surface Temperature and Atmospheric Flow Patterns. J. Climate,, Press,. H., B. P. Flannery, S. A. Teukolsky,. T. Vetterling, 993: umerical Recipes in C [ ],, pp Trenberth K. E., 984: Some effects of finite sample size and persistence on meteorological statistics. Part I: autocorrelations. Mon. ea. Rev.,,

-- Blackman-Tukey FFT MEM Blackman-Tukey MEM MEM MEM MEM Singular Spectrum Analysis Multi-Taper Method (Matlab pmtm) 3... y(t) (Fourier transform) t=

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