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1 41, 1, From Time Series Analysis to Spatio-Temporal Statistical Analysis Yoshihiro Yajima We review developments of time series analysis and spatio-temporal statistical analysis and envisage a future of them. : 1. Gelfand et al. (2010) Finkenstät et al. (2007) ( ) ( yajima@e.u-tokyo.ac.jp)

2 Box and Jenkins (1970) (Autoregressive Moving Average model ARMA ) Box-Jenkins ARMA Hannan (1970) Anderson (1971) Fuller (1976) Priestley (1981) ARMA 2 (Brockwell and Davis (1991) Chapter 8) 1980 (Autoregressive Conditional Heteroskedasticity model) (Stochastic Volatility model) (Tong (1990) Taylor (1994) Guouriéroux (1997) Fan and Yao (2003) (2003))

3 221 (Banerjee et al. (1993) Beran (1994) Doukhan et al. (2002) Johansen (1996) (2003) Robinson (2003)) 3. 2 (strictly stationary process) (weakly stationary process) (Brockwell and Davis (1991)) t t R = (, ) Z = {0, ±1, ±2, } K 3.1 {X t t K} 2 (i){x t } t E(X t ) = µ µ = 0 (ii)cov(x t, X s ) t s {γ(h)} γ(h) = Cov(X t+h, X t ) γ(h) = exp(ihλ)df (λ) (i 2 = 1) (3.1) T K = R T = (, ) K = Z T = [ π, π] F (λ) F (λ) (3.1) f(λ) γ(h) = exp(ihλ)f(λ)dλ T

4 F (λ) f(λ) X t = T exp(itλ)dz(λ) (3.2) {Z(λ)} λ 1 λ 2 (λ 1, λ 2 T ) E Z(λ 2 ) Z(λ 1 ) 2 = λ2 λ 1 df (λ) dz(λ) λ λ E dz(λ) 2 df (λ) f(λ)dλ K = Z 2 γ(h) < (3.3) h=0 {X t } (short-memory) (short range dependent) (weakly dependent) γ(h) = (3.4) h=0 (long-memory) (long range dependent) (strongly dependent) ARMA {X t } X t φ 1 X t 1... φ p X t p = U t θ 1 U t 1... θ q U t q (3.5) {X t } (p, q) ARMA {U t } (white noise) V ar(u t ) = σ 2 Cov(U t, U s ) = 0(t s) {U t } f U (λ) = σ2 2π B BX t = X t 1 (3.5) (3.6) φ(b)x t = θ(b)u t φ(b) = 1 φ 1... φ p B p θ(b) = 1 θ 1... θ q B q

5 223 φ(z) = 0 ( z = 1) ARMA X t U s (s t) (causality) h 0 γ(h) Ka h (0 < a < 1) h (3.7) K (3.3) f(λ) = σ2 θ(e iλ ) 2 2π φ(e iλ ) 2 (3.8) 0 (3.7) (3.8) 2 ARMA ARFIMA (Granger and Joyeux (1980) Hosking (1981)) FI fractionally integrated d (fractional diffrence operator) d d = (1 B) d = j=0 ( ) d d(d 1)... (d j + 1) = = j j! ( ) d ( B) j (3.9) j Γ(d + 1) Γ(j + 1)Γ(d j + 1) Γ(x) d ( d j) = 0(j > d) (3.9) d (Yajima (1985)) d {X t } φ(b) d X(t) = θ(b)u t {X t } (p, d, q) ARFIMA d < 1/2 {X t } f(λ) = σ2 θ(e iλ ) 2 2π φ(e iλ )(1 e iλ ) d 2 (3.10) ARMA (3.8) (3.10) d (1 e iλ ) d 0 < d < 1/2 λ 0 f(λ)

6 (Fractional Gaussian Noise FGN ) (Manderbrot and van Ness (1968)) (Fractional Brownian Motion FBM ) {B H (t) 0 t < } 2 E(B H (t)) = 0 t E( B H (t) B H (s) 2 ) = σ 2 t s 2H (0 < H < 1) FBM H = 1/2 FBM {X t t = 1, 2,...} X t = B H (t) B H (t 1) {X t } FGN H = 1/2( ) FGN N(0, σ 2 ) H 1/2 1/2 < H ARFIMA λ 0 (Sinai (1976)) ARFIMA FGN f(λ) λ α 1 (λ 0) γ(h) h α (h ) α = 1 2d = 2 2H (3.11) 0 < α < 1(0 < d < 1/2, 1/2 < H < 1) f(λ) (λ 0) (3.4) 4. {X t } θ ARMA φ i (i = 1,..., p), θ j (j = 1,..., q) σ 2 = V ar(u t ) ARFIMA FGN

7 225 d H f(λ; θ) γ(h; θ) θ X N = (X 1,, X N ) log L N (θ; X N ) = 1 2 log det(γ N ) 1 2 (X N µ) Γ 1 N (X N µ) µ = (µ, µ,..., µ) Γ N X N (i, j) Cov(X i, X j ) = γ(i j; θ) Γ 1 N det(γ N) θ Whittle µ = 0 2(2π/N) log L N (θ; X N ) = (2π/N)[log detγ N + X N Γ 1 N X N] N N (2π/N) log τ j N + [ π,π] j=1 m n=1 [ π,π] N m n=1 X mx n e i(m n)λ (2π) 2 dλ (4.1) f(λ; θ) ( log f(λ; θ) + I ) N(λ) dλ (4.2) f(λ; θ) = L N(θ; X N ) (say). I N (λ) I N (λ) = P N t=1 Xt exp( itλ) 2 2πN τ i,n Γ N L N (θ; X N ) θ Whittle (4.1) Γ 1 N Γ N = (γmn ) γ mn e i(m n)λ = (2π) 2 f(λ; θ) dλ [ π,π] Γ N Γ 1 N (Shaman (1976)) (3.6) Γ 1 N = Γ N = I N /σ 2 I N N N f(λ) 0 G(x) lim (2π/N) N G(τ j N ) = N j=1 [ π,π] G(f(λ))dλ (4.3) (Grenander and Szegö (1984)) (4.3) G(x) = log x (4.2) Whittle

8 Idea: Whittle (1961) Univariate Short-Memory: Walker (1964) Hannan (1973) Multivariate Short-Memory: Dunsmuir (1979) Univariate Short-Memory (Irregularly Spaced): Dunsmuir and Robinson (1981) Multivariate Short-Memory (Misspecified): Hosoya and Taniguchi (1982) Long-Memory Gaussian: Yajima (1985) Fox and Taqqu (1986) Long-Memory Gaussian (Misspecified): Yajima (1992) Long-Memory Non Gaussian: Giraitis and Surgailis (1990) Hosoya (1997) f(λ) ( log f(λ; θ) + f(λ) ) dλ f(λ; θ) [ π,π] θ θ 0 f(λ) = f(λ; θ), λ θ θ N (i) ˆθ ˆθ θ 0 a.s. (ii) N(ˆθ N θ 0 ) N(0 4πW (θ 0 ) 1 ) W (θ) = h(λ; θ) = f 1 (λ; θ) [ π,π] [ ] [ ] h(λ; θ) h(λ; θ) f(λ) 2 dλ θ θ f(λ) f(λ; θ), θ f(λ; θ)

9 4.2 N 227 (i) ˆθ θ 0 a.s. (ii)(a) (3.11) 1/2 < α < 1 N(ˆθ N θ 0 ) N(0, 4πV (θ 0 ) 1 W (θ 0 )V (θ 0 ) 1 ) [ 2 ] h(λ; θ) V (θ) = θ θ f(λ)dλ [ π,π] (b) α = 1/2 N/ log N(ˆθ N θ 0 ) 1 (c) 0 < α < 1/2 (N α / log N)(ˆθ N θ 0 ) (ii)(a) f(λ) = f(λ; θ 0 ) V (θ 0 ) = W (θ 0 ) (Hosoya and Taniguchi (1982)) 4.2.(ii)(a) 4.1.(ii) 4.1.(ii) 4.2.(ii) Whittle X N 2 Q N = X NB N X N E(X NB N X N ) B N = (b mn ) N N (m, n) ˆb(λ)( π λ π) b mn = e i(m n)λˆb(λ)dλ [ π,π] f(λ)ˆb(λ) [ π, π] 2 NQN 0 [ π,π] (f(λ)ˆb(λ)) 2 dλ (Fox and Taqqu (1987) Giraitis and Surgailis (1990)) 2 1/ N (Noncentral limit theorem) (Fox and Taqqu (1985)) Whittle h(λ; θ)/ θ θ = θ 0 ˆb(λ) f(λ) ˆb(λ) f(λ)ˆb(λ) (ii) ˆb(0) = 0 f(λ) (λ 0) 4.2(ii)(a) f(λ) 2 ˆb(λ) f(λ)ˆb(λ) 2

10 (ii)(b) (c) ˆb(0) (ii) Fox and Taqqu (1987) 1/ N f(λ)ˆb(λ) 2 Lavancier and Philippe (2011) 5. One-parameter Stochastic Process {X t t R Z} ( ) Multi-parameter Stochastic Process {X t t R d Z d }(2 d) (half sapce) Z d (Guyon (1995)) t Z d ARMA (Guyon (1995)) ARMA R d Z = {0, ±1, ±2,...} d K d (site) s( K d ) Y (s) 1 1 d = 2 s

11 229 Y (s) d = 3 s Y (s) d = 4 t Y (s, t) D( K d ) s {Y (s) : s D} (random field) 5.2 {Y (s) : s K d } 2 (i){y (s)} s E(Y (s)) = µ µ = 0 (ii) t s C(t s) = E(Y (t)y (s)), t, s K d {C(h), h = t s K d } d = 1 (3.1) (3.2) Y (s) = exp(is λ)dm(λ) T d C(h) = exp(ih λ)df (λ) T d (Gikhman and Skorokhod (1974) Rosenblatt (1985) Yaglom (1987)) s = (s 1,..., s d ), h = (h 1,..., h d ), λ = (λ 1,..., λ d ) K = Z T = [ π, π] K = R T = (, ) M = {M(λ), λ T d } d F (λ) d E M( ) 2 = F ( )( T d ) E(M( 1 )M( 2 )) = 0( 1, 2 T d, 1 2 = φ) F (λ) F (λ) f(λ) 6. Ma (2008) 3 A Simple Stochastic Representation A Closed-Form Spectral Density Function A Closed-Form of the Covariance Function

12 Ma (2008)..., just as a stationary discrete-time ARMA time series. In general, however, only one or two tractable forms may be available for a spatial or spatio-temporal stationary random field,... (3.5) (3.7) (3.8) ARMA 3 (i)y (s) (ii) ( ) Cressie (1993) Guyon (1995) Gelfand et al. (2010) 2 2 (isotropic) (separable) C(h) h = d i=1 h2 i 1 (positive definite) C 0 (x)(x R) x h C(h) = C 0 ( h ) C 0 (x) d 2 C(h) (Christakos (1984) Cressie (1993 p. 84)) d C(h) C 0 (x) = 0 exp( x 2 u 2 )dg(x) G(x) g(x) = C 0 (x 1/2 ) g(x)(x 0) (completely monotone) g(x) x 0 n n ( 1) n d n g(x)/dx n 0(0 < x < ) (Schoenberg (1938) Feller (1971) Cressie (1993) Stein (1999)) g(x) = exp( αx γ )(0 < α, 0 < γ 1) C 0 (x) Matérn (Matérn class) (Matérn ( ) Handcock and Stein (1993) Stein (1999)) C 0 (x) = π 1/2 φ 2 ν 1 Γ(ν + 1/2)α 2ν (α x )ν K ν (α x ) (Stein (1999)) K ν 2 (Modified Bessel function of the second kind) α, ν, φ φ α ν ν 2 ν = 1/2 C 0 (x) = πφα 1 exp( α x )

13 231 ν = 3/2 C 0 (x) = 1 2 πφα 3 exp( α x )(1 + α x ) ω = λ f(ω) = φ π (d 1)/2 (α 2 + ω 2 ) ν+d/2 Guttorp and Gneiting (2006) Matérn Matérn Matérn 1947 (Matérn (1947)) ( Harald Cramér) 1960 (Matérn (1960)) (Matérn (1986)) von Kármán (1948a b) ν = 1/3 Whittle (1954) ν = 1 1 σ 2 α ν = 0.5, 1.0, 1.5 ( )Matérn ν ν (Separable Model) 2 h C(h) = C 1 ( h 1 )C 2 ( h 2 ) h 1 = (h 1,..., h m ), h2 = (h m+1,..., h d ) C i ( h i )(i = 1, 2, ) C i ( h i )(i = 1, 2, ) f i ( λ i )(i = 1, 2) f(λ) = f 1 ( λ 1 )f 2 ( λ 2 ) λ = (λ 1,..., λ d ) λ 1 = (λ 1,..., λ m ) λ 2 = (λ m+1,..., λ d ) d = 2 f(λ) λ 1 d 1 λ 2 d 2 (0 < d 1, d 2 < 1) f(λ) (λ λ 2 2) d (0 < d < 2) (Ludeña and Lavielle (1999))

232 41 1 2011 1 Simulation of Matérn class (2010) Sherman (2011) 7.

14 Simulation of Matérn class (2010) Sherman (2011) 7. 4 Whittle Spatial Short-Memory (Regularly Spaced): Dahlhaus and Künsch (1987) Spatial Long-Memory (Regularly Spaced): Ludeña and Lavielle (1999) Spatial Short-Memory Gaussian (Irregularly Spaced): Matsuda and Yajima (2009) Spatial Long-Memory (Non)Gaussin (Irregularlry Spaced): Who? When? Whittle

15 233 Regularly Spaced Increasing Domain Asymptotics P n = d i=1 [1, 2,..., n i] n = d i=1 n i n i (i = 1, 2,..., d) Whittle d = 1 I n (λ) = s P n exp( is λ)y (s) 2 (2π) d n d 2 (edge effect) Ĉ(h) = s s+h P n Y (s)y (s + h)/n d E(Ĉ(h)) C(h) = O(n 1/d ) (7.1) (Guyon (1995)) n(ĉ(h) C(h)) d 2 (7.1) d = 1( ) d = 2 d = 3 d tapered Y (s) = 0(s / P n ) taper h(u) : [0, 1] [0, 1] lim u 0,1 h(u) = 0 h(u) Y (s) h(s)y (s) h(s) = d i=1 h(s i/n i ) s P n 0 tapered I T n (λ) = s P n exp( is λ)h(s)y (s) 2 (2π) d H n H n = s P n h 2 (s) h(u) 1 I n (λ) Y n θ L n(θ; Y n ) = ( π,π] d ( log f(λ; θ) + IT n (λ) f(λ; θ) ) dλ d 2 Whittle d d 3 (Dahlhaus and Künsch (1987)) d 4

16 Irregularly Spaced s i s i = (A 1 u i,1,..., A d u i,d ) i = 1,..., n u i = (u i,1,..., u i,d ) (i = 1,..., n) d i.i.d. g(x) [0, 1] d compact support A j (j = 1, 2,..., d) n Infill Asymptotics k k (n = n k ) (A j (k) (j = 1,..., d)) Mixed Domain Asymptotics (discrete Fourier transform) J k (λ) = (2π) d 2 G 1/2 S k 1 2 n 1 k I k (λ) = J k (λ) 2 n k p=1 Y (s p ) exp( is pλ) G = g(s) 2 ds S [0,1] d k S k = [0, A 1 ]... [0, A d ] S k S k = A 1... A d J k (λ) (2π) d 2 S k 1 2 S k Y (s) exp( is λ)ds G Ĝ = m d m i 1 m i d =1 ĝ(x 1,, x d ) = 1 n k δ d ( i1 ĝ m,, i d m n k j=1 ) 2 ( uj,1 x 1 K,, u j,d x d δ δ K R d δ L k (θ) = D [ ] I k (λ) log{f(λ; θ) + c k (θ)} + dλ f(λ; θ) + c k (θ) θ(ˆθ 0 ) Whittle D (λ D λ D) c k (θ) L k (θ) k d ˆθ θ 0 ( ) d 3 S k 1/2 (ˆθ k θ 0 ) L N(0, 2bW (θ 0 ) 1 ) )

17 (Matsuda and Yajima (2009)) 235 g(s) 4 ds [0,1] b = d ( g(s) [0,1] 2 ds) 2 d ( ) ( ) log f(λ; θ) log f(λ; θ) W (θ) = (2π) d dλ θ θ D b 1 g(x) 1 8. (1) 6 4 (Ma (2008)) 3 (a) d = 1 ( ) d dt + α Y (t) = ɛ(t) (8.1) ɛ(t) 0 σ 2 dy (t) = αy (t)dt + σdw (t)

18 {W (t)} 3 (B 1/2 (t), σ = 1) Ornstein-Uhlenbeck Y (t) = Y (0) exp( αt) + σ t 0 exp( α(t s))dw (s) (Karatzas and Shreve (1991)) Y (0) N(0, σ 2 /2α) {Y (t)} C(h) = (σ 2 /2α)e α h f(λ) = σ 2 /(λ 2 +α 2 ) {Y (t)}(t = 0, 1, 2,...) φ = e α AR(1) Whittle (1954) d = 2 ( ) 2 s s 2 φ 2 Y (s 1, s 2 ) = ɛ(s 1, s 2 ) (8.2) 2 6 C(h 1, h 2 ) = φ h K 1 (φ h ) σ 2 f(λ 1, λ) = (λ λ2 2 + φ2 ) 2 (8.2) (8.1) ɛ(s 1, s 2 ) 0 σ 2 Jones and Zhang (1997) d [( d ) p ] 2 φ 2 c Y (s, t) = ɛ(s, t) (8.3) t s 2 i=1 i C(h, t) = h d/2+1 σ 2 2(2π) d/2 c f(λ, τ) = σ 2 ( λ 2 +φ 2 ) 2p + c 2 τ 2 u d/2 e (u2 +φ 2 ) p t/c 0 (u 2 + φ 2 ) p J d/2 1 (u h )du J ν Bessel (8.2) (8.3) Ruiz-Medina et al. (2004) Kelbert et al. (2005) (8.2) (8.3)

19 237 (b) 2 Stein (2005) f(λ τ) = {c 1 (a 2 1+ λ 2 ) α 1 + c 2 (a τ 2 ) α 2 } β (λ τ) R d R c 1 = σ 2, c 2 = c 2 σ 2, a 2 = 0, α 1 = 2p, α 2 = 1, β = 1 Jones and Zhang (1997) Fuentes et al. (2008) f(λ, τ) = γ(α 2 β 2 + β 2 λ 2 +α 2 τ 2 + ɛ λ 2 τ 2 ) ν, (λ, τ) R d R ɛ ɛ = 1 f(λ, τ) = γ(α 2 + λ 2 ) ν (β 2 + τ 2 ) ν ɛ = 0 f(λ, τ) = γ(α 2 β 2 + β 2 λ 2 +α 2 τ 2 ) ν Matérn (c) Gneiting (2002) σ 2 ( ) h 2 C(h, t) = ψ(t 2 ) φ d/2 ψ(t 2, (h, u) R d R ) φ(t)(t 0) ψ(t)(t 0) Stein (2005) a laundry list of potential models (2) 2

20 (a) (intrinsic stationary random field) h 2 {Y (s)} E(Y (s + h) Y (s)) = 0 E[Y (s + h) Y (s)] 2 = 2γ(h) 2γ(h) (variogram) γ(h) (semivariogram) 3 FBM (multidimensional)fbm exp(is λ) 1 Y (s) = S(λ/ λ )dw (λ) R d λ d/2+h W (λ) 5 M(λ) de W (λ) 2 = dλ S (Istas (2007)) 2γ(h) = R d exp(ih λ) 1 2 λ d+2h S 2 (λ/ λ )dλ S(λ/ λ ) 1 2γ(h) = C h 2H C 1 FBM (Samorodnitsky and Taqqu (1994)) exp(is λ) 1 Y (s) = dw (λ) R d λ d/2+h(λ) H(λ) 0 c( 0) H(cλ) = H(λ) (Bonami and Estrade (2003)) exp(ih λ) 1 2 2γ(h) = dλ λ d+2h(λ) R d H λ 2 S(λ/ λ ) H(λ)

21 239 (b) Priestley (1965) (evolutionary) Y (s, t) = R d R exp(is λ + iτt)φs,t(λ, τ)dm(λ, τ) E dm(λ, τ) 2 = dλdτ φs t(λ τ) (Fuentes et al. (2008)) Cov(Y (s 1, t 1 ), Y (s 2, t 2 )) C(s 1, t 1 ; s 2, t 2 ) = R d R exp(i(s 1 s 2 ) λ + i(t 1 t 2 )τ) φs 1,t 1 (λ τ)φs 2,t 2 (λ τ)dλ dτ φs,t(λ τ) φs,t(λ, τ) = φ (1) s (λ)φ(2) t (τ) Cov(Y (s 1, t 1 ), Y (s 2, t 2 )) = C (1) (s 1, s 2 )C (2) (t 1, t 2 ) K s j (j = 1,..., K) φs j (λ, τ) s s j K(s s j ) φs,t(λ, τ) = k K(s s j )φs j (λ, τ) j=1 s j φs,t(λ, τ) (3) n s i (i = 1,..., n) Y (s i ) C = (c ij ) n n, c ij = Cov(Y (s i ), Y (s j )) (Best Linear Unbiased Predictor (BLUP) (Kriging) ( (2010)) C 1 n 3

22 Covariance Taper with theta=0.8 Product distance 2 Overview of Covariance Tapering n = 1, covariance tapering C(h) C θ (h)( C θ (0) = 1) C tap (h) = C(h)C θ (h) θ C θ (h) = 0, h > θ 2(θ = 0.8) n n Σ tap = (σ ij,tap )(n n) σ ij,tap = C tap (s i s j ) σ ij,tap = 0, s i s j > θ Σ tap 0 sparse C

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seminar0220a.dvi

seminar0220a.dvi 1 Hi-Stat 2 16 2 20 16:30-18:00 2 2 217 1 COE 4 COE RA E-MAIL: ged0104@srv.cc.hit-u.ac.jp 2004 2 25 S-PLUS S-PLUS S-PLUS S-code 2 [8] [8] [8] 1 2 ARFIMA(p, d, q) FI(d) φ(l)(1 L) d x t = θ(l)ε t ({ε t }