Modeling and testing long memory in random fields Frédéric Lavancier lavancier@math.univ-lille1.fr Université Lille 1 LS-CREST Paris 24 janvier 6
1 Introduction Long memory random fields Motivations Previous studies : isotropic long memory
Random fields on the lattice Z d Discrete random field : random process on the lattice Z d In dimension 2 : 18 16 14 12 8 6 4 2 2 4 6 8 12 14 16 18 The value at each point of the lattice is represented by a color : > Red Green Blue Stationary random field : the color s generating law is invariant under translations.
A particular stationary random field : the white noise 9 3 3 9
A { weakly dependent short memory random field 9 3 3 9
A { strongly dependent long memory random field 9 3 3 9
Quantification of the dependence of (X n ) n Z d The dependence between two points of X at lag h Z d can be measured by the covariance function r(h) = cov(x k, X k+h ) = cov(x, X h ). An alternative tool to study dependence : the spectral density f defined on [ π, π] d by r(h) = e i<λ,h> f(λ)dλ. [ π,π] d Remark The two points of view are closely related but not equivalent
Long memory Definition A random field X exhibits long memory when its covariance function is not summable, i.e. h Z d r(h) =. Proposition If the spectral density f of X is unbounded then X is long-range dependent. Examples In dimension d = 2, f(x, y) = x α y β, < α < 1, < β < 1 ; f(x, y) = (x 2 + y 2 ) α, < α < 1 ; f(x, y) = x y α, < α < 1.
Example : a short memory random field 9 3 3 9
Example : a short memory random field x 4 9 3 2.5 2 1.5 1 9.5.5 3 3 9 3 Its covariance function. 3 9
Example : a short memory random field x 4 9 3 2.5 2 1.5 1 9.5.5 3 3 9 3 Its covariance function. 3 9 9 3 3 9 Its periodogram.
Example : An isotropic long memory random field 9 3 3 9
Example : An isotropic long memory random field 9 9 3 3 9 3 Its covariance function. 3 9
Example : An isotropic long memory random field 9 9 3 3 9 3 Its covariance function. 3 9 4 3 3 2 9 3 3 9 Its periodogram.
Example : A product-type long memory random field x 4 9 8 6 9 4 2 2 3 3 9 3 Its covariance function. 3 9 9 3 9 3 3 9 Its periodogram.
Example : A non-isotropic long memory random field 9 9 3 3 9 3 Its covariance function. 3 9 9 3 9 3 3 9 Its periodogram.
Motivations Are there some natural mathematical models leading to long memory random field? Consider a realisation of a long memory random field, can we test the strong dependence property? can we estimate this long memory (direction, intensity,...)? Can we use the usual statistical methods (in regression, in forecasting,...)? What is the asymptotic behaviour of the main tools (partial sums, empirical process,...)?
Previous studies In image analysis : Some texture models (Kashyap and Lapsa, 1984, Bennett and Khotanzad, 1998 and Eom, 1 ). Asymptotic results, under isotropic long memory : Partial sums (Dobrushin and Major, 1979 et Surgailis, 1982 ). Empirical process (Doukhan, Lang and Surgailis, 2 ). Quadratic forms (Heyde and Gay, 1993 and Doukhan, Leon and Soulier, 1996 ). Local times (Doukhan and Leon, 1996 ).
Isotropic long memory Definition A stationary random field exhibits isotropic long memory if one of the two following conditions is fulfilled its covariance function behaves as ( ) n r(n) = n α b L( n ), < α < d. n its spectral density is continuous everywhere but at zero where : ( ) ( ) x 1 f(x) x α d b L, < α < d. x x where L is a slowly varying function at infinity and b is a continuous function on the unit sphere of R d.
2 Modeling Filtering Aggregation Long memory in Statistical Mechanics
Filtering : autoregressive fields Let (ɛ n ) n Z d be a white noise, P (L 1,..., L d )X n1,...,n d = ɛ n1,...,n d admits a unique stationary solution iff 2 1 P (e iλ1,..., e iλ d ) dλ <. [ π,π] d If d = 1 λ, P (e iλ ) π π 1 P (e iλ ) 2 dλ < If d 2 λ, P (e iλ1,..., e iλ d ) = [ π,π] d 1 Example P (e iλ 1,...,e iλ d ) 2 dλ <. X n1,...,n 5 1 5 (X n 1 1,n 2,...,n 5 + + X n1,...,n 5 1) = ɛ n1,...,n 5 admits a stationary solution with spectral density 1 f X (λ 1,..., λ 5 ) 1 1 5 + + e (eiλ1 iλ5 ) 2.
Filtering : general case Let (ɛ n ) n Z d be a white noise with the spectral representation ɛ n = e i<n,λ> dw (λ), [ π,π] d where W is the random spectral measure of ɛ. Property The field constructed by filtering ɛ through a, defined by X n = e i<n,λ> a(λ)dw (λ) = â j1,...,j d ɛ n1 j 1,...,n d j d, [ π,π] d j 1,...,j d where â are the Fourier coefficients of a, has the spectral density f X (λ) a(λ) 2. X is long-range dependent if the filter a is unbounded
Some examples in dimension d = 2 Example (Non-isotropic long memory) a(λ 1, λ 2 ) = λ 1 λ 2 α where < α < 1 2. Example (Long memory depending on the directions) ( X n1,n 2 = 1 L ) 1 + L α 2 ɛ n1,n2 2, < α < 1/2, admits the spectral density ( f(λ 1, λ 2 ) = σ2 4π 2 4 sin 2 λ 1 λ 2 sin2 2 2 + sin2 ( )) λ1 + λ α 2. 2
Aggregation Construct N independent copies of the autoregressive field X : P (L 1, L 2 )X n1,n 2 = ɛ n1,n 2, where P is a polynomial function with random coefficients, independent from ɛ. The aggregated field is obtained thanks to the classical CLT : Z n,m = lim N 1 N N k=1 The Gaussian field Z has the spectral density f(λ) = X (k) n,m. σ2 (2π) d E P 1 ( e iλ1, e iλ2) 2. If one chooses properly the law generating the coefficients of P, then Z is a long memory Gaussian random field.
Example (Product-type long memory) P (L 1, L 2 ) = (1 β 1 L 1 )(1 β 2 L 2 ) where β 1 and β 2 are generated by the same law with density 1 x. Then f(λ 1, λ 2 ) 1 at (λ 1, λ 2 ) = (, ) λ1 λ 2 r(h, l) is non-summable. Example (Long memory in one particual direction) P (L 1, L 2 ) = 1 βl 1 L 2 where β is generated by a density 1 x. Then 1 f(λ 1, λ 2 ) at (λ 1, λ 2 ) = (, ) λ1 +λ 2 r(h, l) = if h l and r(h, h) is non-summable.
Long memory in Statistical mechanics The Ising model : The spins takes value in { 1, 1} and the interaction potential is { βxi x Φ i,j (x i, x j ) = j if d k=1 i k j k = 1 otherwise, where β > represents the inverse of the temperature. When d 2, there is phase transition if β β c and r(h) h (d 2+µ), at β = β c, h where µ [, 2] is a parameter depending on d.
Long memory in Statistical Mechanics -The homogeneous systems : x i R and { 1 Φ i,j (x i, x j ) = 2 J()x2 i if i = j J(i j)x i x j if i j, where (J(i)) i Z d is an even positive definite sequence of l 1 (Z d ). Theorem (Dobrushin 19, Künsch 19 ) If [ π,π] d Ĵ 1 (λ)dλ <, the pure phases are Gaussian with spectral measure Ĵ 1. Moreover, there is phase transition iff Ĵ admit one root. For homogeneous systems, System in phase transition Long memory random field.
Example (The harmonic potential in dimension d 3) 1 2d if n = 1 J(n) = 1 if n = otherwise. One has Ĵ(λ) 1 ( 1 2 1 d λk) 2, when λ. k=1 Example (In dimension d = 2, particular direction of l.m.) j 1 α <j k j+α if l = θk, k > 1 J(k, l) = 1 if k = l = otherwise, where α ], 1/2[ et θ R. One can prove that «2α Ĵ(λ 1, λ 2) 1 2 sin λ1 + θλ 2. 2
3 Testing long memory
Expected test Null hypothesis : X exhibits short memory Alternative hypothesis : X exhibits long memory The test statistic is based on the empirical variance of the partial sums S j = j 1 i 1 =1... j d i d =1 ( Xi1,...,i d X n ). Why? The asymptotic behaviour of the partial sums allows to distinguish between weak dependence and strong dependence.
4 Asymptotic study of the partial sums Partial sums under weak dependence Previous results under isotropic long memory Spectral scheme Application to partial sums
Partial sums under weak dependence Theorem (Wichura (1971), Dedecker (1) ) Let X be a second order stationary random field such that j Z d r(j) <. Under moments hypothesis on X and assuming σ 2 = j Z d r(j), 1 σn d/2 [nt 1 ] k 1 =1... [nt d ] k d =1 X k1,...,k d D([,1] d ) B(t 1,..., t d ), where B is the Brownian Sheet on [, 1] d i.e. the Gaussian field with covariance function γ(s, t) = d i=1 s i t i.
Results under isotropic long memory Let (ξ k ) k Z d be a strong white noise and X k = j Z d a j ξ k j, where ( ) j a j = j β L( j )b, j d 2 < β < d. L is a slowly varying function at infinity and b a continuous function on the unit sphere of R d. The field X exhibits isotropic long memory. Theorem (Dobrushin and Major (1979), Surgailis (1982), Avram and Taqqu (1987) ) 1 n d m(β d 2 ) L(n) m [nt 1] k 1=1 [nt d ]... k d =1 P m (X k1,...,k d ) fidi n Z m(t), where P m is the Appell polynomial of degree m and where Z m is the Hermite process of order m.
Spectral scheme for linear fields Let the linear field X k1,...,k d = â j1,...,j d ξ k1 j 1,...,k d j d = e i<k,λ> a(λ)dw (λ), j [ π,π] 1,...,j d d where â is the Fourier transform of filter a and where ξ is a noise with spectral representation : ξ k = e i<k,λ> dw (λ). The partial sums of X, can be rewritten Sn X (t) = [nt 1] 1 Sn X (t 1,..., t d ) =n d/2 [ nπ,nπ] d a where W n (A) = n d/2 W (n 1 A). k 1= [nt d ] 1... k d = X k1,...,k d, ( ) d λ e iλj[ntj]/n 1 n n(e iλj/n 1) dw n(λ), j=1
Theorem (Lang and Soulier () when d = 1, Lavancier when d 1 ) Let W n (A) = n d/2 W (n 1 A) where W is the random spectral measure of an i.i.d white noise. If L Φ 2 (R d ) n Φ, then Φ n dw n L ΦdW, where W is the Gaussian white noise spectral field. Remark The result is still true when W comes from a non-i.i.d white noise provided its spectral density is bounded above and it follows a fonctional CLT.
Application to partial sums when d 2 Recall : X k = â j ξ k j = [ π,π] d e i<k,λ> a(λ)dw (λ), therefore f X (λ) a 2 (λ). Proposition (Lavancier, CLT ) If a is continuous at and a(), [nt 1] [nt 1 d ]... n d/2 k 1= k d = where B is the Brownian Sheet. Remark The result is concerned with X k1,...,k d fidi n a()b(t), Weakly dependent random fields (when a is continuous everywhere) Long memory random fields involving non-zero spectral singularities (ex : a(λ 1, λ 2 ) = λ 1 λ 2 1 α, < α < 1/2)
Application to partial sums when d 2 Proposition (Lavancier, Non-CLT) If a(λ) ã(λ) at with ã(cλ) = c α ã(λ) ( < α < 1), [nt 1] [nt 1 d ]... n d/2+α k 1= k d = X k1,...,k d fidi n ã(λ) d j=1 e itjλj 1 dw (λ). iλ j Remark This result is concerned with long memory random field : isotropic l.m., ex : ã(λ 1, λ 2 ) = (λ 2 1 + λ2 2 ) α, < α < 1. non-isotropic l.m., ex : ã(λ 1, λ 2 ) = λ 1 + θλ 2 α, < α < 1 2, θ R.
Recall : X n = e i<n,λ> a(λ)dw (λ). [ π,π] d Remark (Convergence to the Fractional Brownian Sheet) If a(λ) = d a j (λ j ), j=1 where, for all j, a j (λ j ) λ j α j, < α j < 1/2, then 1 n (d/2+p j α j) [nt 1 ] k 1 =... [nt d ] k d = X k1,...,k d fidi n R d d j=1 e it jλ j 1 iλ j λ j α j dw (λ).
5 Testing procedure and simulations Test hypothesis and test statistic Consistency Simulations in dimension d = 2
Test hypothesis Null hypothesis : weak dependence j Z d r(j) < n d/2 σ [nt 1] k 1=1 [nt d ]... k d =1 D([,1] X d ) k B(t), where σ 2 := j Z d r(j) and B is the Brownian Sheet on [, 1]d. moments hypothesis. Alternative hypothesis : long memory n γ [nt 1] L(n) k 1=1 [nt d ]... k d =1 D([,1] X d ) k Y (t), where γ > d/2, L is a slowly varying function at infinity and Y is a measurable random field.
Test statistic Let q n be a integer in [1, n], an estimator of σ 2 := P j Z r(j) is d ( X ŝ 2 ˆr : empirical covariance function q n,n = ω qn,j ˆr(j), where ω qn,j = Q d j { q n,...,q n} d i=1 (1 j i q n ) Let S j = j 1 X i 1 =1... j d X i d =1 `Xi1,...,i d X n. Definition (Extension of the V/S statistic to d > 1) Let A n = {1,..., n} d, the V/S statistic is defined by hence M n = n 2d ŝ 2 q n,n V M n = n d ar ( Sj, j A ) n, j A n S j ŝ 2 q n,n 2 n d j A n S j 2.
Consistency Proposition (Lavancier, Under the null hypothesis) Under the null hypothesis, choose q n such that lim n q n = and lim n q n /n =, then M n L Z,1] d B(t)! 2 " dy Z t i B(1)! dt B(t) i=1 [,1] d!! 2 dy t i B(1) dt#, i=1 where B is the Brownian Sheet on [, 1] d. Proposition (Lavancier, Under the alternative hypothesis) Under the alternative hypothesis, choose q n such that lim n q n = and for all δ >, lim n q n /n δ =, then M n P.
Simulation of the limit law under the null hypothesis, when d = 2 Average =, 897 V ariance =, 18 c 9% =, 1448 c 95% =, 1692.5.1.15.2.25.3.35.4.45
Simulations based on random fields with q = 8 9 f(λ 1, λ 2) λ 1 γ λ 2 γ, < γ < 1. 3 γ, 75, 5, 25 ˆp 81% 75, 4% 62, 4% 3 9
Simulations based on random fields with q = 8 9 f(λ 1, λ 2) λ 1 γ λ 2 γ, < γ < 1. 3 γ, 75, 5, 25 ˆp 81% 75, 4% 62, 4% 3 9 9 f(λ 1, λ 2) (λ 2 1 + λ 2 2) γ, < γ < 1. 3 γ, 96, 7, 4 ˆp 75% 58, 1% 32, 7% 3 9
Simulations based on random fields with q = 8 9 f(λ 1, λ 2) λ 1 γ λ 2 γ, < γ < 1. 3 γ, 75, 5, 25 ˆp 81% 75, 4% 62, 4% 3 9 9 f(λ 1, λ 2) (λ 2 1 + λ 2 2) γ, < γ < 1. 3 γ, 96, 7, 4 ˆp 75% 58, 1% 32, 7% 3 9 9 f(λ 1, λ 2) λ 1 λ 2 γ, < γ < 1. 3 γ, 75, 5, 25 ˆp 27% 27% 24, 5% 3 9