White noise processes

Transcription

1 Whie noise processes A d-dimensional ime series X = {, Z} is said o be whie noise wih mean 0 and covariance marix Σ, if WN(0, Σ), E = 0, E( ) = Σ, E( X( + u) ) = 0 for u 0. Example: iid N (0, 1) Saionary Processes, Oc 7,

2 Moving-average processes Mulivariae MA(q) process: = q Θ(j)ε( j), ε() WN(0, Σ) j=0 wih Θ(0) = 1 d. Compac form: = Θ(B)ε() wih Θ(z) = 1 + Θ(1)z Θ(q)z q. (B is he backward shif operaor defined by B j = X( j), j Z) Auocorrelaion funcion: q u Θ(j + u)σθ(j) if 0 u q Γ(u) = cov ( X(+u), ) j=0 q u = Θ(j)ΣΘ(j + u ) if q u < 0 j=0 0 oherwise Example: MA(1) = θε( 1) + ε(), ε() = WN(0, σ 2 ) (1 + θ 2 )σ 2 if u = 0 γ(u) = θσ 2 if u = ±1 0 if u > 1 Saionary Processes, Oc 7,

3 Moving-average processes Example: MA(1) = θε( 1) + ε() θ = θ = Saionary Processes, Oc 7,

4 Auoregressive processes Mulivariae AR(p) process: = p Φ(j)X( j) + ε(), ε() WN(0, Σ) Compac form: Φ(B) = ε() wih Φ(z) = 1 Φ(1)z... Φ(p)z p (B is he backward shif operaor defined by B j = X( j), j Z) Causal crierion: If de Φ(z) 0 for all z C such ha z 1 hen he equaion Φ(B) = ε() has exacly one saionary soluion, = Ψ(j)ε( j) j=0 where he marices Ψ(j) are deermined uniquely by Ψ(z) = Ψ(j)z j = Φ(z) 1. j=0 Example: AR(1) = φx( 1) + ε(), ε() = WN(0, σ 2 ) φ u γ(u) = σ 2 1 φ 2 Saionary Processes, Oc 7,

5 Auoregressive processes Example: AR(1) = φx( 1) + ε() φ = φ = Saionary Processes, Oc 7,

6 Auoregressive processes Example: AR(2) = φ 1 X( 1) + φ 2 X( 2) + ε() φ 1 = 2 cos(λ)/r, φ 2 = 1/r 2 wih r = 5 and λ = φ 1 = 2 cos(λ)/r, φ 2 = 1/r 2 wih r = 5 and λ = Saionary Processes, Oc 7,

7 Auoregressive processes Example: AR(2) = φ 1 X( 1) + φ 2 X( 2) + ε() φ 1 = 2 cos(λ)/r, φ 2 = 1/r 2 wih r = 1.5 and λ = φ 1 = 2 cos(λ)/r, φ 2 = 1/r 2 wih r = 1.5 and λ = Saionary Processes, Oc 7,

8 ARMA processes Mulivariae ARMA(p,q) process: = p Φ(j)X( j) + q Θ(j)ε( j) + ε(), ε() WN(0, Σ) Compac form: wih Φ(B) = Θ(B)ε() Φ(z) = 1 Φ(1)z... Φ(p)z p and Θ(z) = 1 + Θ(1)z Θ(p)z p (B is he backward shif operaor defined by B j = X( j), j Z) Causal crierion: If de Φ(z) 0 for all z C such ha z 1 hen he equaion Φ(B) = Θ(B)ε() has exacly one saionary soluion, = Ψ(j)ε( j) j=0 where he marices Ψ(j) are deermined uniquely by Ψ(z) = Ψ(j)z j = Φ(z) 1 Θ(z). j=0 Saionary Processes, Oc 7,

9 Harmonic processes Le X = {, Z} be he sochasic process given by = n R(λ j ) cos(λ j + φ j ), where φ j iid U[ π, π] R(λ j ), j = 1,..., n are independen wih mean 0 and variance σ j φ j and R(λ k ) independen Since cos(λ j + φ j ) = cos(λ j ) cos(φ j ) sin(λ j ) sin(φ j ) we have wih = n [ A(λj ) cos(λ j ) + B(λ j ) sin(λ j ) ] A(λ j ) = R(λ j ) cos(φ j ) B(λ j ) = R(λ j ) sin(φ j ) Saionary Processes, Oc 7,

10 Harmonic processes Since cos(λ) = 1 2 can be wrien as wih = 2n C(λ j )e iλ j ( e iλ + e iλ) and sin(λ) = 1 2i( e iλ e iλ) λ 2n+1 j = λ j for j = 1,..., 2n C(λ 2n+1 j ) = C(λ j ) for j = 1,..., 2n C(λ j ) = 1 2( A(λj ) ib(λ j ) ) = 1 2 R(λ j)e iφ j for j = 1,..., n Noe: EC(λ j ) = 0 E( C(λj )C(λ j ) ) = 1 4 σ2 j E( C(λj )C(λ j ) ) = 0 for all j = 1,..., 2n Variance decomposiion ino frequency componens: where var ( ) = 2n σ 2 j = var ( C(λ j )e iλ j ) ( E C(λj )C(λ j ) ) = 1 2 n σj 2 is he variance of he jh frequency componen. Saionary Processes, Oc 7,

11 Harmonic processes Auocorrelaion funcion: γ(u) = 2n i=1 = 2n ( E C(λj )C(λ j ) ) e iλ ju 1 i=1 4 σ2 j e iλ ju = 1 2 i=1 n σj 2 cos(λ j u) I follows ha X is (weakly) saionary. Example: = A cos(λ 0 ) + B sin(λ 0 ) wih λ o =, A, B N (0, 1): ρ(u) = cos(λ 0 ) Saionary Processes, Oc 7,

12 Harmonic processes Example: = n A j cos(λ j ) + B j sin(λ j ) wih λ j iid U[ π, π] and A j, B j iid N (0, 1) n = n = Saionary Processes, Oc 7,

13 Harmonic processes Example: = 2000 A j cos(λ j ) + B j sin(λ j ) wih λ j iid U[ π, π] and A j, B j iid N (0, σ(λ j ) 2 ) σ(λ) λ X similar o AR(2) process = φ(1)x( 1) + φ(2)x( 2) + ε() wih φ 1 = 2 cos(λ)/r, φ 2 = 1/r 2, r = 5 and λ =. Saionary Processes, Oc 7,