Durbin-Levinson recursive method

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1 Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again from scratch; the procedure is a method for computing important theoretical quantities. 9 ottobre / 19

2 Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because Idea it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again from scratch; the procedure is a method for computing important theoretical quantities. ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( X 1 P L(X2,...,X n)x 1 ) Note ( X 1 P L(X2,...,X n)x 1 ) is orthogonal to the previous. 9 ottobre / 19

3 Durbin-Levinson, 2 ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 Check orthogonality condition to find a: i > 1 : ˆX n+1 X n+1, X i = P L(X2,...,X n)x n+1 X n+1, X i + a X 1 P L(X2,...,X n)x 1, X i = last step coming from the definitions of projections (i = 2... n). 9 ottobre / 19

4 Durbin-Levinson, 3 ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( X 1 P L(X2,...,X n)x 1 ) Check orthogonality condition with i = 1: 9 ottobre / 19

5 Durbin-Levinson, 3 ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( X 1 P L(X2,...,X n)x 1 ) Check orthogonality condition with i = 1: 0 = ˆX n+1 X n+1, X 1 P L(X2,...,X n)x 1 9 ottobre / 19

6 Durbin-Levinson, 3 ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 Check orthogonality condition with i = 1: 0 = ˆX n+1 X n+1, X 1 P L(X2,...,X n)x 1 = P L(X2,...,X n)x n+1 X n+1, X 1 P L(X2,...,X n)x 1 +a X 1 P L(X2,...,X n)x ottobre / 19

7 Durbin-Levinson, 3 ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 Check orthogonality condition with i = 1: 0 = ˆX n+1 X n+1, X 1 P L(X2,...,X n)x 1 = P L(X2,...,X n)x n+1 X n+1, X 1 P L(X2,...,X n)x 1 +a X 1 P L(X2,...,X n)x 1 2 = X n+1, X 1 P L(X2,...,X n)x 1 + a X 1 P L(X2,...,X n)x ottobre / 19

8 Durbin-Levinson, 3 ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( X 1 P L(X2,...,X n)x 1 ) Check orthogonality condition with i = 1: 0 = ˆX n+1 X n+1, X 1 P L(X2,...,X n)x 1 = P L(X2,...,X n)x n+1 X n+1, X 1 P L(X2,...,X n)x 1 +a X 1 P L(X2,...,X n)x 1 2 = X n+1, X 1 P L(X2,...,X n)x 1 + a X 1 P L(X2,...,X n)x 1 2 = a = X n+1, X 1 P L(X2,...,X n)x 1 X 1 P L(X2,...,X n)x ottobre / 19

9 Durbin-Levinson. 4 We tried ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 and found a = X n+1, X 1 P L(X2,...,X n)x 1 X 1 P L(X2,...,X n)x 1 2 = X n+1, X 1 P L(X2,...,X n)x 1 v 1 n 1 with v n 1 = E( ˆX n X n 2 ) = X n P L(X1,...,X n 1 )X n 2 = X 1 P L(X2,...,X n)x ottobre / 19

10 Durbin-Levinson. 4 We tried ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 and found a = X n+1, X 1 P L(X2,...,X n)x 1 X 1 P L(X2,...,X n)x 1 2 = X n+1, X 1 P L(X2,...,X n)x 1 v 1 n 1 with v n 1 = E( ˆX n X n 2 ) = X n P L(X1,...,X n 1 )X n 2 = X 1 P L(X2,...,X n)x 1 2. We write ˆX n+1 = ϕ n,1 X n + + ϕ n,n X 1 = n ϕ n,j X n+1 j 9 ottobre / 19

11 Durbin-Levinson. 4 We tried ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 and found a = X n+1, X 1 P L(X2,...,X n)x 1 X 1 P L(X2,...,X n)x 1 2 = X n+1, X 1 P L(X2,...,X n)x 1 v 1 n 1 with v n 1 = E( ˆX n X n 2 ) = X n P L(X1,...,X n 1 )X n 2 = X 1 P L(X2,...,X n)x 1 2. We write ˆX n+1 = ϕ n,1 X n + + ϕ n,n X 1 = n ϕ n,j X n+1 j so that P L(X2,...,X n)x n+1 = n 1 ϕ n 1,j X n+1 j 9 ottobre / 19

12 Durbin-Levinson. 4 We tried ˆX n+1 = P L(X1,...,X n)x n+1 = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 and found a = X n+1, X 1 P L(X2,...,X n)x 1 X 1 P L(X2,...,X n)x 1 2 = X n+1, X 1 P L(X2,...,X n)x 1 v 1 n 1 with v n 1 = E( ˆX n X n 2 ) = X n P L(X1,...,X n 1 )X n 2 = X 1 P L(X2,...,X n)x 1 2. We write ˆX n+1 = ϕ n,1 X n + + ϕ n,n X 1 = n ϕ n,j X n+1 j so that P L(X2,...,X n)x n+1 = n 1 and substituting we get a recursion. ϕ n 1,j X n+1 j 9 ottobre / 19

13 Durbin-Levinson algorithm. 5 ˆX n+1 = n ϕ n,j X n+1 j = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 Hence ϕ n,n = a = X n+1, X 1 P L(X2,...,X n)x 1 vn 1 1 n 1 = γ(n) ϕ n 1,j γ(n j) v 1 n 1. 9 ottobre / 19

14 Durbin-Levinson algorithm. 6 Then from n n 1 n 1 ϕ n,j X n+1 j = ϕ n 1,j X n+1 j + a(x 1 ϕ n 1,j X j+1 ) n 1 n 1 = ϕ n 1,j X n+1 j + a(x 1 ϕ n 1,n k X n+1 k ) k=1 one sees ϕ n,j = ϕ n 1,j aϕ n 1,n j = ϕ n 1,j ϕ n,n ϕ n 1,n j j = 1... n 1 9 ottobre / 19

15 Durbin-Levinson algorithm. 6 Then from n n 1 n 1 ϕ n,j X n+1 j = ϕ n 1,j X n+1 j + a(x 1 ϕ n 1,j X j+1 ) n 1 n 1 = ϕ n 1,j X n+1 j + a(x 1 ϕ n 1,n k X n+1 k ) k=1 one sees ϕ n,j = ϕ n 1,j aϕ n 1,n j = ϕ n 1,j ϕ n,n ϕ n 1,n j j = 1... n 1 We need also a recursive procedure for v n. 9 ottobre / 19

16 Durbin-Levinson algorithm. 7 n v n = E( ˆX n+1 X n+1 2 ) = γ 0 ϕ n,j γ(j) n 1 = γ 0 ϕ n,n γ(n) (ϕ n 1,j ϕ n,n ϕ n 1,n j )γ(j) n 1 n 1 = γ 0 ϕ n 1,j γ(j) ϕ n,n γ(n) ϕ n 1,n j γ(j) ( ) = v n 1 ϕ n,n ϕ n,n v n 1 = v n 1 1 ϕ 2 n,n. The terms in red are equal because of the definition ϕ n,n. 9 ottobre / 19

17 Durbin-Levinson algorithm. 7 v n = E( ˆX n+1 X n+1 2 ) = γ 0 n ϕ n,j γ(j) n 1 = γ 0 ϕ n,n γ(n) (ϕ n 1,j ϕ n,n ϕ n 1,n j )γ(j) n 1 n 1 = γ 0 ϕ n 1,j γ(j) ϕ n,n γ(n) ϕ n 1,n j γ(j) = v n 1 ϕ n,n ϕ n,n v n 1 = v n 1 ( 1 ϕ 2 n,n ). The terms in red are equal because of the definition ϕ n,n. The final formula v n = ( 1 ϕ 2 n,n) vn 1 shows that ϕ n,n determines the decrease of predictive error with increasing n. 9 ottobre / 19

18 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) 9 ottobre / 19

19 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) ϕ 1,1 = γ(1) v 0 = ρ(1) 9 ottobre / 19

20 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) ϕ 1,1 = γ(1) = ρ(1) v 0 v 1 = ( 1 ϕ 2 ) 1,1 v0 = γ(0) ( 1 ρ(1) 2) 9 ottobre / 19

21 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) ϕ 1,1 = γ(1) = ρ(1) v 0 v 1 = ( 1 ϕ 2 ) 1,1 v0 = γ(0) ( 1 ρ(1) 2). n 1 ϕ n,n = γ(n) ϕ n 1,j γ(n j) v 1 n 1 9 ottobre / 19

22 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) ϕ 1,1 = γ(1) = ρ(1) v 0 v 1 = ( 1 ϕ 2 ) 1,1 v0 = γ(0) ( 1 ρ(1) 2). n 1 ϕ n,n = γ(n) ϕ n 1,j γ(n j) v 1 n 1 ϕ n,j = ϕ n 1,j ϕ n,n ϕ n 1,n j j = 1... n 1 9 ottobre / 19

23 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) ϕ 1,1 = γ(1) = ρ(1) v 0 v 1 = ( 1 ϕ 2 ) 1,1 v0 = γ(0) ( 1 ρ(1) 2). n 1 ϕ n,n = γ(n) ϕ n 1,j γ(n j) v 1 n 1 ϕ n,j = ϕ n 1,j ϕ n,n ϕ n 1,n j j = 1... n 1 v n = ( 1 ϕ 2 n,n) vn 1. 9 ottobre / 19

24 Durbin-Levinson algorithm. Summary v 0 = E( X 1 ˆX 1 2 ) = E( X 1 2 ) = γ(0) ϕ 1,1 = γ(1) = ρ(1) v 0 v 1 = ( 1 ϕ 2 ) 1,1 v0 = γ(0) ( 1 ρ(1) 2). n 1 ϕ n,n = γ(n) ϕ n 1,j γ(n j) v 1 n 1 ϕ n,j = ϕ n 1,j ϕ n,n ϕ n 1,n j j = 1... n 1 v n = ( 1 ϕ 2 n,n) vn 1. One could divide everything by γ(0) and work with ACF instead of ACVF 9 ottobre / 19

25 Durbin-Levinson algorithm for AR(1) X t stationary with X t = φx t 1 + Z t, Z t WN(0, σ 2 ) and E(X s Z t ) = 0 if s < t 9 ottobre / 19

26 Durbin-Levinson algorithm for AR(1) X t stationary with X t = φx t 1 + Z t, Z t WN(0, σ 2 ) and E(X s Z t ) = 0 if s < t = γ(h) = σ2 φ h 1 φ 2. 9 ottobre / 19

27 Durbin-Levinson algorithm for AR(1) X t stationary with X t = φx t 1 + Z t, Z t WN(0, σ 2 ) and E(X s Z t ) = 0 if s < t = γ(h) = σ2 φ h 1 φ 2. v 0 = σ2 1 φ 2, ϕ 1,1 = φ, v 1 = σ 2, 9 ottobre / 19

28 Durbin-Levinson algorithm for AR(1) X t stationary with X t = φx t 1 + Z t, Z t WN(0, σ 2 ) and E(X s Z t ) = 0 if s < t = γ(h) = σ2 φ h 1 φ 2. ϕ 2,2 = σ2 v 0 = 1 φ 2, ϕ 1,1 = φ, v 1 = σ 2, [ σ 2 φ 2 1 φ 2 ϕ σ2 φ 1 φ 2 ] v 1 1 = 0. ϕ 2,1 = ϕ 1,1, v 2 = v 1, ϕ n,1 = φ, ϕ n,j = 0 j > 1, v n = v 1 = σ 2. 9 ottobre / 19

29 Durbin-Levinson algorithm for MA(1) X t = Z t ϑz t 1, Z t WN(0, σ 2 ), γ(0) = σ 2 (1 + ϑ 2 ), γ(1) = σ 2 ϑ. 9 ottobre / 19

30 Durbin-Levinson algorithm for MA(1) X t = Z t ϑz t 1, Z t WN(0, σ 2 ), γ(0) = σ 2 (1 + ϑ 2 ), γ(1) = σ 2 ϑ. v 0 = σ 2 (1 + ϑ 2 ) ϕ 1,1 = ϑ 1 + ϑ 2 9 ottobre / 19

31 Durbin-Levinson algorithm for MA(1) X t = Z t ϑz t 1, Z t WN(0, σ 2 ), γ(0) = σ 2 (1 + ϑ 2 ), γ(1) = σ 2 ϑ. v 0 = σ 2 (1 + ϑ 2 ) ϕ 1,1 = ϑ 1 + ϑ 2 v 1 = σ2 (1 + ϑ 2 + ϑ 4 ) 1 + ϑ 2 ϕ 2,2 = 1 + ϑ 2 + ϑ 4... v 2 = σ2 (1 + ϑ 2 + ϑ 4 + ϑ 6 ) 1 + ϑ 2 + ϑ 4... ϑ 2 9 ottobre / 19

32 Durbin-Levinson algorithm for MA(1) X t = Z t ϑz t 1, Z t WN(0, σ 2 ), γ(0) = σ 2 (1 + ϑ 2 ), γ(1) = σ 2 ϑ. v 0 = σ 2 (1 + ϑ 2 ) ϕ 1,1 = ϑ 1 + ϑ 2 v 1 = σ2 (1 + ϑ 2 + ϑ 4 ) 1 + ϑ 2 ϕ 2,2 = 1 + ϑ 2 + ϑ 4... v 2 = σ2 (1 + ϑ 2 + ϑ 4 + ϑ 6 ) 1 + ϑ 2 + ϑ 4... Remarks: Computations are long and tedious. v n converges (slowly) towards σ 2 (the white-noise variance) if ϑ < 1. ϑ 2 9 ottobre / 19

33 Durbin-Levinson for sinusoidal wave X t = B cos(ωt) + C sin(ωt), with ω R, E(B) = E(C) = E(BC) = 0, V(B) = V(C) = σ 2. 9 ottobre / 19

34 Durbin-Levinson for sinusoidal wave X t = B cos(ωt) + C sin(ωt), with ω R, E(B) = E(C) = E(BC) = 0, V(B) = V(C) = σ 2. Then γ(h) = σ 2 cos(ωh). 9 ottobre / 19

35 Durbin-Levinson for sinusoidal wave X t = B cos(ωt) + C sin(ωt), with ω R, E(B) = E(C) = E(BC) = 0, V(B) = V(C) = σ 2. Then γ(h) = σ 2 cos(ωh). v 0 = σ 2 ϕ 1,1 = cos(ω) 9 ottobre / 19

36 Durbin-Levinson for sinusoidal wave X t = B cos(ωt) + C sin(ωt), with ω R, E(B) = E(C) = E(BC) = 0, V(B) = V(C) = σ 2. Then γ(h) = σ 2 cos(ωh). v 0 = σ 2 v 1 = σ 2 (1 cos 2 (ω)) = σ 2 sin 2 (ω) ϕ 1,1 = cos(ω) ϕ 2,2 = cos(2ω) cos2 (ω) sin 2 (ω) = 1 9 ottobre / 19

37 Durbin-Levinson for sinusoidal wave X t = B cos(ωt) + C sin(ωt), with ω R, E(B) = E(C) = E(BC) = 0, V(B) = V(C) = σ 2. Then γ(h) = σ 2 cos(ωh). v 0 = σ 2 v 1 = σ 2 (1 cos 2 (ω)) = σ 2 sin 2 (ω) v 2 = 0 = X n+1 = P L(Xn,Xn 1 )X n+1. ϕ 1,1 = cos(ω) ϕ 2,2 = cos(2ω) cos2 (ω) sin 2 (ω) = 1 9 ottobre / 19

38 Partial auto-correlation For a stationary process {X t } α(h) the partial auto-correlation represents the correlation between X t and X t+h, after removing the effect of intermediate values. 9 ottobre / 19

39 Partial auto-correlation For a stationary process {X t } α(h) the partial auto-correlation represents the correlation between X t and X t+h, after removing the effect of intermediate values. Definition: α(1) = ρ(x t, X t+1 ) = ρ(1). α(h) = ρ(x t P L(Xt+1,...,X t+h 1 )X t, X t+h P L(Xt+1,...,X t+h 1 )X t+h ) h > 1. 9 ottobre / 19

40 Partial auto-correlation For a stationary process {X t } α(h) the partial auto-correlation represents the correlation between X t and X t+h, after removing the effect of intermediate values. Definition: α(1) = ρ(x t, X t+1 ) = ρ(1). α(h) = ρ(x t P L(Xt+1,...,X t+h 1 )X t, X t+h P L(Xt+1,...,X t+h 1 )X t+h ) h > 1. α(h) = E((X t P L(Xt+1,...,X t+h 1 )X t )(X t+h P L(Xt+1,...,X t+h 1 )X t+h )) V(X t P L(Xt+1,...,X t+h 1 )X t ) = X 1 P L(X2,...,X h )X 1, X h+1 P L(X2,...,X h )X h+1 X 1 P L(X2,...,X h )X 1 2 = X 1, X h+1 P L(X2,...,X h )X h+1 X 1 P L(X2,...,X h )X 1 2 = ϕ h,h. 9 ottobre / 19

41 Partial auto-correlation For a stationary process {X t } α(h) the partial auto-correlation represents the correlation between X t and X t+h, after removing the effect of intermediate values. Definition: α(1) = ρ(x t, X t+1 ) = ρ(1). α(h) = ρ(x t P L(Xt+1,...,X t+h 1 )X t, X t+h P L(Xt+1,...,X t+h 1 )X t+h ) h > 1. α(h) = E((X t P L(Xt+1,...,X t+h 1 )X t )(X t+h P L(Xt+1,...,X t+h 1 )X t+h )) V(X t P L(Xt+1,...,X t+h 1 )X t ) = X 1 P L(X2,...,X h )X 1, X h+1 P L(X2,...,X h )X h+1 X 1 P L(X2,...,X h )X 1 2 = X 1, X h+1 P L(X2,...,X h )X h+1 X 1 P L(X2,...,X h )X 1 2 = ϕ h,h. Durbin-Levinson s algorithm is a method to compute α( ). 9 ottobre / 19

42 Remember in fact Durbin-Levinson algorithm. 5 ˆX n+1 = n ϕ n,j X n+1 j = P L(X2,...,X n)x n+1 + a ( ) X 1 P L(X2,...,X n)x 1 Hence ϕ n,n = a = X n+1, X 1 P L(X2,...,X n)x 1 vn 1 1 n 1 = γ(n) ϕ n 1,j γ(n j) v 1 n 1. 9 ottobre / 19

43 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). 9 ottobre / 19

44 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). {X t } AR(p), i.e. stationary proces s.t. p X t = φ k X t k + Z t, {Z t } WN(0, σ 2 ). k=1 9 ottobre / 19

45 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). {X t } AR(p), i.e. stationary proces s.t. p X t = φ k X t k + Z t, {Z t } WN(0, σ 2 ). If t p, k=1 P L(X1,...,X t)x t+1 = p k=1 φ kx t+1 k (check). 9 ottobre / 19

46 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). {X t } AR(p), i.e. stationary proces s.t. p X t = φ k X t k + Z t, {Z t } WN(0, σ 2 ). If t p, k=1 P L(X1,...,X t)x t+1 = p k=1 φ kx t+1 k (check). Then ϕ p,p = α(p) = φ p, ϕ h,h = 0 if h > p, i.e. α(h) = 0 for h > p. 9 ottobre / 19

47 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). {X t } AR(p), i.e. stationary proces s.t. p X t = φ k X t k + Z t, {Z t } WN(0, σ 2 ). If t p, k=1 P L(X1,...,X t)x t+1 = p k=1 φ kx t+1 k (check). Then ϕ p,p = α(p) = φ p, ϕ h,h = 0 if h > p, i.e. α(h) = 0 for h > p. {X t } MA(1) = α(h) = ϑ h /(1 + ϑ ϑ 2h ) (long computation) 9 ottobre / 19

48 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). {X t } AR(p), i.e. stationary proces s.t. p X t = φ k X t k + Z t, {Z t } WN(0, σ 2 ). If t p, k=1 P L(X1,...,X t)x t+1 = p k=1 φ kx t+1 k (check). Then ϕ p,p = α(p) = φ p, ϕ h,h = 0 if h > p, i.e. α(h) = 0 for h > p. {X t } MA(1) = α(h) = ϑ h /(1 + ϑ ϑ 2h ) (long computation) PACF of AR processes has finite support, while PACF of MA is always non-zero. This is the opposite as for ACF. 9 ottobre / 19

49 Examples of PACF {X t } AR(1), = α(1) = φ, α(h) = 0 for h > 1 (seen before). {X t } AR(p), i.e. stationary proces s.t. p X t = φ k X t k + Z t, {Z t } WN(0, σ 2 ). If t p, k=1 P L(X1,...,X t)x t+1 = p k=1 φ kx t+1 k (check). Then ϕ p,p = α(p) = φ p, ϕ h,h = 0 if h > p, i.e. α(h) = 0 for h > p. {X t } MA(1) = α(h) = ϑ h /(1 + ϑ ϑ 2h ) (long computation) PACF of AR processes has finite support, while PACF of MA is always non-zero. This is the opposite as for ACF. Sample PACF. Apply Durbin-Levinson algorithm to ˆγ( ). 9 ottobre / 19

50 Sample ACF and PACF Oveshort data ACF Lag Partial ACF Lag 9 ottobre / 19

51 Sample ACF of Huron: AR(1) fit ACF of detrended Huron data ACF Lag 9 ottobre / 19

52 Sample ACF of Huron: AR(1) fit ACF of detrended Huron data ACF Add theoretical ACF of AR(1) with φ = Lag 9 ottobre / 19

53 Sample ACF of Huron: AR(1) fit ACF of detrended Huron data ACF Lag Add confidence intervals, assuming φ = 0.79 (different from book). 9 ottobre / 19

54 Sample ACF and PACF of Huron data Huron data ACF Lag Partial ACF PACF suggests use of an AR(2) model. Lag 9 ottobre / 19

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