Akaike criterion: Kullback-Leibler discrepancy

Transcription

1 Model choice. Akaike s criterion Akaike criterion: Kullback-Leibler discrepancy Given a family of probability densities {f ( ; ), 2 }, Kullback-Leibler s index of f ( ; ) relativetof ( ; ) is Z ( ) =E ( 2 log(f (X ; ))) = 2 log(f (x; ))f (x; ) dx. Kullback-Leibler s discrepancy between f ( ; ) andf ( ; ) is Z f (x; ) d( ) = ( ) ( ) = 2 log f (x; ) dx. f (x; ) R n R n 24 novembre / 29

2 Model choice. Akaike s criterion Akaike criterion: Kullback-Leibler discrepancy Given a family of probability densities {f ( ; ), 2 }, Kullback-Leibler s index of f ( ; ) relativetof ( ; ) is Z ( ) =E ( 2 log(f (X ; ))) = 2 log(f (x; ))f (x; ) dx. Kullback-Leibler s discrepancy between f ( ; ) andf ( ; ) is Z f (x; ) d( ) = ( ) ( ) = 2 log f (x; ) dx. f (x; ) Jensen s inequality implies E(log(Y )) apple log(e(y )) for any random variable. Hence Z f (x; ) d( ) 2 log f (x; ) dx =0 R n f (x; ) with equality only if f (x; ) =f (x; ) a.e.[f ( ; )]. R n R n 24 novembre / 29

3 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. 24 novembre / 29

4 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). 24 novembre / 29

5 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). Let =(, #, 2 ) the parameters of an ARMA(p,q) model and ˆ the MLE based on X 1,...,X n.lety an independent realization of the same process. Then 2 log L Y ( ˆ, ˆ#, ˆ2 )=n log(2 )+n log( ˆ2 ) + log(r 0...r n 1 )+ S Y ( ˆ, ˆ#) ˆ2 24 novembre / 29

6 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). Let =(, #, 2 ) the parameters of an ARMA(p,q) model and ˆ the MLE based on X 1,...,X n.lety an independent realization of the same process. Then 2 log L Y ( ˆ, ˆ#, ˆ2 )=n log(2 )+n log( ˆ2 ) + log(r 0...r n 1 )+ S Y ( ˆ, ˆ#) Indeed remember that for an ARMA(p,q) process L(, #, 1 2 )=(2 2 ) n/2 (r 0...r n 1 ) 1/2 exp S(, #) 2 2 P with S(, #) = n (x j ˆx j ) 2. j=1 r j 1 r 0,...,r n 1 depend only on parameters (, #) and not on observed data. Data enter likelihood only through the terms (x j ˆx j ) 2 in S(, #). ˆ2 24 novembre / 29

7 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). Let =(, #, 2 )the parameters of an ARMA(p,q) model and ˆ the MLE based on X 1,...,X n. Let Y an independent realization of the same process. Then 2 log L Y ( ˆ, ˆ#, ˆ2 )=nlog(2 )+nlog( ˆ2 ) + log(r 0...r n 1 )+ S Y ( ˆ, ˆ#) ˆ2 24 novembre / 29

8 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). Let =(, #, 2 )the parameters of an ARMA(p,q) model and ˆ the MLE based on X 1,...,X n. Let Y an independent realization of the same process. Then 2 log L Y ( ˆ, ˆ#, ˆ2 )=nlog(2 )+nlog( ˆ2 ) + log(r 0...r n 1 )+ S Y ( ˆ, ˆ#) = 2 log L X ( ˆ, ˆ#, ˆ2 )+ S Y ( ˆ, ˆ#) ˆ2 S X ( ˆ, ˆ#) ˆ2 ˆ2 24 novembre / 29

9 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). Let =(, #, 2 )the parameters of an ARMA(p,q) model and ˆ the MLE based on X 1,...,X n. Let Y an independent realization of the same process. Then 2 log L Y ( ˆ, ˆ#, ˆ2 )=nlog(2 )+nlog( ˆ2 ) + log(r 0...r n 1 )+ S Y ( ˆ, ˆ#) = 2 log L X ( ˆ, ˆ#, ˆ2 )+ S Y ( ˆ, ˆ#) ˆ2 = 2 log L X ( ˆ, ˆ#, ˆ2 )+ S Y ( ˆ, ˆ#) ˆ2 S X ( ˆ, ˆ#) ˆ2 n =) ˆ2 24 novembre / 29

10 Model choice. Akaike s criterion Approximating Kullback-Leibler discrepancy Given observations X 1,...,X n, we would like to minimize d( ) among all candidate models, given the true model. As the true model is unknown, we estimate d( ). Let =(, #, 2 )the parameters of an ARMA(p,q) model and ˆ the MLE based on X 1,...,X n. Let Y an independent realization of the same process. Then 2 log L Y ( ˆ, ˆ#, ˆ2 )=nlog(2 )+nlog( ˆ2 ) + log(r 0...r n 1 )+ S Y ( ˆ, ˆ#) = 2 log L X ( ˆ, ˆ#, ˆ2 )+ S Y ( ˆ, ˆ#) ˆ2 S X ( ˆ, ˆ#) = 2 log L X ( ˆ, ˆ#, ˆ2 )+ S Y ( ˆ, ˆ#) n =) ˆ2! E ( ( ˆ )) = E (,#, 2 )( 2 log L X ( ˆ, ˆ#, ˆ2 S Y ( ˆ, ˆ#) )) + E (,#, 2 ) ˆ2 ˆ2 ˆ2 n. 24 novembre / 29

11 Model choice. Akaike s criterion Kullback-Leibler discrepancy and AICC Using linear approximations, and asymptotic distributions of estimators, one arrives at S Y ( ˆ, ˆ#) 2 (n + p + q). E (,#, 2 ) Similarly n ˆ2 = S X ( ˆ, ˆ#) for large n is distributed as 2 2 (n p q 2) and is asymptotically independent of ( ˆ, ˆ#). 24 novembre / 29

12 Model choice. Akaike s criterion Kullback-Leibler discrepancy and AICC Using linear approximations, and asymptotic distributions of estimators, one arrives at S Y ( ˆ, ˆ#) 2 (n + p + q). E (,#, 2 ) Similarly n ˆ2 = S X ( ˆ, ˆ#) for large n is distributed as 2 2 (n p q 2) and is asymptotically independent of ( ˆ, ˆ#). Hence! S Y ( ˆ, ˆ#) 2 (n + p + q) E (,#, 2 ) ˆ2 2 (n p q 2)/n 24 novembre / 29

13 Model choice. Akaike s criterion Kullback-Leibler discrepancy and AICC Using linear approximations, and asymptotic distributions of estimators, one arrives at S Y ( ˆ, ˆ#) 2 (n + p + q). E (,#, 2 ) Similarly n ˆ2 = S X ( ˆ, ˆ#) for large n is distributed as 2 2 (n p q 2) and is asymptotically independent of ( ˆ, ˆ#). Hence! S Y ( ˆ, ˆ#) 2 (n + p + q) E (,#, 2 ) ˆ2 2 (n p q 2)/n From E ( ( ˆ )) = E (,#, 2 )( 2 log L X ( ˆ, ˆ#, ˆ2 )) + E (,#, 2 ) SY ( ˆ, ˆ#) ˆ2 n AICC = 2 log L X ( ˆ, ˆ#, ˆ2 2(p + q + 1)n )+ n p q 2 is an approximate unbiased estimate of (ˆ ). 24 novembre / 29

14 Model choice. Akaike s criterion Criteria for model choice The order is chosen by minimizing the value of AICC (Corrected Akaike s Information Criterion): 2 log L X ( ˆ, ˆ#, ˆ2 )+ 2(p+q+1)n n p q 2. The second term can be considered a penalty for models with a large number of parameters. 24 novembre / 29

15 Model choice. Akaike s criterion Criteria for model choice The order is chosen by minimizing the value of AICC (Corrected Akaike s Information Criterion): 2 log L X ( ˆ, ˆ#, ˆ2 )+ 2(p+q+1)n n p q 2. The second term can be considered a penalty for models with a large number of parameters. For n large it is approximately the same as Akaike s information Criterion (AIC): 2 log L X ( ˆ, ˆ#, ˆ2 ) + 2(p + q + 1), but carries a higher penalty for finite n, and thus is somewhat less likely to overfit. In R: AICC <- AIC(myfit,k=2*n/(n-p-q-2)) 24 novembre / 29

16 Model choice. Akaike s criterion Criteria for model choice The order is chosen by minimizing the value of AICC (Corrected Akaike s Information Criterion): 2 log L X ( ˆ, ˆ#, ˆ2 )+ 2(p+q+1)n n p q 2. The second term can be considered a penalty for models with a large number of parameters. For n large it is approximately the same as Akaike s information Criterion (AIC): 2 log L X ( ˆ, ˆ#, ˆ2 ) + 2(p + q + 1), but carries a higher penalty for finite n, and thus is somewhat less likely to overfit. In R: AICC <- AIC(myfit,k=2*n/(n-p-q-2)) A rule of thumb is the fits of model 1 and model 2 are not significantly di erent if AICC 1 AICC 2 < 2(only the di erence matters, not the absolute value of AICC). Hence, we may decide to choose model 1 if it simpler than 2 (or its residuals are closer to white-noise) even if AICC 1 > AICC 2 as long as AICC 1 < AICC novembre / 29

17 Model choice. Akaike s criterion Tests on residuals ˆX t (ˆ', ˆ#) predicted value of X t given the estimates ( ˆ', ˆ#). Ŵ t = X t ˆX t (ˆ', ˆ#) 1/2 r t 1 (ˆ', ˆ#) standardized residuals. Portmanteau tests on ACF of Ŵ t : Box-Pierce; Ljung-Box; Test on turning points Rank tests novembre / 29

18 Autocovariance Amutivariatestochasticprocess{X t 2 R m }, t 2 Z is weakly stationary if E(X 2 t,i) < 18t, i E(X t ) µ, Cov(X t+h, X t ) (h). In particular ij(h) =Cov(X t+h,i, X t,j ) = E((X t+h,i µ i )(X t,j µ j )). 24 novembre / 29

19 Autocovariance Amutivariatestochasticprocess{X t 2 R m }, t 2 Z is weakly stationary if E(X 2 t,i) < 18t, i E(X t ) µ, Cov(X t+h, X t ) (h). In particular ij(h) =Cov(X t+h,i, X t,j ) = E((X t+h,i µ i )(X t,j µ j )). Note that in general ij(h) 6= ji (h), while ij(h) =Cov(X t+h,i, X t,j ) = (stationarity) = Cov(X t,i, X t h,j ) =(symmetry)=cov(x t h,j, X t,i )= ji ( h). 24 novembre / 29

20 Autocovariance Amutivariatestochasticprocess{X t 2 R m }, t 2 Z is weakly stationary if E(X 2 t,i) < 18t, i E(X t ) µ, Cov(X t+h, X t ) (h). In particular ij(h) =Cov(X t+h,i, X t,j ) = E((X t+h,i µ i )(X t,j µ j )). Note that in general ij(h) 6= ji (h), while ij(h) =Cov(X t+h,i, X t,j ) = (stationarity) = Cov(X t,i, X t h,j ) =(symmetry)=cov(x t h,j, X t,i )= ji ( h). Another simple property is i,j (h) apple( ii (0) jj (0)) 1/2. The ACF ij (h) = ij(h) ( ii (0) jj (0)) 1/2. 24 novembre / 29

21 Multivariate White-noise and MA Amutivariatestochasticprocess{Z t 2 R m } is a white-noise with covariance S, {Z t } WN(0, S), if ( S h =0 {Z t } is stationary with mean 0 and ACVF (h) = 0 h 6= novembre / 29

22 Multivariate White-noise and MA Amutivariatestochasticprocess{Z t 2 R m } is a white-noise with covariance S, {Z t } WN(0, S), if ( S h =0 {Z t } is stationary with mean 0 and ACVF (h) = 0 h 6= 0. {X t 2 R m } is a linear process if X t = and C k are matrices s.t. +1X k= 1 +1P k= 1 C k Z t k {Z t } WN(0, S) (C k ) ij < +1 for all i, j =1...m. 24 novembre / 29

23 Multivariate White-noise and MA Amutivariatestochasticprocess{Z t 2 R m } is a white-noise with covariance S, {Z t } WN(0, S), if ( S h =0 {Z t } is stationary with mean 0 and ACVF (h) = 0 h 6= 0. {X t 2 R m } is a linear process if X t = and C k are matrices s.t. +1X k= 1 +1P k= 1 {X t } is stationary and X (h) = 1 P C k Z t k {Z t } WN(0, S) (C k ) ij < +1 for all i, j =1...m. k= 1 C k+h SC t k. 24 novembre / 29

24 Estimation of mean The mean µ can be estimated through X n. From the univariate theory, we know E( X n )=µ, V(( X n ) i )! 0(asn!1), if ii(h) h!1! 0 nv(( X n ) i )! +1X h= 1 ii(h) if +1X h= 1 ii (h) < +1. Moreover ( X n ) i is asymptotically normal. Stronger assumptions are required for the vector X n to be asymptotically normal Theorem then n 1/2 ( X n If X t = µ + µ)=) N(0, +1X k= 1 C k Z t k {Z t } WN(0, S) 1P k= 1 C k+h SC t k ). 24 novembre / 29

25 Confidence intervals for the mean In principle, from X n N(µ, 1 1P n m-dimensional confidence ellipsoid. But... k= 1 C k+h SCk t ) one could build an 24 novembre / 29

26 Confidence intervals for the mean In principle, from X n N(µ, 1 1P C k+h SCk t ) one could build an n k= 1 m-dimensional confidence ellipsoid. But... not intuitive, C k and S not known and have to be estimated... Instead, build confidence intervals from ( X n ) i N(µ i, 1 n +1P h= 1 ii(h)). 24 novembre / 29

27 Confidence intervals for the mean In principle, from X n N(µ, 1 1P C k+h SCk t ) one could build an n k= 1 m-dimensional confidence ellipsoid. But... not intuitive, C k and S not known and have to be estimated... Instead, build confidence intervals from ( X n ) i N(µ i, 1 n +1P h= 1 h= r +1P h= 1 ii(h)). ii(h) =2 f i (0) can be consistently estimated from rx h 2 ˆf i (0) = 1 ˆii (h) where r n!1and r n r n! novembre / 29

28 Confidence intervals for the mean In principle, from X n N(µ, 1 1P C k+h SCk t ) one could build an n k= 1 m-dimensional confidence ellipsoid. But... not intuitive, C k and S not known and have to be estimated... Instead, build confidence intervals from ( X n ) i N(µ i, 1 n +1P h= 1 h= r +1P h= 1 ii(h)). ii(h) =2 f i (0) can be consistently estimated from rx h 2 ˆf i (0) = 1 ˆii (h) where r n!1and r n r n! 0. Componentwise confidence intervals can be combined. If we found u i ( ) s.t. P( µ i ( X n ) i < u i (a)) 1, then mx P( µ i ( X n ) i <u i (a), i =1, m) 1 P µ i ( X n ) i u i (a) 1 m. i=1 24 novembre / 29

29 Confidence intervals for the mean In principle, from X n N(µ, 1 1P C k+h SCk t ) one could build an n k= 1 m-dimensional confidence ellipsoid. But... not intuitive, C k and S not known and have to be estimated... Instead, build confidence intervals from ( X n ) i N(µ i, 1 n +1P h= 1 h= r +1P h= 1 ii(h)). ii(h) =2 f i (0) can be consistently estimated from rx h 2 ˆf i (0) = 1 ˆii (h) where r n!1and r n r n! 0. Componentwise confidence intervals can be combined. If we found u i ( ) s.t. P( µ i ( X n ) i < u i (a)) 1, then mx P( µ i ( X n ) i <u i (a), i =1, m) 1 P µ i ( X n ) i u i (a) 1 m. Choosing = 0.05 m i=1, one has a 95%-confidence m-rectangle. 24 novembre / 29

30 Estimation of ACVF (bivariate case, m = 2) 8 1 np h >< (X t+h X n )(X t X n ) t 0 apple h < n n t=1 ˆ(h) = 1 np >: (X t+h Xn )(X t Xn ) t n < h < 0. n t= h+1 ˆ ij (h) =ˆij (h)(ˆii (0)ˆjj (0)) 1/2. 24 novembre / 29

31 Estimation of ACVF (bivariate case, m = 2) 8 1 np h >< (X t+h X n )(X t X n ) t 0 apple h < n n t=1 ˆ(h) = 1 np >: (X t+h Xn )(X t Xn ) t n < h < 0. n t= h+1 ˆ ij (h) =ˆij (h)(ˆii (0)ˆjj (0)) 1/2. Theorem If X t = µ + +1X k= 1 C k Z t k {Z t } IID(0, S) then 8 h ˆij (h)! P ij (h) ˆ ij (h)! P ij (h) as n!1. 24 novembre / 29

32 An example: Southern Oscillation Index Southern Oscillation Index (an environmental measure) compared to fish recruitment in South Pacific (1950 to 1985) Southern Oscillation Index Recruitment novembre / 29

33 ACF of Southern Oscillation Index soi soi & rec ACF Lag rec & soi Lag rec Bottom left panel is negative lags. 12 of ACF Lag Lag 24 novembre / 29

34 An example from Box and Jenkins Sales (V2) with a leading indicator (V1) sales V V Time 24 novembre / 29

35 ACF of sales data V1 V1 & V2 ACF Lag V2 & V Lag V2 Data are not stationary. ACF Lag Lag 24 novembre / 29

36 Di erenced sales data dsales V V Time 24 novembre / 29

37 ACF of sales data V1 V1 & V2 ACF ACF Lag V2 & V Lag V2 Only crosscorrelation relevant only at lags 2, Lag Lag 24 novembre / 29

38 Testing for independence of time-series: basis Generally asymptotic distribution of ˆij (h) is complicated. But 24 novembre / 29

39 Testing for independence of time-series: basis Generally asymptotic distribution of ˆij (h) is complicated. But Theorem Let X t,1 = 1X j= 1 with {Z t,1 } WN(0, j Z t j,1 X t,2 = 2 1), {Z t,2 } WN(0, 1X j= 1 jz t j,2 2 2) and independent. 24 novembre / 29

40 Testing for independence of time-series: basis Generally asymptotic distribution of ˆij (h) is complicated. But Theorem Let X t,1 = 1X j= 1 with {Z t,1 } WN(0, j Z t j,1 X t,2 = 2 1), {Z t,2 } WN(0, 1X j= 1 jz t j,2 2 2) and independent. Then nv(ˆ12 (h)) n!1! 1X j= 1 11(j) 22 (j). 24 novembre / 29

41 Testing for independence of time-series: basis Generally asymptotic distribution of ˆij (h) is complicated. But Theorem Let X t,1 = 1X j= 1 with {Z t,1 } WN(0, j Z t j,1 X t,2 = 2 1), {Z t,2 } WN(0, 1X j= 1 jz t j,2 2 2) and independent. Then 1X nv(ˆ12 (h)) n!1! 11(j) 22 (j). 0 n 1/2 ˆ 12 (h) =) j= 1 1X j= (j) 22 (j) A. 24 novembre / 29

42 Testing for independence of time-series: an example Suppose {X t,1 } and {X t,2 } are independent AR(1) processes with i,i (h) =0.8 h. Then asymptotic variance of ˆ 12 (h) is X 1 n 1 h= h n novembre / 29

43 Testing for independence of time-series: an example Suppose {X t,1 } and {X t,2 } are independent AR(1) processes with i,i (h) =0.8 h. Then asymptotic variance of ˆ 12 (h) is n 1 1 X h= h n Values of ˆ 12 (h) quite larger than 1.96n 1 should be common even if the two series are independent. 24 novembre / 29

44 Testing for independence of time-series: an example Suppose {X t,1 } and {X t,2 } are independent AR(1) processes with i,i (h) =0.8 h. Then asymptotic variance of ˆ 12 (h) is n 1 1 X h= h n Values of ˆ 12 (h) quite larger than 1.96n 1 should be common even if the two series are independent. Instead, if one series is white-noise, then V(ˆ 12 (h)) 1 n. 24 novembre / 29

45 Testing for independence of time-series: an example Suppose {X t,1 } and {X t,2 } are independent AR(1) processes with i,i (h) =0.8 h. Then asymptotic variance of ˆ 12 (h) is n 1 1 X h= h n Values of ˆ 12 (h) quite larger than 1.96n 1 should be common even if the two series are independent. Instead, if one series is white-noise, then V(ˆ 12 (h)) 1 n. Hence, in testing for independence, it is often recommended to prewhiten one series. 24 novembre / 29

46 Pre-whitening a time series Instead of testing ˆ 12 (h) of the original series, one trasforms them into white noise. If {X t,1 } and {X t,2 } are invertible ARMA, then where 1 P k=0 Z t,i = 1X k=0 (i) k X t k,i WN(0, (i) k zk = (i) (z) = (i) (z)/ (i) (z). 2 i ), i =1, 2 24 novembre / 29

47 Pre-whitening a time series Instead of testing ˆ 12 (h) of the original series, one trasforms them into white noise. If {X t,1 } and {X t,2 } are invertible ARMA, then where 1 P k=0 Z t,i = 1X k=0 (i) k X t k,i WN(0, (i) k zk = (i) (z) = (i) (z)/ (i) (z). 2 i ), i =1, 2 {X t,1 } and {X t,2 } are independent if and only if {Z t,1 } and {Z t,2 },hence one test for ˆ Z1,Z 2 (h). 24 novembre / 29

48 Pre-whitening a time series Instead of testing ˆ 12 (h) of the original series, one trasforms them into white noise. If {X t,1 } and {X t,2 } are invertible ARMA, then where 1 P k=0 Z t,i = 1X k=0 (i) k X t k,i WN(0, (i) k zk = (i) (z) = (i) (z)/ (i) (z). 2 i ), i =1, 2 {X t,1 } and {X t,2 } are independent if and only if {Z t,1 } and {Z t,2 },hence one test for ˆ Z1,Z 2 (h). As (i) (z) and (i) (z) not known, one fits an ARMA to the series, and uses the residuals Ŵ t,i in place of Z t,i. It may be enough doing this just to one series. 24 novembre / 29

49 Siimulated data 1st series is AR(1) with ' =0.9; 2nd series is AR(2) with ' 1 =0.7, ' 2 =0.27. dat_sim dat dat Time 24 novembre / 29

50 ACF of simulated data dat1 dat1 & dat2 ACF Lag Lag dat2 & dat1 dat2 ACF Lag Lag 24 novembre / 29

51 ACF of residuals MLE fits the correct model to both series. 24 novembre / 29

52 ACF of residuals MLE fits the correct model to both series. fitunk1$res fitunk1$res & fitunk2$res ACF ACF Lag fitunk2$res & fitunk1$res Lag fitunk2$res A few crosscorrelation coe cient may appear slightly significant Lag Lag 24 novembre / 29

53 Bartlett s formula More generally Theorem If {X t } is a bivariate Gaussian time series with lim ncov (ˆ 12(h), ˆ 12 (k)) = n!1 +1X j= 1 +1 P h= 1 ij (h) < 1, then h 11 (j) 22 (j + k h) + 12 (j + k) 21 (j h) 12 (h)( 11 (j) 12 (j + k)+ 22 (j) 21 (j k)) 12 (k)( 11 (j) 12 (j + h)+ 22 (j) 21 (j h)) (h) 12 (k) (j)+ 2 12(j)+ 1 i (j) 24 novembre / 29

54 Spectral density of multivariate series If +1P h= 1 ij (h) < 1, one can define f ( )= 1 1X e ih (h), 2 [, ] 2 and one obtains h= 1 Z (h) = e i h f ( ) d 24 novembre / 29

55 Spectral density of multivariate series If +1P h= 1 ij (h) < 1, one can define f ( )= 1 1X e ih (h), 2 [, ] 2 and one obtains and h= 1 Z (h) = X t = Z e i h f ( ) d e i h dz( ) where Z i ( ) are (complex) processes with independent increments s.t. Z 2 1 f ij ( ) d = E (Z i ( 2 ) Z i ( j ))(Z j ( 2 ) Z j ( 1 )). 24 novembre / 29

56 Coherence of series For a bivariate series the coherence at frequency is X 12 ( )= f 12 ( ) [f 11 ( )f 22 ( )] 1/2 and represents the correlation between dz 1 ( )anddz 2 ( ). The squared coherency function is X 12 ( ) 2 satisfies 0 apple X 12 ( ) 2 apple novembre / 29

57 Coherence of series For a bivariate series the coherence at frequency is X 12 ( )= f 12 ( ) [f 11 ( )f 22 ( )] 1/2 and represents the correlation between dz 1 ( )anddz 2 ( ). The squared coherency function is X 12 ( ) 2 satisfies 0 apple X 12 ( ) 2 apple 1. Remark. If X t,2 = +1 P k= 1 kx t k,1,then X 12 ( ) novembre / 29

58 Periodogram nx Define J(! j )=n 1/2 X t e it! j, t=1 for j between [(n 1)/2] and [n/2].! j =2 j/n Then I n (! j )=J(! j )J (! j )where means transpose and complex conjugate.!! I 12 (! j )= 1 nx nx X t1 e it! j X t2 e it! j n is the cross periodogram. t=1 t=1 24 novembre / 29

59 Estimation of spectral density and coherence Again, one estimates f ( )by ˆf ( )= 1 2 Xm n k= m n W n (k)i n g(n, )+2 k n. If X t = P +1 k= 1 C kz t k {Z t } IID(0, S) then 0 1 Xm n ˆf ij ( ) ij ( ), f ij ( ) Wn 2 (k) A 0 < <. k= m n The natural estimator of X 12 ( ) 2 is ˆ212( )= ˆf 12 ( ) 2 ˆf 11 ( )ˆf 22 ( ). 24 novembre / 29

60 An example of coherency estimation Squared coherency between SOI and recruitment squared coherency The horizontal line represents a (conservative) test of the assumption X 12 ( ) 2 = 0. Strong coherency at period 1 yr. and longer than frequency 24 novembre / 29