SF2943: A FEW COMMENTS ON SPECTRAL DENSITIES

Transcription

1 SF2943: A FEW COMMENTS ON SPECTRAL DENSITIES FILIP LINDSKOG The aim of this brief note is to get some basic understanding of spectral densities by computing, plotting and interpreting the spectral densities of the stationary time series (1) (2) (3) (4) X t = 0.8X t 1 + Z t + 0.9Z t 1, X t = 0.8X t 1 0.9X t 2 + Z t, X t = 0.8X t 1 + Z t, X t = 0.8X t 1 + Z t, where {Z t is an iid sequence of standard normal random variables. The processes are examples of causal linear time series, the roots of the autoregressive polynomials are all located outside the unit circle in the complex plane. In order to get some very preliminary feeling for the four systems above, we study how the linear systems transforms the input (z 1, z 2, z 3,... ) = (1, 0, 0,... ) into output values (x 1, x 2, x 3,... ). The result (with linear interpolation) is shown in Figure 1. The AR(1) process (3) shows a periodic behavior, although a rather trivial one (plus, minus, plus, minus, etc.). The AR(2) process (2) shows a more interesting periodic behavior, with cycle lengths of approximately 3 time steps. For a zero-mean stationary time series with ACVF γ satisfying spectral density is given by (5) f(λ) = 1 e ihλ γ(h). γ(h) <, the The assumption γ(h) < ensures that the sum in (5) is absolute convergent (makes sense). Since e ihλ = cos( hλ) + i sin( hλ) and since cos and sin both have period, it is sufficient to consider only λ in (, π]. Since sin is an odd function, sin( x) = sin(x), and the sum in (5) is computed over a set of h-values that is symmetric about 0, f(λ) = 1 e ihλ γ(h) = 1 cos( hλ)γ(h). In particular, f is a real-valued even function, f(λ) = f( λ). Therefore, it is sufficient to consider only λ in [0, π]. Moreover, it can be shown that f(λ) 0. 1

2 2 FILIP LINDSKOG Let us compute the spectral density of a causal AR(1) process with γ(h) = σ 2 φ ( h )/(1 φ 2 ): σ 2 ( ) f(λ) = (1 φ φ h (e ihλ + e ihλ ) ) h=0 h=1 { = z h = 1 1 z if z < 1 = σ 2 (1 (1 φ 2 + φe iλ ) φeiλ + ) 1 φe iλ 1 φe iλ σ 2 1 φ 2 e iλ e iλ = (1 φ 2 ) 1 φ(e iλ + e iλ ) + φ 2 e iλ e iλ = σ2 (1 2φ cos λ + φ2 ) 1. For a causal ARMA(p, q) process it can be shown that f(λ) = σ2 θ(e iλ ) 2 φ(e iλ ) 2, λ [, π]. In particular, for a causal AR(2) process f(λ) is maximized for λ = φ 1 (φ 2 1)/(4φ 2 ). For the AR(2) process in (2) we get the λ arccos( 0.42) 2.0 maximising the spectral density. This corresponds to the cycle length /λ 3. The spectral densities of (1) to (4), for λ [0, π], are shown in Figure 2. The AR(1) process (3) has a spectral density with a maximum at λ = π. This is consistent with the cycle length of /π = 2 shown in Figure 1. The AR(2) process (2) has a spectral density with a maximum at λ 2. This is consistent with the cycle length of /2 = π 3 shown in Figure 1. The processes (1) and (4) have spectral densities with maxima at λ = 0. They show no periodic behavior (cycles of infinite length). Notice that π e ikλ f(λ)dλ = π = 1 = γ(k) 1 γ(h) e i(k h)λ γ(h)dλ π e i(k h)λ dλ since the integral vanishes for h k. We notice that γ(0) = π f(λ)dλ. In particular, the comparison of the spectral densities might be visualized more clearly by dividing the spectral densities by the stationary variance γ(0) of the respective processes. Finally, we want to see if the sample analogues of the spectral densities, based on simulated samples {x 1,..., x n, are sufficiently accurate to draw the same conclusions as if we knew the spectral densities. The result is shown in Figure 4. The sample analogue of

3 SF2943: A FEW COMMENTS ON SPECTRAL DENSITIES 3 (5) is obtained by replacing the autocovariance function γ by the sample autocovariance function γ in (5): f(λ) = 1 n 1 h= n+1 e ihλ γ(h). The sample analogue of the spectral density corresponds to what is called the periodogram of {x 1,..., x n : I n (λ) = 1 n x t e itλ 2. n It can be shown that I n (ω k ) = f(ω k ), t=1 ω k = k n (, π].

4 4 FILIP LINDSKOG R-code ar2specdens<-function(l,s,phi1,phi2) { s^2/(2*pi*(1+phi1^2+phi2^2+2*phi2+2*(phi1*phi2-phi1)*cos(l)-4*phi2*cos(l)^2)) arma11specdens<-function(l,s,phi,theta) { s^2*(1+theta^2+2*theta*cos(l))/(2*pi*(1+phi^2-2*phi*cos(l))) xvals<-(0:100)*pi/100 plot(xvals,arma11specdens(xvals,1,0.8,0.9),type="l",xlab="",ylab="", ylim=c(0,max(ar2specdens(xvals,1,-0.8,-0.9)))) lines(xvals,ar2specdens(xvals,1,-0.8,-0.9)) lines(xvals,ar2specdens(xvals,1,-0.8,0)) lines(xvals,ar2specdens(xvals,1,0.8,0)) ar2roots<-polyroot(c(1,0.8,0.9)) (ar2roots[1]*ar2roots[2])^2/((ar2roots[1]*ar2roots[2]-1)*(ar2roots[2]-ar2roots[1]))* (ar2roots[1]/(ar2roots[1]^2-1)-ar2roots[2]/(ar2roots[2]^2-1)) g01<-(1+1.7^2/(1-0.8^2)) g02<-(ar2roots[1]*ar2roots[2])^2/((ar2roots[1]*ar2roots[2]-1)*(ar2roots[2]-ar2roots[1])) (ar2roots[1]/(ar2roots[1]^2-1)-ar2roots[2]/(ar2roots[2]^2-1)) g03<-1/(1-0.8^2) g04<-1/(1-0.8^2) plot(xvals,arma11specdens(xvals,1,0.8,0.9)/g01,type="l",xlab="",ylab="", ylim=c(0,max(ar2specdens(xvals,1,-0.8,-0.9)/re(g02)))) lines(xvals,ar2specdens(xvals,1,-0.8,-0.9)/re(g02)) lines(xvals,ar2specdens(xvals,1,-0.8,0)/g03) lines(xvals,ar2specdens(xvals,1,0.8,0)/g04) z<-1 tmpmat<-matrix(0,4,20) tmpmat[1,1]<-z tmpmat[2,1]<-z tmpmat[3,1]<-z tmpmat[4,1]<-z

5 SF2943: A FEW COMMENTS ON SPECTRAL DENSITIES 5 tmpmat[1,2]<-0.8*tmpmat[1,1]+0.9*z tmpmat[2,2]<--0.8*tmpmat[2,1] tmpmat[3,2]<--0.8*tmpmat[3,1] tmpmat[4,2]<-0.8*tmpmat[4,1] for(i in (3:20)) { tmpmat[1,i]<-0.8*tmpmat[1,i-1] tmpmat[2,i]<--0.8*tmpmat[2,i-1]-0.9*tmpmat[2,i-2] tmpmat[3,i]<--0.8*tmpmat[3,i-1] tmpmat[4,i]<-0.8*tmpmat[4,i-1] plot(tmpmat[1,],type="l",ylim=c(-1,1.8),xlab="",ylab="",col="red",xaxt="n") axis(1,(0:20)) lines(tmpmat[2,],col="blue") lines(tmpmat[3,],col="green") lines(tmpmat[4,],col="black") arma1.sim<-arima.sim(model=list(ar=c(-0.8,-0.9)),n=200) arma2.sim<-arima.sim(model=list(ar=c(-0.8)),n=200) arma3.sim<-arima.sim(model=list(ar=c(0.8)),n=200) arma4.sim<-arima.sim(model=list(ar=c(0.8),ma=c(0.9)),n=200) (arma1.sim) lines(xvals/(2*pi),ar2specdens(xvals,1,-0.8,-0.9)) (arma2.sim) lines(xvals/(2*pi),ar2specdens(xvals,1,-0.8,0)) (arma3.sim) lines(xvals/(2*pi),ar2specdens(xvals,1,0.8,0)) (arma4.sim) lines(xvals/(2*pi),arma11specdens(xvals,1,0.8,0.9))

6 6 FILIP LINDSKOG Figures Figure 1. Illustration of how the systems transform input to output Figure 2. Spectral densities

7 SF2943: A FEW COMMENTS ON SPECTRAL DENSITIES Figure 3. Spectral densities normalized by dividing by the stationary variance of the time series 1e 02 1e+01 5e 03 5e 01 1e 03 1e+00 1e 03 1e+01 Figure 4. Periodograms based on samples of size 200 and the corresponding spectral densities