Question

Time series analysis

2. Set n 100 and generate and plot the time series xt 2 cos(2π.06t) + 3 sin(2π.06t) Ý,-4 cos(2n. 10t) + 5 sin(2m10) z, 6 cos(2π·40t) + 7 sin(2π·40t) (a) Use the periodogram function in R to plot the periodogram of Vi. Can you explain the spikes? (b) Now let wi ~ N(0, 25) be iid and plot the periodogram of the series V +w. Does it still pick out the periodic components?

Answer 1

The R- code for all both (a) and (b) is as follows

n<-100
xt<-rep(0,n)
yt<-rep(0,n)
zt<-rep(0,n)
vt<-rep(0,n)

for(i in 1:n)
{
xt[i]<-2*cos(2*pi*0.06*i)+3*sin(2*pi*0.06*i)
yt[i]<-2*cos(2*pi*0.06*i)+3*sin(2*pi*0.06*i)
zt[i]<-2*cos(2*pi*0.06*i)+3*sin(2*pi*0.06*i)
vt[i]<-xt[i]+yt[i]+zt[i]
}
ts.plot(xt)
ts.plot(yt)
ts.plot(zt)
ts.plot(vt)

spectrum(vt)

#Let at=wt+vt, where wt ~ N(0,25)
at<-vt+rnorm(n,0,5)
spectrum(at)

The time series plots for xt, yt, zt, vt are as follows

CN 0 20 40 60 80 100 Time

CN 0 20 40 60 80 100 Time

CN 0 20 40 60 80 100 Time

20 40 60 80 100 0 Time

The periodogram for series Vt is obtained as

Series: x Raw Periodogram 0.0 0.1 0.2 0.3 0.4 0.5 frequency bandwidth 0.00289

For the series Vt + Wt, where Wt ~ N(0,25) the periodogram is obtained as -

Series: x Raw Periodogram 0.0 0.1 0.2 0.3 0.4 0.5 frequency bandwidth 0.00289