# library(ts)

n_512
x.sim_harmonic.proc(w=pi*c(1/10,1/17),A=c(5,7),t=n-1,plot=F)
x_x.sim$x+rnorm(n,sd=25)
ts.plot(x)

x.pgram_pgram(x)
plot(x.pgram$freq,x.pgram$pgram,type="l")
# freq = w/(2*pi) 
#   w1 = pi/10 ; freq = 1/20
#   w2 = pi/17 ; freq = 1/34
abline(v=1/34,lty=2)
abline(v=1/20,lty=2)

# estimate the amplitudes by least squares
w1_pi/10 ; w2_pi/17
time_0:(n-1)
y_cbind(cos(w1*time),sin(w1*time),cos(w2*time),sin(w2*time))
x.fit_lsfit(y,x,intercept=F)
x.fit$coef
# ts plot of residuals
ts.plot(x.fit$resid)

# periodograms of residuals
x.resid.pgram_pgram(x.fit$resid)
plot(x.resid.pgram$freq,x.resid.pgram$pgram,type="l")

# fitted sinusoidal trend and periodogram plots of fit
x.hat_x-x.fit$resid
x.hat.pgram_pgram(x.hat)
plot(x.hat.pgram$freq,x.hat.pgram$pgram,type="l")

plot(x.pgram$pgram,x.hat.pgram$pgram)

#round(x.pgram$pgram-x.hat.pgram$pgram,2)
ts.plot(x.pgram$pgram-x.hat.pgram$pgram)

ts.plot(cbind(x.pgram$pgram,x.hat.pgram$pgram),gpars=list(lty=1:2))

#***************


f.hat.plt.eg1_function(x.pgram,m=3){
 txt_paste( c("Daniell Smooth, m = ",m),collapse="")
 k.daniell_kernel("daniell",m=m)
 f.hat_kernapply(x.pgram$pgram/(2*pi),k.daniell)
 freq_x.pgram$freq[m:(length(x.pgram$freq)-1-m)]
 f.ci.up_f.hat*(1+2*sqrt(var.exp/(2*m+1)))
 f.ci.low_f.hat*(1-2*sqrt(var.exp/(2*m+1)))

 ylim_c(0, max(f.hat,f.ci.up))
 plot(freq,f.hat,type="l",main=txt,ylim=ylim)
 lines(freq,f.ci.up,lty=2)
 lines(freq,f.ci.low,lty=2)
}

m_10
f.hat.plt.eg1(x.pgram,m=m)