# Ch3_S+

> # generate time series with 3 cycles
> t_1:128
> x1_2*cos(2*pi*t*6/128) +3*sin(2*pi*t*6/128)
> x2_4*cos(2*pi*t*10/128) + 5*sin(2*pi*t*10/128)
> x3_3*cos(2*pi*t*40/128) + 4*sin(2*pi*t*40/128)
> x_x1+x2+x3
> min(x)
[1] -15.36613
> max(x)
[1] 13.88798
> win.graph()
> par(mfrow=c(2,2))
> tsplot(x1,main="nu=6/128, amp^2=13", ylim=c(-16,16))
# in above amp^2=2^2+3^3=13 is the square of the amplitude
> tsplot(x2,main="nu=10/128, amp^2=41", ylim=c(-16,16))
> tsplot(x3,main="nu=40/128, amp^2=25", ylim=c(-16,16))
> tsplot(x,main="sum", ylim=c(-16,16))

# the periodogram of x
> par(mfrow=c(1,1))
> d_fft(x)/length(x)
> per_abs(d)^2
> freq_(0:64)/128
> per_per[1:65]
> plot(freq,per,type="l")


# try it this way 
> spectrum(x, method="pgram", plot=T)
# what is that? 
# ok try this
> spec.pgram(x, detrend=F, demean=F, plot=T)
# Note dB=decibel, and the value S in dBs is  10log_10 S

# spectrum of recruits
> help(spec.pgram) # to get details
# see the difference
> par(mfrow=c(2,2))
> spec.pgram(rec, spans=1, taper=0.1, pad=0, detrend=F, demean=T, plot=T)
> spec.pgram(rec, spans=3, taper=0.1, pad=0, detrend=F, demean=T, plot=T)
> spec.pgram(rec, spans=c(3,3), taper=0.1, pad=0, detrend=F, demean=T, plot=T)
> spec.pgram(rec, spans=c(3,5), taper=0.1, pad=0, detrend=F, demean=T, plot=T)
# note spectrum() does the same thing as spec.pgram

# AR spectral estimate for recruits
> rec.ar <- ar(rec)          # Fit an AR model using AIC
> rec.ar$aic             # to see which model is chosen (p=13)
> par(mfrow=c(1,1))
> rec.arspec <- spec.ar(rec.ar,plot=T) #  superimpose the AR spectrum for this
> rec.per <- spec.pgram(rec)          # data and its periodogram.  
> spec.plot(rec.per, add=T)


# Coherency between SOI and Recruits
> x1_matrix(soi)
> x2_matrix(rec)
> x_cbind(x1,x2)
> spec_spec.pgram(x, spans=c(3,5), taper=0.1, pad=0, detrend=F, demean=T, plot=T)
> rho_spec$coh
> length(rho)
[1] 229
> length(x1)
[1] 453
> freq_(0:length(rho)-1)/length(x1)
> plot(freq,rho,type="l")