### Simulate MA(3) data and plot. R

library(ts)

# simulate white Gaussian noise and 3pt MA
# two graphs one on top of other

win.graph()

x <- rnorm(250, mean=0, sd=1)
x.fil <- filter(x, c(1,1,1)/3)
par(mfrow=c(2,1))
ts.plot(x)
ts.plot(x.fil)

# plot of data with MA superimposed

win.graph()

par(mfrow=c(1,1))
t <- 1:length(x)
plot(t,x)
lines(x.fil)

# Random walk simulation

win.graph()

par(mfrow=c(2,2))
w <- rnorm(250, mean=0, sd=1)
x <- cumsum(w)
ts.plot(x, gpars=list(main="Random Walk"))
acf(x)
xd <- diff(x)
ts.plot(xd,gpars=list(main="Differenced"))
acf(xd)

# Cosine signal plus noise

win.graph()

par(mfrow=c(2,2))
t <- 1:200
x <- sqrt(2)*cos(2*pi*t/50)
w <- rnorm(200,0,1)
ts.plot(x, gpars=list(main="Sinusoid (one cylce every 50 points)"))
ts.plot(x+w, gpars=list(main="SNR=1"))
ts.plot(x+2*w, gpars=list(main="SNR=1/4"))
ts.plot(x+4*w, gpars=list(main="SNR=1/16"))

# NOTES:
# SNR signal-to-noise ratio = var(signal)/var(noise)
# Here var(x)=sum(x^2)/n, in this case n=200
# and sum[cos(2*pi*nu*t)^2]=n/2  except when nu=0 or n/2.
# Here nu=1/50  so in this example
# var(signal)/var(noise) = 1/1, 1/4 and 1/16.

# check the sample values
var(x)/var(w)
var(x)/var(2*w)
var(x)/var(4*w)

# modulated signal + noise
w <- rnorm(200,0,1)
t <- 1:100
y <- cos(2*pi*t/10)
e <- exp(-t/50)
s <- c(rep(0,100),10*y*e )
x <- s+w

par(mfrow=c(2,1))
ts.plot(s, gpars=list(main="signal"))
ts.plot(x, gpars=list(main="signal+noise"))