# to illustrate concept of stationarity, undifferencing (integration)
# given linear input trend for Yt: (1-B)^d*(1-B)^D*Yt = St

# First differenced 1-backshifted linear trend Yt with noise (1-B)^1.Yt = St = constant ~ 1
# Twice differenced 1-backshifted linear trend Yt with noise (1-B)^2.Yt = St ~ 0

x <- c(1:144) + rnorm(144);
plot(x)
diff1 <- diff(x,1)
plot(diff1)
mean(diff1)
diff2 <- diff(diff1,1)
plot(diff2)
mean(diff2)

# First differenced 12-backshifted linear trend Yt with noise (1-B^12)^1.Yt = St = constant ~ 12
# Twice differenced 12-backshifted linear trend Yt with noise (1-B^12)^2.Yt = St ~ 0

x <- c(1:144) + rnorm(144);
plot(x)
diff1 <- diff(x,12)
plot(diff1)
mean(diff1)
diff2 <- diff(diff1,12)
plot(diff2)
mean(diff2)

# First differenced 12-backshifted linear trend Yt with noise (1-B^12).Yt = St = constant ~ 12
# First differenced 1-backshifted of this: (1-B)(1-B^12).Yt = St ~ 0

x <- c(1:144) + rnorm(144);
plot(x)
diff1 <- diff(x,12)
plot(diff1)
mean(diff1)
diff2 <- diff(diff1,1)
plot(diff2)
mean(diff2)

# CONCLUSIONS: 1. The "integrated" series can be 0, a const or at most a linear function
                  for all orders of differencing s.t. d + D = 2 to give a stationary
                  series with <St> ~ 0.

               2. The "integrated" series can be 0 or at most a constant for orders of
                  differencing where d + D = 1 to give a stationary series with <St> ~ 0.