## Modeling structure

In this lecture we model observed time series data $$\{y_1,\ldots,y_n\}$$ as a normal dynamic linear model (NDLM) with observation equation given by $y_t = x_t + v_t \qquad v_t \sim N(0,V),$ and system equation given by $x_t = \phi x_{t-1} + w_t \qquad w_t \sim N(0,W),$ and initial distribution $$x_0 \sim N(m_0,C_0)$$.

The R funtion sim.dlm simulates data from the above NDLM:

sim.dlm = function(n,phi,V,W){
sV  = sqrt(V)
sW  = sqrt(W)
x = rep(0,n)
x[1] = rnorm(1,0,sW)
for (t in 2:n)
x[t] = rnorm(1,phi*x[t-1],sW)
y = rnorm(n,x,sV)
return(list(y=y,x=x))
}

## Simulating time series data

We simulate a time series of $$n=200$$ observations from $$x_0=0$$, $$V=1$$, $$W=0.25$$ and $$\phi=0.95$$.

set.seed(12345)
n   = 200
V   = 1.00
W   = 0.25
phi = 0.95
sim = sim.dlm(n,phi,V,W)
y   = sim$y x = sim$x

par(mfrow=c(1,1))
ts.plot(y,type="b",ylab="",main="y(t) vs x(t)")
lines(x,col=2,type="b")