Let’s imagine that our data look like so:

It is quite smooth (not like noise). Let’s fit a ETS with level and trend.

What should the level be? The one-step ahead (\(h=1\)) prediction model for an ETS is, where \(l_t\) is the level to use in our prediction.

\[\hat{y}_{t+1|1:t} = l_t + b_t\] “level” is where to “start” our prediction and then we’ll add our estimated trend \(b_t\).

No error in the data and it is smooth, so \(l_t\) should be \(y_t\)!

The estimated level for the ETS is in fit$states:

library(forecast)
fit <- ets(dat, model="AAN", damped=FALSE)
plot(dat, type="p",pch=2)
points(fit$states[2:13,1],pch=3)
legend("topright",c("data","l(t)"),pch=c(2,3))

The estimated trend to use depends on the \(\beta\) (weighting). In this case, it is an average of past trends (diff(dat)) which make sense as the trend keeps changing.

plot(diff(dat), type="p",pch=2)
points(fit$states[3:13,2],pch=3)
legend("topright",c("data","b(t)"),pch=c(2,3))

Let’s imagine that our data look like so:

\[\hat{y}_{t+1|1:t} = l_t + b_t\]

The best trend \(b_t\) will take into account the estimated level and will be a smoothed value of the diff(dat).