Like the regression forecast, the double exponential smoothing
forecast is based on the assumption of a model consisting of
a constant plus a linear trend.
For the purposes of a forecast where the parameters of the
model may change, it is more convenient to express the model
as a function of ,
where
is the positive displacement from a reference time T.
The estimate a and b at time T are
based in the observation at time T and the estimates
for the previous period, T 1.
Here we have both the constant and trend coefficients estimated
by exponential smoothing. The forecasting parameters,
for the constant term and
for the trend term can be set independently. Both paremeters
must be between 0 and 1.
The forecast for the expected value for future periods is the
constant plus a linear term that depends on the number of periods
into the future.
With a linear term as part of the forecast, this method will
track trends in the time series. We use the same data as for
the other forecasting methods for illustration. We repeat the
data below. Recall that the simulated data begins with a constant
mean of 10. At time 11 the mean increases with a trend of 1
until time 20 when the mean becomes a constant again with value
20. The noise is simulated using a normal distribution with
mean 0 and standard deviation 3. The values are rounded to the
nearest integer.
At any time T, only three pieces of information
are necessary to compute the estimates, ,
,
and .
We illustrate the computations for time 20, using the estimated
coefficients for time 19 and the data for time 20.
