As for the moving
average, this method assumes that the time series follows a
constant model.
The value of b is estimated as the weighted average
of the last observation and the last estimate. Here
is a parameter in the interval [0, 1].
Rearranging, obtains an alternative form.
The new estimate is the old estimate plus a proportion of the
observed error.
Because we are supposing a constant model, the forecast is the
same as the estimate.
We illustrate the method using the parameter value
= 0.2 and the data below.
The first 10 observations are used to warm up the procedure
with the average of the observations providing the estimate
for time 10. The average is 11.1. The value of the estimate
for time 11 is computed below.
Subsequent estimates are computed with the exponential smoothing
formula and are shown in the table.
Only two data elements are required to compute each estimate,
the observed data and the old estimate. This contrasts with
the moving average which requires the previous m observations
for the computation.
Replacing
with its equivalent, we find that the estimate is
Continuing in this fashion, we discover that the estimate is
really a weighted sum of all past data.
The larger values of
provide relatively greater weight to more recent data than smaller
values of .
With the value of 1,
is the last data point. With the value 0,
is the same as .
The figure shows the parameter estimates obtained for three
different values of
together with the mean of the time series. Although the model
for this method is a constant, we illustrate the response to
a time series with a trend. The simulated example includes a
trend of 1 from 20 to 30.
A lag characteristic, similar to the one associated
with the moving average estimate, can also be seen in the figure.
The lag and bias for the exponential smoothing estimate can
be expressed as a function of .
The quantity a in the expression is the linear trend
value.
For smaller values of
we obtain a greater lag in response to the trend.
The error is the difference between the actual
data and the forecasted value. If the time series is truly a
constant value, the expected value of the error is zero and
the variance of the error is comprised of a term that is a function
of
and a second term that is the variance of the noise, .
The variance of the error increases as
increases. To minimize the effect of noise, we would like to
make
as small as possible (0), but this makes the forecast unresponsive
to a change in the underlying time series. To make the forecast
responsive to changes, we want as large as possible (1), but
this increases the error variance. Practical forecasting requires
an intermediate value.
We equate the approximating error for the moving average and
exponential smoothing methods.
Solving for ,
we find the value providing the same approximation error as
the moving average.
Using this relation between the parameters of the two methods,
we find that the lag and bias introduced by the trend will also
be the same.
The parameters used in the moving average illustrations of
the last page (m = 5, 10, 20) are roughly comparable
to the parameters used for exponential smoothing in figure above
(
= 0.4, 0.2, 0.1).
