http://www2.chass.ncsu.edu/garson/pa765/time.htm
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Exponential Smoothing
Exponential smoothing is a form of time series analysis in which the
researcher is interested less in modeling than in sheer prediction of the
next period based on data about the current and recent periods. It is used
for short-term, "one step ahead" forecasting in practical situations, and is
not appropriate for long or even medium-term forecasting.
Weighting. A key question in forming an exponential smoothing line is
how much to count the current period, the previous period, and earlier
periods when estimating the value in the next period. Exponential
smoothing software allows up to four weighting parameters to be
assigned:
1. Alpha: This parameter affects level estimates and varies from 0
(old observations count just as much as the current observation) to
1 (the current observation is used exclusively).
2. Gamma: This parameter affects trend estimates and varies from 0
(trends are based on all observations counting equally) to 1 (the
trend is only based on the most recent observations in the series).
Gamma is only used if there is a trend in the series.
3. Phi: This parameter affects damping estimates and is used instead
of gamma when it is thought that the series is dying out. A value of
phi near .1 uses all observations to estimate any trend toward dying
out, whereas phi values near .9 respond rapidly to any recent
observations indicating the trend is dying.
4. Delta: This parameter affects seasonality estimates and is used
only when the data shows seasonal cycles. Delta values near 0
cause all points to count equally while delta values near 1 estimate
seasonality primarily from recent observations.
Selecting which parameters are needed. The researcher begins by
considering which parameters need to be set. Only alpha needs to be set if
there is no seasonality, the trend is not dying (damping), and the there is
no trend (the series varies randomly about its mean). For alpha to apply,
there does need to be autocorrelation (memory), meaning that adjacent
points are not random but tend to be relatively close together, even if there
is no overall trend. Autocorrelation, trends, damping, and seasonality can
all be assessed by looking at a sequence plot of the time series, obtained in
SPSS by selecting Graphs, Sequence. (The Sequence Plot dialog box also
provides for logarithmic transformations, differencing, and seasonality
differencing).
Grid search for needed parameters. To help estimate smoothing
parameters, SPSS provides a grid search function. One chooses Analyze,
Time Series, Exponential Smoothing from the menus, then in the
Exponential Smoothing dialog box. The grid search function causes SPSS
to create a sequence of equally spaced values for alpha and for each value
calculates a measure of how well the predictions agreed with the actual
values. The parameters that produce the smallest SSE (sum square of
errors) are the best-fitting parameters. By default, SPSS displays the 10
best-fitting sets of parameters and their corresponding SSE values.
(Warning: if you are estimating more than one parameter, the size of the
grid grows exponentially). After the grid search comes up with the optimal
parameter setting, SPSS adds two new series to your file. The series fit_1
contains the predicted values from the exponential smoothing, and err_1
contains the errors. Select Graphs, Sequence, from the SPSS menus to
obtain a sequence plot of the new smoothed series fit_1. The original
series amount and the fit_1 forecasts are both show, and their
correspondence indicates the degree to which the exponential smoothing
forecasts are tracking actual values.
Models. SPSS provides exponential smoothing in the menu system under
Analyze, Time Series, Exponential Smoothing. The Exponential
Smoothing dialog box allows the user to select from four models:
5. Simple. The series has no overall trend and shows no seasonal
variation.
6. Holt. The series has a linear trend but shows no seasonal variation.
7. Winters. The series has a linear trend and shows multiplicative
seasonal variation.You cannot select this option unless you have
defined seasonality with the Define Dates command.
8. Custom. You can specify the form of the trend component and the
way in which the seasonal component is applied.
Residual analysis. The SPSS exponential smoothing procedure
automatically adds the variable err_1 variable to the file. This series is
simply the difference between the actual value and the prediction value. It
can be plotted by selecting Graphs. Sequence from the SPSS menus. This
plot is inspected to assure that residuals are randomly distributed. A
finding of non-randomness indicates the model is inadequate.