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D.2 Preparing Your Data for Modeling 399
(with period 12). They must therefore be transformed in order to be represented as
a realization of a stationary time series using one or more of the transformations
available for this purpose in ITSM.
Box–Cox Transformations
Box–Cox transformations are performed by selecting Transform>Box-Cox and
specifying the value of the Box–Cox parameter λ. If the original observations are
Y1,Y2,...,Yn, the Box–Cox transformation fλ converts them to fλ(Y1), fλ(Y2),...,
fλ(Yn), where
⎧
⎨ y
fλ(y)
⎩
λ − 1
, λ 0,
λ
log(y), λ 0.
These transformations are useful when the variability of the data increases or
decreases with the level. By suitable choice of λ, the variability can often be made
nearly constant. In particular, for positive data whose standard deviation increases
linearly with level, the variability can be stabilized by choosing λ 0.
The choice of λ can be made visually by watching the graph of the data when
you click on the pointer in the Box–Cox dialog box and drag it back and forth along
the scale, which runs from zero to 1.5. Very often it is found that no transformation
is needed or that the choice λ 0 is satisfactory.
Example D.2.6 For the series AIRPASS.TSM, the variability increases with level, and the data are
strictly positive. Taking natural logarithms (i.e., choosing a Box–Cox transformation
with λ 0) gives the transformed data shown in Figure D.1.
Notice how the amplitude of the fluctuations no longer increases with the level of
the data. However, the seasonal effect remains, as does the upward trend. These will
be removed shortly. The data stored in ITSM now consist of the natural logarithms
of the original data.
Classical Decompositon
There are two methods provided in ITSM for the elimination of trend and seasonality.
These are:
i. “classical decomposition” of the series into a trend component, a seasonal component,
and a random residual component, and
ii. differencing.