|Forecasting Process Details
For pure ARIMA models, transforming the response time series may
aid in obtaining stationary noise series. For general ARIMA models
with inputs, transforming the response time series or
one or more of the input time series may provide a better model fit.
Similarly, the fit of smoothing models may improve when the response series
There are four transformations available,
for strictly positive series only.
Let yt > 0 be the original time series,
and let wt be the transformed series.
The transformations are defined as follows:
- is the logarithmic transformation.
wt = ln(yt)
- is the logistic transformation.
wt = ln(c yt / (1-c yt))
where the scaling factor c is
and ceil(x) is the smallest integer
greater than or equal to x.
c = (1-e-6) 10 - ceil( log10(max( yt) ))
- Square Root
- is the square root transformation.
- Box Cox
- is the Box-Cox transformation.
Parameter estimation is performed using the transformed series.
The transformed model predictions and confidence limits
are then obtained from the transformed time-series and
these parameter estimates.
The transformed model predictions
are used to obtain either the
minimum mean absolute error (MMAE) or
minimum mean squared error (MMSE) predictions
,depending on the setting of the forecast options.
The model is then evaluated based on the residuals of the original
time series and these predictions.
The transformed model confidence limits are inverse-transformed
to obtain the forecast confidence limits.
Predictions for Transformed Models
Since the transformations described in the previous section are
monotonic, applying the inverse-transformation to the transformed model
predictions results in the median of the conditional probability
density function at each point in time.
This is the minimum mean absolute error (MMAE) prediction.
If wt = F(yt) is the transform with
inverse-transform yt = F-1(wt), then
The minimum mean squared error (MMSE) predictions
are the mean of the conditional probability density function
at each point in time.
Assuming that the prediction errors are normally distributed
with variance ,the MMSE predictions for each of the transformations are as follows:
- is the conditional expectation of inverse-logarithmic transformation.
- is the conditional expectation of inverse-logistic transformation.
where the scaling factor
c = (1-e-6)10 - ceil( log10(max( yt))).
- Square Root
- is the conditional expectation of the inverse-square root transformation.
- Box Cox
- is the conditional expectation of the inverse Box-Cox transformation.
The expectations of the inverse logistic and
Box-Cox ( ) transformations do not
generally have explicit solutions and are computed using
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.