Forecasting Process Details 
Series Transformations
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
is transformed.
There are four transformations available,
for strictly positive series only.
Let y_{t} > 0 be the original time series,
and let w_{t} be the transformed series.
The transformations are defined as follows:
 Log
 is the logarithmic transformation.

w_{t} = ln(y_{t})
 Logistic
 is the logistic transformation.

w_{t} = ln(c y_{t} / (1c y_{t}))
where the scaling factor c is

c = (1e^{6}) 10 ^{ ceil( log10(max( yt) ))}
and ceil(x) is the smallest integer
greater than or equal to x.
 Square Root
 is the square root transformation.
 Box Cox
 is the BoxCox transformation.
Parameter estimation is performed using the transformed series.
The transformed model predictions and confidence limits
are then obtained from the transformed timeseries 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 inversetransformed
to obtain the forecast confidence limits.
Predictions for Transformed Models
Since the transformations described in the previous section are
monotonic, applying the inversetransformation 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 w_{t} = F(y_{t}) is the transform with
inversetransform y_{t} = F^{1}(w_{t}), 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:
 Log
 is the conditional expectation of inverselogarithmic transformation.
 Logistic
 is the conditional expectation of inverselogistic transformation.
where the scaling factor
c = (1e^{6})10 ^{ ceil( log10(max( yt)))}.
 Square Root
 is the conditional expectation of the inversesquare root transformation.
 Box Cox
 is the conditional expectation of the inverse BoxCox transformation.
The expectations of the inverse logistic and
BoxCox ( ) transformations do not
generally have explicit solutions and are computed using
numerical integration.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.