Forecasting Process Details

Series Transformations

Subsections:

For pure ARIMA models, transforming the response time series can 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 can provide a better model fit. Similarly, the fit of smoothing models can 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,

\[ w_{t} = \mr{ln}(y_{t}) \]
Logistic

is the logistic transformation,

\[ w_{t} = \mr{ln}(c y_{t} / (1-c y_{t})) \]

where the scaling factor ${c}$ is

\[ c = (1-10^{-6}) 10 ^{- \mr{ceil}( \mr{log}_{10}({max}( y_{t}) ))} \]

and ${\mr{ceil}(x)}$ is the smallest integer greater than or equal to x.

Square Root

is the square root transformation,

\[ w_{t} = \sqrt {y_{t}} \]
Box Cox

is the Box-Cox transformation,

\[ w_{t} = \begin{cases} \frac{y_{t}^{{\lambda }} - 1}{\lambda }, & \lambda {\ne } 0 \\ \mr{ln}( y_{t}), & \lambda = 0 \end{cases} \]

Parameter estimation is performed by 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 ${\hat{w}_{t}}$ are used to obtain either the minimum mean absolute error (MMAE) or minimum mean squared error (MMSE) predictions ${\hat{y}_{t}}$, 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 ${w_{t} = \mr{F}(y_{t})}$ is the transform with inverse-transform ${y_{t}=\mr{F}^{-1}(w_{t})}$, then

\[ \mr{median}( \hat{y}_{t} ) = \mr{F}^{-1}( {E} \left[ w_{t} \right] ) = \mr{F}^{-1}( \hat{w}_{t}) \]

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 ${ {\sigma }_{t}^{2}}$, the MMSE predictions for each of the transformations are as follows:

Log

is the conditional expectation of inverse-logarithmic transformation,

\[ \hat{y}_{t} = {E} {\left[ e^{w_{t} } \right]} = \mr{exp} \left( \hat{w}_{t} + {\sigma }^{2}_{t}/{2} \right) \]
Logistic

is the conditional expectation of inverse-logistic transformation,

\[ \hat{y}_{t}= {E}\left[ \frac{1}{c(1+exp(-w_{t}))} \right] \]

where the scaling factor ${c = (1-e^{-6})10 ^{- \mr{ceil}( \mr{log}_{10}(\mr{max}( y_{t})))}}$.

Square Root

is the conditional expectation of the inverse-square root transformation,

\[ \hat{y}_{t} = {E} \left[ w^{2}_{t} \right] = \hat{w}^{2}_{t} + {\sigma }^{2}_{t} \]
Box Cox

is the conditional expectation of the inverse Box-Cox transformation,

\[ \hat{y}_{t} = \begin{cases} {E} \left[ ({\lambda }w_{t}+1)^{1 / {\lambda } } \right], & \lambda {\ne } 0 \\ {E} \left[ e^{w_{t}} \right] = \mr{exp} (\hat{w}_{t} + \frac{1}{2} {\sigma }^{2}_{t}), & \lambda = 0 \end{cases} \]

The expectations of the inverse logistic and Box-Cox ( ${{\lambda } {\ne } 0}$ ) transformations do not generally have explicit solutions and are computed by using numerical integration.