ARIMA time-series forecasting assumes that the time series mean, variance, and autocorrelation are stationary over time. This characteristic is called stationarity. If a time series statistic has nonstationarity, it must be adjusted:
Nonstationarity in the mean—In this case, the mean is not constant but drifts slowly. This can be true for both seasonal and non-seasonal series and is removed by differencing the series. The automatic ARIMA implementation of Predictor determines the amount of non-seasonal differencing required to make a series stationary by using repeated KPSS (Kwiatkowski-Phillips-Schmidt-Shin) tests with appropriate alpha values. For seasonal series, repeated Canova-Hansen tests with appropriate alpha values are used.
Nonstationarity in variance—In this case, the time series is heteroscedastic; the variance of the data around the mean changes over time. This nonstationarity in variance is removed by applying the Box-Cox transformation, a special type of power transformation:
, if lambda is not equal to 0
, if lambda equals 0Where the original series is {xt}, the transformed series is {zt}, and the power transformation constant is lambda (λ).
Predictor determines a suitable value of lambda with an algorithm that uses the seasonality information to divide the dataset into groups, and then tries to fina a lambda value that renders the variance stationary across groups.
For users who want more control over the Box-Cox transformation, Predictor provides commonly used power-transformation options, such as log transformation (lambda = 0) or square-root transformation (lambda = 0.5), and even a custom transformation with a user-selected lambda between –5 and +5 (inclusive). However, Predictor prevents the use of custom lambda values that would result in transformed values being too large or too small.