Stationarity

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 nonseasonal series and is removed by differencing the series. The automatic ARIMA implementation of Predictor determines the amount of nonseasonal 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:

    Box-Cox transformation, if lambda is not equal to 0

    Box-Cox transformation, if lambda equals 0

    Where the original series is {xt}, the transformed series is {zt}, and the power transformation constant is lambda (λ).

    Predictive Planning 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, Predictive Planning 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, Predictive Planning prevents the use of custom lambda values that would result in transformed values being too large or too small.