Coherent Forecasting of Ordinal Time Series With Application to Air-Quality Data

Abstract

Motivated by an application to air quality data, we provide a comprehensive investigation of coherent forecasting techniques for ordinal time series. Here, the coherency requirement expresses that the computed forecast values have to be consistent with the ordered qualitative range of the data-generating process (DGP). Three classes of coherent forecasts are proposed, namely ordinal point forecasts, prediction intervals and probability mass function (pmf) forecasts. In addition, corresponding criteria for evaluating their predictive performance are established. An extensive simulation study analyses the performance of the coherent forecasting techniques across various ordinal DGPs, taking into account the impact of estimated model parameters and model misspecification. The proposed forecasting approaches are then applied to our central motivating application, the real-world time series on ordinal air quality levels and practical implications are discussed. In this context, it is also demonstrated that the coherent point and pmf forecasts can be combined visually in a meaningful and interpretable way. While centrally motivated by environmental monitoring, the proposed framework applies more broadly to ordinal time series arising in other settings where ordered categorical outcomes are used for assessment and decision making.