Seasonal modulation mixed models for time series forecasting

We propose an extension of a seasonal modulation smooth model with P-splines for times series data using a mixed model formulation. A smooth trend with seasonality decomposition can be estimated simultaneously. We extend the model to consider the forecasting of new future observations in the mixed model framework. Two different approaches are used for forecasting in the context of mixed models, and the equivalence of both methods is shown. The methodology is illustrated with monthly sulphur dioxide (SO2) levels in a selection of monitoring sites in Europe from January 1990 to December 2001.