Nonlinear modelling of periodic threshold autoregressions using TSMARS

We present new methods for modelling nonlinear threshold-type autoregressive behaviour in periodically correlated time series. The methods are illustrated using a series of average monthly flows of the Fraser River in British Columbia. Commonly used nonlinearity tests of the river flow data in each month indicate nonlinear behaviour in certain months. The periodic nonlinear correlation structure is modelled nonparametrically using TSMARS, a time series version of Friedman's extended multivariate adaptive regression splines (MARS) algorithm, which allows for categorical predictor variables. We discuss two methods of using the computational algorithm in TSMARS for modelling and fitting periodically correlated data. The first method applies the algorithm to data from each period separately. The second method models data from all periods simultaneously by incorporating an additional predictor variable to distinguish different behaviour in different periods, and allows for coalescing of data from periods with similar behaviour. The models obtained using TSMARS provide better short-term forecasts for the Fraser River data than a corresponding linear periodic AR model.