Parametric and non-parametric modelling of time series - An empirical study

Time series modelling methods can be loosely classified as (i) parametric methods and (ii) non-parametric methods. Within a usually quite flexible but well structured family of models, the parametric modelling process typically consists of model identification, parameter estimation, model diagnostic checking, and forecasting. On the other hand, within a much less structured framework, different non-parametric smoothing techniques are usually used to bring out the features of the observed time series, however, few serious and systematic attempts have been made to model time series non-parametrically. We concentrate on a non-parametric method based on multivariate adaptive regression splines (MARS). Parallel to the parametric modelling process, we systemize a non-parametric modelling process as (i) model perception (where a very large spline expansion of a very large family of models is specified), (ii) model search (forward plus backward search to come up with a model), (iii) model diagnostic checking, and (iv) forecasting. The major difference between the MARS and the parametric methods is that the potential models for the MARS method form a family which is much larger than any family of parametric time series models, and the local structures found in the data are used to guide the search for a fitted model. Also, unlike most non-parametric methods, MARS time series models can be analytically written down. In this paper, we present the results of an empirical comparison of parametric (ARIMA) and non-parametric (MARS) time series modelling methods. Eight environmental time series are used for the comparison.