Nonlinear Modeling of Time Series Using Multivariate Adaptive Regression Splines (MARS)

Abstract

Abstract Multivariate Adaptive Regression Splines (MARS) is a new methodology, due to Friedman, for nonlinear regression modeling. MARS can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions. Given a set of predictor variables, MARS fits a model in the form of an expansion in product spline basis functions of predictors chosen during a forward and backward recursive partitioning strategy. MARS produces continuous models for high-dimensional data that can have multiple partitions and predictor variable interactions. Predictor variable contributions and interactions in a MARS model may be analyzed using an ANOVA style decomposition. By letting the predictor variables in MARS be lagged values of a time series, one obtains a new method for nonlinear autoregressive threshold modeling of time series. A significant feature of this extension of MARS is its ability to produce models with limit cycles when modeling time series data that exhibit periodic behavior. In a physical context, limit cycles represent a stationary state of sustained oscillations, a satisfying behavior for any model of a time series with periodic behavior. Analysis of the yearly Wolf sunspot numbers with MARS appears to give an improvement over existing nonlinear threshold and bilinear models. A graphical representation for the models is given.