An investigation of multivariate adaptive regression splines for modeling and analysis of univariate and semi-multivariate time series systems

Abstract

This dissertation investigates the use of multivariate adaptive regression splines (MARS), due to Friedman, for nonlinear regression modeling and analysis of time series systems. MARS can be conceptualized as a generalization of recursive partitioning that uses spline fitting in lieu of other simple fitting functions. MARs is a computationally intensive methodology that fits a nonparametric regression model in the form of an expansion in product spline basis functions of predictor variables chosen during a forward and backward recursive partitioning strategy. The MARS algorithm produces continuous nonlinear regression models for high-dimensional data using a combination of predictor variable interactions and partitions of the predictor variable space. By letting the predictor variables in the MARS algorithm be lagged values of a time series system, one obtains a univariate (ASTAR) or semi-multivariate (SMASTAR) adaptive spline threshold autoregressive model for nonlinear autoregressive threshold modeling and analysis of time series, thereby extending the threshold autoregression (TAR) time series methodology developed by Tong. The model seem well suited for taking into account the complex interactions among multivariate, cross-correlated, lagged predictor variables of a time series system. A significant feature of this time series application 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 difficulty faced during regression modeling is the problem of model selection, i.e., choosing the appropriate model dimension and model predictable variables. Currently, a modified form of generalized cross-validation (GCV*), first suggested by Craven and Wahba, is used for model selection within the MARS algorithm. However, one question that immediately develops is whether GCV* Is the 'best' criterion for model selection when using serially and cross-correlated time series data. Using MSE as a measure of performance, simulations show that othr model selection criteria, in particular, the Schwarz-Rissanen (SC) criterion can improve model election over GCV*).