Variants Of Multivariate Adaptive Regression Splines (MARS): Convex vs. Non-convex, Piecewise-linear vs. Smooth And Sequential Algorithms.

Abstract

Multivariate Adaptive Regression Splines (MARS) is a statistical modeling method used to represent high-dimensional data with interactions. It uses different algorithms to select the terms to be included in the approximation model that best represent the data. In addition, it performs a variable selection, therefore the most significant predictors are shown in the final model. Design and analysis of computer experiments (DACE) is a statistical technique for creating approximations (called metamodels) of computer models. For optimization problems in which there is an unknown function that must be approximated, DACE approach could be applied. In stochastic dynamic programming (SDP) for example, a metamodel can be used to approximate the unknown future value function.The goal of DACE is to efficiently predict the response value of a computer model. MARS has been used as a metamodel in DACE technique. MARS is a flexible model, however in optimization, certain characteristics may be desired, such as a convex or piecewise-linear structure. To satisfy these characteristics, different variants of MARS have been developed. By enabling these variants, MARS modeling facilitates the optimization process. These variations include the ability to model a convex function, a piecewise-linear function and to provide a smoothing option using a quintic routine.DACE has had an enormous contribution for studying complex system, however one of consistent concerns for the researchers is computational time. As researchers seek to study more and more complex systems, corresponding computer models continue to push the limits of computing power. To overcome this drawback, efficient sequential approaches have been studied to reduce the computational effort. This research work focuses its efforts on the development of sequential approaches based on MARS model. The objective is to sequentially update the approximation function using current and new input data points. Additionally, by using less input data points, an accurate prediction of the unknown function could be obtained in a faster manner, and thus the complexity of the model structure is less. This could also facilitate the optimization process.Different case studies are shown in order to test the different MARS variants and sequential MARS approaches proposed in this dissertation. These cases include an inventory forecasting problem, an automotive crash safety design problem and an air pollution SDP problem.