TY - JOUR

T1 - A convex version of multivariate adaptive regression splines

AU - Martinez, Diana L.

AU - Shih, Dachuan T.

AU - Chen, Victoria C.P.

AU - Kim, Seoung Bum

N1 - Funding Information:
This research was partially supported by the Dallas-Fort Worth International Airport contract #8002058 and National Science Foundation grant ECCS-0801802 .

PY - 2015/1

Y1 - 2015/1

N2 - Multivariate adaptive regression splines (MARS) provide a flexible statistical modeling method that employs forward and backward search algorithms to identify the combination of basis functions that best fits the data and simultaneously conduct variable selection. In optimization, MARS has been used successfully to estimate the unknown functions in stochastic dynamic programming (SDP), stochastic programming, and a Markov decision process, and MARS could be potentially useful in many real world optimization problems where objective (or other) functions need to be estimated from data, such as in surrogate optimization. Many optimization methods depend on convexity, but a non-convex MARS approximation is inherently possible because interaction terms are products of univariate terms. In this paper a convex MARS modeling algorithm is described. In order to ensure MARS convexity, two major modifications are made: (1) coefficients are constrained, such that pairs of basis functions are guaranteed to jointly form convex functions and (2) the form of interaction terms is altered to eliminate the inherent non-convexity. Finally, MARS convexity can be achieved by the fact that the sum of convex functions is convex. Convex-MARS is applied to inventory forecasting SDP problems with four and nine dimensions and to an air quality ground-level ozone problem.

AB - Multivariate adaptive regression splines (MARS) provide a flexible statistical modeling method that employs forward and backward search algorithms to identify the combination of basis functions that best fits the data and simultaneously conduct variable selection. In optimization, MARS has been used successfully to estimate the unknown functions in stochastic dynamic programming (SDP), stochastic programming, and a Markov decision process, and MARS could be potentially useful in many real world optimization problems where objective (or other) functions need to be estimated from data, such as in surrogate optimization. Many optimization methods depend on convexity, but a non-convex MARS approximation is inherently possible because interaction terms are products of univariate terms. In this paper a convex MARS modeling algorithm is described. In order to ensure MARS convexity, two major modifications are made: (1) coefficients are constrained, such that pairs of basis functions are guaranteed to jointly form convex functions and (2) the form of interaction terms is altered to eliminate the inherent non-convexity. Finally, MARS convexity can be achieved by the fact that the sum of convex functions is convex. Convex-MARS is applied to inventory forecasting SDP problems with four and nine dimensions and to an air quality ground-level ozone problem.

KW - Convexity

KW - Regression splines

UR - http://www.scopus.com/inward/record.url?scp=84906512170&partnerID=8YFLogxK

U2 - 10.1016/j.csda.2014.07.015

DO - 10.1016/j.csda.2014.07.015

M3 - Article

AN - SCOPUS:84906512170

VL - 81

SP - 89

EP - 106

JO - Computational Statistics and Data Analysis

JF - Computational Statistics and Data Analysis

SN - 0167-9473

ER -