Bounded optimal knots for regression splines

• Published in: Computational statistics & data analysis Vol. 45; no. 2; pp. 159 - 178
• Main Authors: Molinari, Nicolas, Durand, Jean-François, Sabatier, Robert
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2004; Elsevier Science; Elsevier
• Series: Computational Statistics & Data Analysis
• Subjects: Additive models; AIC; BIC; Bound constrained optimization; Exact sciences and technology; Free knots selection; Linear inference, regression; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Numerical methods in mathematical programming, optimization and calculus of variations; Numerical methods in optimization and calculus of variations; Probability and statistics; Sciences and techniques of general use; Statistics; Surface estimation; Free knots selection; Surface estimation; AIC; Bound constrained optimization; BIC; Additive models; B spline; Statistical method; Free knot selection; Additive process; Least squares method; Regression spline; AIC and BIC; Spline approximation; Constrained optimization
• ISSN: 0167-9473; 1872-7352
• DOI: 10.1016/S0167-9473(02)00343-2

Using a B-spline representation for splines with knots seen as free variables, the approximation to data by splines improves greatly. The main limitations are the presence of too many local optima in the univariate regression context, and it becomes even worse in multivariate additive modeling. When the number of knots is a priori fixed, we present a simple algorithm to select their location subject to box constraints for computing least-squares spline approximations. Despite its simplicity, or perhaps because of it, the method is comparable with other more sophisticated techniques and is very attractive for a small number of variables, as shown in the examples. In a complete algorithm, the BIC and AIC criteria are evaluated for choosing the number of knots as well as the degree of the splines.