Approximation to Data by Splines with Free Knots

Abstract

Approximations to data by splines improve greatly if the knots are free variables. Using the B-spline representation for splines, and separating the linear and nonlinear aspects, the approximation problem reduces to nonlinear least squares in the variable knots. We describe the problems encountered in this formulation caused by the “lethargy” theorem, and how a logarithmic transformation of the knots can lead to an effective method for computing free knot spline approximations.