Least squares approximation by splines with free knots

Abstract

Suppose we are given noisy data which are considered to be perturbed values of a smooth, univariate function. In order to approximate these data in the least squares sense, a linear combination of B-splines is used where the tradeoff between smoothness and closeness of the fit is controlled by a smoothing term which regularizes the least squares problem and guarantees unique solvability independent of the position of knots. Moreover, a subset of the knot sequence which defines the B-splines, the so-calledfree knots, is included in the optimization process. The resulting constrained least squares problem which is linear in the spline coefficients but nonlinear in the free knots is reduced to a problem that has only the free knots as variables. The reduced problem is solved by a generalized Gauss-Newton method. The method developed can be combined with a knot removal strategy in order to obtain an approximating spline with as few parameters as possible.