Recursive Construction of Optimal Smoothing Splines Generated by Linear Control Systems

Abstract

This paper considers the problem of constructing smoothing spline curves recursively each time when a new set of data is observed. The spline curves are constituted by employing an approach based on linear control systems. Based on the basic problem of optimal smoothing splines and an idea of recursive least squares method, a recursive design algorithm of optimal smoothing splines is developed. Assuming that the data for smoothing is obtained by sampling a curve, we analyze asymptotical and statistical properties of smoothing spline curve when the number of iterations tends to infinity. The algorithm and analyses are extended to the case of periodic splines. We demonstrate the effectiveness and usefulness by numerical examples.