An Estimated General Cross Validation Function for Periodic Control Theoretic Smoothing Splines

Abstract

In this paper, a method is developed for estimating the optimal smoothing parameter ε for periodic control theoretic smoothing splines. The procedure is based on general cross validation (GCV ) and requires no a priori information about the underlying curve or level of noise in the measurements. The optimal ε is the minimizer of a GCV cost function, which is derived based on a discretization of the L 2 smoothing problem for periodic control theoretic smoothing splines. Keywords: general cross validation, optimal smoothing, influence matrix, periodic control theoretic smoothing splines E.1 IntroductionIn this paper, we consider the problem of estimating representations of objects or contours using a type of continuous closed curves, the periodic control theoretic smoothing splines. The splines are retrieved from noisy measurements of an unknown, underlying contour. Intended applications include mapping, identification and path planning for autonomous agents. The focus of this paper is the issue of the level of smoothing, determined by the magnitude of the so called smoothing parameter ε.It is well known that an interpolating spline generated from noisy measurements yields a poor estimate of the underlying curve, as the spline will pass through every measurement point. An interpolating spline may be regarded as a smoothing spline with ε = ∞, while a periodic smoothing spline with ε → 0 approaches a circle. The theory of regular smoothing splines, and the particular issue of choice of smoothing parameter, is treated in [13][14][15]. Control theoretic smoothing splines have been studied in [1][2][3][4][5][6][7] and it has been shown in [5] that such splines, where the curve is found through minimizing a cost function, act as filters and are better suited for noisy measurements. A thorough treatment of control theoretic smoothing splines is provided in the book [8].The particular type of periodic control theoretic smoothing spline explored in this paper has been previously presented in [9][10][11]. These publications cover error convergence properties for a recursive formulation of the smoothing spline problem. Experimental results indicate that the convergence is fairly robust with respect to the choice of smoothing parameter ε, but a formal 109