An algorithm for data smoothing using spline functions

Abstract

In this paper we consider the problem of fitting a smooth curve to a set of experimental or tabulated data points. Given the set of data points (x _i, y_i),i=1, ...n, we determine the smooth curveg(x) from the condition that$\int\limits_{x_1 }^{x_n } {(g^{(m)} (x))^2 dx}$ (g ^(m)(x))² dx is minimal for allg(x) satisfying the smoothing constraint$\sum\limits_{i = 1}^n {\left( {\frac{{g(x_i ) - y_i }}{{\delta y_i }}} \right)^2 } \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} S$, wherem is a given positive integer, whereδy _i is usually an estimate of the standard deviation of the ordinatey _i and whereS is a constant usually chosen in the range (n+1)±$\sqrt {2(n + 1)}$.

It is shown that the smooth curveg(x) is a piecewise polynomial of degree 2m-1, having continuity of function values and first 2m-2 derivatives.

The problem was first outlined by Schoenberg [1]. Reinsch [2] gave an algorithm form=2. Anselone and Laurent [3] considered the problem for generalm using the methods of Functional Analysis. In this paper we produce an algorithm arising from the solution of the problem using a Lagrangian parameter.