Smoothing noisy data with spline functions

Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m–1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm²n operations and therefore permits efficient orderm²n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern³ operations.