Error Estimates Derived from the Data for Least-Squares Spline Fitting

摘要

This is the first in a series of papers on a particular class of practical methods for extracting an accurate estimate of a signal from noisy measurements. The problem, in the simplest form that will be considered, is that a signal s(t) is measured at uniformly spaced discrete times ti for i = 1 to N. The measurements have random noise with known statistics. Throughout this paper it will assumed that the measurement noise is white. However, for a few years, the author been successfully using these methods for problems in which the noise is not white and not even stationary, and the sampling very nonuniform. This problem was first systematically studied in its modern form in (1)-(3), though closely related problems were studied by Gauss (4) as far back as 1804. The measured signal is represented as y = s + e, where s is the true signal and e is the vector of measurement errors.