Smoothing noisy data with spline functions - Estimating the correct degree of smoothing by the method of generalized cross-validation

Abstract

Smoothing splines are well known to provide nice curves which smooth discrete, noisy data. We obtain a practical, effective method for estimating the optimum amount of smoothing from the data. Derivatives can be estimated from the data by differentiating the resulting (nearly) optimally smoothed spline. We consider the model yi(ti)+εi, i=1, 2, ..., n, ti∈[0, 1], where g∈W2(m)={f:f, f′, ..., f(m-1) abs. cont., f(m)∈ℒ2[0,1]}, and the {εi} are random errors with Eεi=0, Eεiεj=σ2δij. The error variance σ2 may be unknown. As an estimate of g we take the solution gn, λ to the problem: Find f∈W2(m) to minimize {Mathematical expression}. The function gn, λ is a smoothing polynomial spline of degree 2 m-1. The parameter λ controls the tradeoff between the "roughness" of the solution, as measured by {Mathematical expression}, and the infidelity to the data as measured by {Mathematical expression}, and so governs the average square error R(λ; g)=R(λ) defined by {Mathematical expression}. We provide an estimate {Mathematical expression}, called the generalized cross-validation estimate, for the minimizer of R(λ). The estimate {Mathematical expression} is the minimizer of V(λ) defined by {Mathematical expression}, where y=(y1, ..., yn)t and A(λ) is the n×n matrix satisfying (gn, λ (t1), ..., gn, λ (tn))t=A (λ) y. We prove that there exist a sequence of minimizers {Mathematical expression} of EV(λ), such that as the (regular) mesh {ti}i=1n becomes finer, {Mathematical expression}. A Monte Carlo experiment with several smooth g's was tried with m=2, n=50 and several values of σ2, and typical values of {Mathematical expression} were found to be in the range 1.01-1.4. The derivative g′ of g can be estimated by {Mathematical expression}. In the Monte Carlo examples tried, the minimizer of {Mathematical expression} tended to be close to the minimizer of R(λ), so that {Mathematical expression} was also a good value of the smoothing parameter for estimating the derivative. © 1979 Springer-Verlag.