Smoothing noisy data with spline functions

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 modely i (t i )+ε i ,i=1, 2, ...,n,t i∈[0, 1], whereg∈W 2 () ={f:f,f′, ...,f (m−1) abs. cont.,f (m)∈ℒ2[0,1]}, and the {ε i } are random errors withEε i =0,Eε i ε j =σ2δ ij . The error variance σ2 may be unknown. As an estimate ofg we take the solutiong n, λ to the problem: Findf∈W 2 (m) to minimize $$\frac{1}{n}\sum\limits_{j = 1}^n {(f(t_j ) - y_j )^2 + \lambda \int\limits_0^1 {(f^{(m)} (u))^2 du} }$$ . The functiong n, λ is a smoothing polynomial spline of degree 2m−1. The parameter λ controls the tradeoff between the “roughness” of the solution, as measured by $$\int\limits_0^1 {[f^{(m)} (u)]^2 du}$$ , and the infidelity to the data as measured by $$\frac{1}{n}\sum\limits_{j = 1}^n {(f(t_j ) - y_j )^2 }$$ , and so governs the average square errorR(λ; g)=R(λ) defined by $$R(\lambda ) = \frac{1}{n}\sum\limits_{j = 1}^n {(g_{n,\lambda } (t_j ) - g(t_j ))^2 }$$ . We provide an estimate $$\hat \lambda$$ , called the generalized cross-validation estimate, for the minimizer ofR(λ). The estimate $$\hat \lambda$$ is the minimizer ofV(λ) defined by $$V(\lambda ) = \frac{1}{n}\parallel (I - A(\lambda ))y\parallel ^2 /\left[ {\frac{1}{n}{\text{Trace(}}I - A(\lambda ))} \right]^2$$ , wherey=(y 1, ...,y n)t andA(λ) is then×n matrix satisfying(g n, λ (t 1), ...,g n, λ (t n))t=A (λ) y. We prove that there exist a sequence of minimizers $$\tilde \lambda = \tilde \lambda (n)$$ ofEV(λ), such that as the (regular) mesh{t i} i=1 n becomes finer, $$\mathop {\lim }\limits_{n \to \infty } ER(\tilde \lambda )/\mathop {\min }\limits_\lambda ER(\lambda ) \downarrow 1$$ . A Monte Carlo experiment with several smoothg's was tried withm=2,n=50 and several values of σ2, and typical values of $$R(\hat \lambda )/\mathop {\min }\limits_\lambda R(\lambda )$$ were found to be in the range 1.01–1.4. The derivativeg′ ofg can be estimated by $$g'_{n,\hat \lambda } (t)$$ . In the Monte Carlo examples tried, the minimizer of $$R_D (\lambda ) = \frac{1}{n}\sum\limits_{j = 1}^n {(g'_{n,\lambda } (t_j ) - } g'(t_j ))$$ tended to be close to the minimizer ofR(λ), so that $$\hat \lambda$$ was also a good value of the smoothing parameter for estimating the derivative.