Discrete smoothing splines and digital filtration. Theory and applications

Hřebíček, Jiří ; Šik, František ; Veselý, Vítězslav

Applications of Mathematics, Tome 35 (1990) no. 1, pp. 28-50

Résumé

Two universally applicable smoothing operations adjustable to meet the specific properties of the given smoothing problem are widely used: 1. Smoothing splines and 2. Smoothing digital convolution filters. The first operation is related to the data vector $r={(r_0,..., r_{n-1})}^T$ with respect to the operations $\Cal{A}$, $\Cal{L}$ and to the smoothing parameter $\alpha$. The resulting function is denoted by $\sigma_\alpha(t)$. The measured sample $r$ is defined on an equally spaced mesh $\Delta=\{t_i=ih\}^{n-1}_{i=0}$ $T=nh$. The smoothed data vector $y$ is then $y=\{\sigma_\alpha(t_i)\}^{n-1}_{i=0}$. The other operation gives $y\in E^n$ computed by $\bold {y=h*r}$, where $\bold *$ stands for the discrete convolution, the running weighted mean by $h$. The main aims of the present contribution: to prove the existence of close interconnection between the two smoothing approaches (Cor. 2.6 and [11]), to develop the transfer function, which characterizes the smoothing spline as a filter in terms of $\alpha$ and $\lambda_{ik}$ (the eigenvalues of the discrete analogue of $Cal {L}$) (Th. 2.5), to develop the reduction ratio between the original and the smoothed data in the same terms (Th. 3.1).

Détail

DOI : 10.21136/AM.1990.104385

Classification : 41A15, 65D07, 65D10, 65K10, 93E11, 93E14
Keywords: discrete smoothing spline CDS-spline; smoothing parameter; digital convolution filter; transfer function; sinusoidal wave; saw-like waves; rectangular pulse train