Aka Whittaker-Henderson, Whittaker-Eilers Smoother is known as the perfect smoother. Its a discrete-time version of spline smoothing for equally spaced data. It minimizes the functional
$$\sum{i=0}^n (zi - yi)^2 + \lambda \sum{i=0}^n (\delta ^p z)_i ^2 $$
where y are the datapoints, z is the smoothed function, and $\delta^2 z$ is the pth derivative of $z_i$, which is evaluated numerically. A penalty is imposed on nonsmooth functions, with higher values of $\lambda$ increasing the penalty and leading to a smoother output.
The smoothed output can be obtained by solving the linear system
$$x = (W + \lambda * D^T D )^{-1} W y $$
Where W is the weight matrix (Identity matrix in practice) and D is the difference matrix
(See difference_matrix for its construction).
Here we see the wood dataset smoothed whith both order 2 and 3.
Compared to a moving average smoother, this method does not suffer from a group-delay.
Compared to a convolution kernel such as the savitzky-golay filter, the values at the edge are well defined and don't need to be interpolated. The savitzky-golay filter does have a nice flat passband, but suffers from unsatisfactory high-frequency noise, which is not sufficiently suppressed. This is a particular problem when the derivative of the data is of importance.
To use this smoother in you project, add this to your Cargo.toml:
toml
[dependencies]
whittaker_smoother = "0.1"
See the papers folder for two papers showing additional details of the method.
This implementation was inspired by A python implementation.
Copyright (C) 2020
This program is free software: you can redistribute it and/or modify it under the terms of the GNU Affero General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.
This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Affero General Public License for more details.
You should have received a copy of the GNU Affero General Public License along with this program. If not, see https://www.gnu.org/licenses/.
