Whittaker Smoother

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

$$\sum_{i=0}^n (z_i - y_i)^2 + \lambda \sum_{i=0}^n (\delta ^p z)_i ^2 $$

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).

Examples

Here we see the wood dataset smoothed whith both order 2 and 3.

Comparison to Moving Averages and Convolution Kernels

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.

However, this smoother takes future values into account and therefore suffers from a look-ahead bias.

Usage

To use this smoother in you project, add this to your Cargo.toml:

[dependencies]
whittaker_smoother = "0.1"

Now you can use the smoothing function as such:

use whittaker_smoother::whittaker_smoother;

// Here we use the WOOD_DATASET, but this can be any series that you would like to smooth
let raw = Vec::from_iter(WOOD_DATASET.iter().map(|v| *v as f64));
let lambda = 2e4;
let order = 3;
let smoothed = whittaker_smoother(&raw, lambda, order).unwrap();

And BAM, that's it! There is you perfectly smoothed series.

Further Reading:

See the papers folder for two papers showing additional details of the method.

This implementation was inspired by A python implementation.

Potential Upgrades:

• Add benchmarks
• Use sparse matrices if available
• Add function for computing the optimal lambda based on cross-validation (See eilers2003)

License

Copyright (C) 2020 <Mathis Wellmann wellmannmathis@gmail.com>

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/.