A fast, stand-alone Theil-Sen estimator for robust regression in Matlab.
A stand-alone Theil-Sen estimator for robust regression in Matlab.
(Stand-alone: No toolbox required.)
### Theil-Sen estimator
### Theil-Sen estimator
A [Theil-Sen estimator](https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator) provides robust linear regression in the 2D plane:
A [Theil-Sen estimator](https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator) provides robust linear regression in the 2D plane:
The resulting estimates of slope and intercept are relatively insensitive to outliers.
The resulting estimates of slope and intercept are relatively insensitive to outliers.
FIXME: Mention main idea (data point pairs), and resulting complexity.
The present implementation of [TheilSen.m](TheilSen.m) is exact but "naive":
It generates the set of all pairs of the _n_ input samples, resulting in an overall complexity of _O(n²)_ in both speed and space.
The resulting slope and offset are the median slope and offset of all lines defined by the data point pairs.
(Note that other implementations of the algorithm achieve better complexity, and are thus much faster for large amounts of data points.)
### No dependency on Matlab toolboxes
### No toolbox required
This code is based on [Theil-Sen Robust Linear Regression](https://mathworks.com/matlabcentral/fileexchange/48294-theil-sen-robust-linear-regression), version 1.2.0.0, by [Zachary Danziger](https://mathworks.com/matlabcentral/profile/authors/1044524).
This code is based on [Theil-Sen Robust Linear Regression](https://mathworks.com/matlabcentral/fileexchange/48294-theil-sen-robust-linear-regression), version 1.2.0.0, by [Zachary Danziger](https://mathworks.com/matlabcentral/profile/authors/1044524).
A key modification was to use `median(X, 'omitnan')` instead of `nanmedian(X)` to avoid dependency on the (commercially licensed) [Statistics Toolbox](https://mathworks.com/products/statistics.html).
A key modification is to use `median(X, 'omitnan')` instead of `nanmedian(X)` to avoid dependency on the (commercially licensed) [Statistics Toolbox](https://mathworks.com/products/statistics.html).
See the [changelog](#changelog) below for further modifications.
(Note that there are several other implementations of the Theil-Sen estimator on Mathworks's [File Exchange](https://mathworks.com/matlabcentral/fileexchange).
(Note that there are several other implementations of the Theil-Sen estimator on Mathworks's [File Exchange](https://mathworks.com/matlabcentral/fileexchange).
Unfortunately all were found to depend on the Statistics Toolbox, [except one](https://mathworks.com/matlabcentral/fileexchange/43135-regression-utilities), which was judged to be slower and less versatile.)
Unfortunately all were found to depend on the Statistics Toolbox, [except one](https://mathworks.com/matlabcentral/fileexchange/43135-regression-utilities), which was judged to be slower and less versatile.)
