@@ -11,15 +11,15 @@ A stand-alone Theil-Sen estimator for robust simple regression in Matlab.
A [Theil-Sen estimator](https://en.wikipedia.org/wiki/Theil%E2%80%93Sen_estimator) provides robust, simple linear regression in the 2D plane:
The resulting estimates of slope and intercept are relatively insensitive to outliers.
The 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 implementation of [TheilSen.m](TheilSen.m) is exact but naïve:
It generates the set of all pairs of the _n_ input samples, resulting in an overall complexity of _O(n²)_ in speed and space.
The resulting slope and offset are the median slope and offset of the lines defined by all data point pairs.
(Note that other implementations of the algorithm achieve better complexity, and are thus much faster for large amounts of data points.)
(Note that alternative implementations of the algorithm have lower complexity, and thus much faster for large amounts of input samples.)
### 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.
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.
...
...
@@ -36,9 +36,10 @@ Please refer to the comments in the header lines of [TheilSen.m](TheilSen.m).
### Example
The script [example.m](example.m) simulates data based on known, true values.
It then fits and compares the Least-Squares with the Theil-Sen estimator.
Note how a few "unlucky" outliers can bias the least-squares estimate, but have little effect on the Theil-Sen estimator.
The script [example.m](example.m) simulates data based on known "true" values with minor, additive Gaussian noise.
The data are then corrupted with a small percentage of outliers.
It then fits and compares the least squares with the Theil-Sen estimator.
Note how a few "unlucky" outliers can bias the least squares estimate (LS), but have little effect on the Theil-Sen estimator (TS).
<imgsrc="example.svg"alt="plot from example.m"width=500px/>
...
...
@@ -46,17 +47,17 @@ Note how a few "unlucky" outliers can bias the least-squares estimate, but have
- October 2014 by Z. Danziger: Original version.
- September 2015 by Z. Danziger: Updated help, speed increase for 2D case
- March 2022 by J. Keyser: Adjusted formatting, added documentation, improved example and added plot, replace `nanmedian(X)` with `median(X, 'omitnan')`, ...
- March 2022 by J. Keyser: Adjusted formatting, added documentation, improved example and added plot, replaced`nanmedian(X)` with `median(X, 'omitnan')`, removed 2D special case, restructured input and output parameters.
## Contributing and project status
This project is relatively unmaintained, and only shared as-is, in the hope to be helpful.
This project is relatively unmaintained, and only shared as-is in the hope to be helpful.
If you find a bug, feel free to let the author(s) know.
Feature requests should be directed to the original author.
Feature requests should be directed to the original author (see below).
## Authors
1. Zachary Danziger, original author ([Matlab profile](https://de.mathworks.com/matlabcentral/profile/authors/1044524), [Lab webpage](https://anil.fiu.edu/))
1. Zachary Danziger, original author ([Matlab profile](https://de.mathworks.com/matlabcentral/profile/authors/1044524), [lab webpage](https://anil.fiu.edu/))
