Robust locally weighted regression (Loess / Lowess)
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Key: MATH-278
URL: https://issues.apache.org/jira/browse/MATH-278
Project: Commons Math
Issue Type: New Feature
Reporter: Eugene Kirpichov
Attached is a patch that implements the robust Loess procedure for smoothing
univariate scatterplots with local linear regression (
http://en.wikipedia.org/wiki/Local_regression) described by William Cleveland
in http://www.math.tau.ac.il/~yekutiel/MA%20seminar/Cleveland%201979.pdf , with
tests.
(Also, the patch fixes one missing-javadoc checkstyle warning in the
AbstractIntegrator class: I wanted to make it so that the code with my patch
does not generate any checkstyle warnings at all)
I propose to include the procedure into commons-math because commons-math, as
of now, does not possess a method for robust smoothing of noisy data: there is
interpolation (which virtually can't be used for noisy data at all) and there's
regression, which has quite different goals.
Loess allows one to build a smooth curve with a controllable degree of
smoothness that approximates the overall shape of the data.
I tried to follow the code requirements as strictly as possible: the tests
cover the code completely, there are no checkstyle warnings, etc. The code is
completely written by myself from scratch, with no borrowings of third-party
licensed code.
The method is pretty computationally intensive (10000 points with a bandwidth
of 0.3 and 4 robustness iterations take about 3.7sec on my machine; generally
the complexity is O(robustnessIters * n^2 * bandwidth)), but I don't know how
to optimize it further; all implementations that I have found use exactly the
same algorithm as mine for the unidimensional case.
Some TODOs, in vastly increasing order of complexity:
- Make the weight function customizable: according to Cleveland, this is
needed in some exotic cases only, like, where the desired approximation is
non-continuous, for example.
- Make the degree of the locally fitted polynomial customizable: currently the
algorithm does only a linear local regression; it might be useful to make it
also use quadratic regression. Higher degrees are not worth it, according to
Cleveland.
- Generalize the algorithm to the multidimensional case: this will require A
LOT of hard work.
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