Fitting and Repeated Median Regression

Introduction

The repeated median algorithm is a robust method for fitting a line to sample points, and forms the basis for most data science applications in the AFA platform.

In this approach, the steps are:

1. The median slope, m_i, at some point (x_i,y_i) is calculated from taking the median of the slope against every other point in the dataset (y_j-y_i)(x_j-x_i).

2. The slope from the repeated median algorithm, M, is the median of m_i.

3. The y-intercept can be calculated from the median of (y_i - Mx_i)

Discussion

The estimator has a beak-down of 50%. In practical terms, the method can handle data with up to 50% erroneous values (or outliers) without substantial impact to the calculated output. For comparison, in traditional least squares reduction, the break-down point is 1/n, meaning that a single outlier can have substantial impact on results (unless perhaps using a massive data set compared to the number of outljiers).

Below is an idealised example of the impact of a few outliers on fitting using least squares as well as the Repeated Medians approach.

References

• https://en.wikipedia.org/wiki/Repeated_median_regression

• https://extremelearning.com.au/the-siegel-and-theil-sen-non-parametric-estimators-for-linear-regression/

• Siegel, Andrew F. (1982), "Robust regression using repeated medians", Biometrika, 69 (1): 242–244,