Robustness of higher-order regression using repeated medians

In this work, we extend the concept of robust regressions via repeatedly taking the median to nonlinear functions. In the presented method, two nested medians – the Siegel slope – are taken of the n th data derivative for each evaluated polynomial order n , beginning at the highest one. Lower-order coefficients are determined consecutively, by correcting the data for the previously determined ones. We demonstrate and quantify that this algorithm has a high robustness against outliers, with a breakdown characteristic that matches the original linear implementation. We also quantify and rationalize that varying polynomial orders and noise influence the algorithm’s accuracy. The presented method exhibited an increased accuracy and robustness over a wide parameter field, when compared to those of an implementation of the Levenberg–Marquardt algorithm, as well as the method proposed by Theil, namely applying consecutive single median regressions.