Robustness of kernel based regression: Influence and weight functions

It has been shown that kernel based regression (KBR) with a least squares loss has some undesirable properties from robustness point of view. KBR with more robust loss functions, e.g. Huber or Logistic losses, often give rise to more complicated computations. In classical statistics, robustness is improved by reweighting the original estimate. We study the influence of reweighting the LS-KBR estimate using three well-known weight functions and one new weight function called Myriad. Our results give practical guidelines in order to choose the weights, providing robustness and fast convergence. It turns out that Logistic and Myriad weights are suitable reweighting schemes when outliers are present in the data. In fact, the Myriad shows better performance over the others in the presence of extreme outliers (e.g. Cauchy distributed errors). These findings are then illustrated on toy example as well as on a real life data sets. Finally, we establish an empirical maxbias curve to demonstrate the ability of the proposed methodology.