Weighted Repeated Median Smoothing and Filtering

Abstract

We propose weighted repeated median filters and smoothers for robust non-parametric regression in general and for robust signal extraction from time series in particular. The proposed methods allow to remove outlying sequences and to preserve discontinuities (shifts) in the underlying regression function (the signal) in the presence of local linear trends. Suitable weighting of the observations according to their distances in the design space reduces the bias arising from non-linearities. It also allows to improve the efficiency of (unweighted) repeated median filters using larger bandwidths, keeping their properties for distinguishing between outlier sequences and long-term shifts. Robust smoothers based on weighted L 1 -regression are included for the reason of comparison.

Robust Forecasting of Non-Stationary Time Series

“…The following four methods are included in this comparison: the regular Local Polynomial Regression (LPR), the Weighted Repeated Median (WRM) technique of Fried, Einbeck, and Gather (2007), the local polynomial M-smoother of Grillenzoni (2009), and finally the local polynomial MM-estimator proposed in this paper. In each method, we use the asymmetric exponential kernel, defined as K(x) = exp(x)I {x<0} .…”

“…, T . The signal m(t) is a sinusoidal function defined as m(t) = 12.5 sin(tπ/200), similar to Fried, Einbeck, and Gather (2007). We set σ(t) = m(t)/6 to bring in heteroscedasticity.…”

“…Local polynomial regression (see e.g. Fan and Gijbels, 1996), for instance, is a flexible technique but not robust; furthermore, like local polynomial M-regression (Grillenzoni, 2009) and the weighted repeated median (Fried, Einbeck, and Gather, 2007), it does not explicitly take a possible change of the variance over time into account.…”

An Adaptive Variable Block for Median Filter

“…The speckle noise or periodic noise can be removed in frequency domain by using band reject filtering or notch filtering [11][12][13]. Among them, the median filtering method has been the most widely used for denoising noisy images [14][15][16][17].…”

Discussion of ‘Beyond mean regression’

“…The modal smoother is compared with a local median smoother (see e.g. Fried et al, 2007) and a local constant mean (Nadaraya-Watson) smoother, using in either case the same horizontal bandwidth h = 8.…”