Simple consistent cluster methods based on redescending M-estimators with an application to edge identification in images

Abstract

We use the local maxima of a redescending M-estimator to identify cluster, a method proposed already by Morgenthaler (in: H.D. Lawrence, S. Arthur (Eds.), Robust Regression, Dekker, New York, 1990, pp. 105-128) for finding regression clusters. We work out the method not only for classical regression but also for orthogonal regression and multivariate location and show that all three approaches are special cases of a general approach which includes also other cluster problems. For the general case we show consistency for an asymptotic objective function which generalizes the density in the multivariate case. The approach of orthogonal regression is applied to the identification of edges in noisy images. r

Robust fitting of mixture regression models

“…Markatou (2000) and Shen, Yang, and Wang (2004) proposed using a weight factor for each data to robustify the estimation procedure for mixture regression models. There are also some related robust methods for linear clustering; see, for example, Hennig (2002Hennig ( , 2003, Mueller and Garlipp (2005), García-Escudero, Gordaliza, San Martín, Van Aelst, andZamar (2009), andGarcía-Escudero, Gordaliza, Mayo-Iscara, andSan Martín (2010).…”

Automatic Mapping of Forest Stands Based on Three-Dimensional Point Clouds Derived from Terrestrial Laser-Scanning

“…3D-points were then projected onto the horizontal plane, resulting in a two-dimensional point cloud with xy-coordinates. In a first step, a circle was fitted to the 2D-point cloud, using the circular cluster method of Müller and Garlipp [39], which is based on edge identification by redescending M-estimators (these are non-decreasing near the origin, but decreasing toward 0 far from the origin). The method is implemented in the R-package edci [40].…”

“…We used a dense grid of 25 × 25 starting points (i.e., possible circle centers) with a distance of 3.2 cm between two neighboring starting points, and 5 starting radii per starting point providing in total 3125 starting values (i.e., combinations of radius and centerpoint) for the optimization algorithm. The initial diameter estimate (obtained from the Müller and Garlipp algorithm [39]) is henceforward referred to as DBH 1 . In a second step, all data points with distance of greater than or equal to 5 cm to the circular arc associated with DBH 1 were removed.…”

A new method for robust mixture regression

“…Yao, Wei, & Yu () and Song, Yao, & Xing () considered robust mixture regression using a t ‐distribution and a Laplace distribution, respectively. There has also been extensive work in linear clustering; see, for example, Henning (, ), Mueller & Garlipp (), and García‐Escudero et al. (, ).…”