An exact algorithm for weighted-mean trimmed regions in any dimension

Trimmed regions are a powerful tool of multivariate data analysis. they describe a probability distribution in euclidean d-space regarding location,dispersion,and shape,and they order multivariate data with respect to their centrality. dyckerhoff and mosler (2011) have introduced the class of weighted-mean trimmed regions,which possess attractive properties regarding continuity,subadditivity,and monotonicity. we present an exact algorithm to compute the weighted-mean trimmed regions of a given data cloud in arbitrary dimension d. these trimmed regions are convex polytopes in r d. to calculate them,the algorithm builds on methods from computational geometry. a characterization of a region's facets is used,and information about the adjacency of the facets is extracted from the data. a key problem consists in ordering the facets. it is solved by the introduction of a tree-based order,by which the whole surface can be traversed efficiently with the minimal number of computations. the algorithm has been programmed in c++ and is available as the r package wmtregions.