ℓ1-norm Based Major Component Detection and Analysis for Asymmetric Radial Data

Abstract

ℓ1-norm based major component detection and analysis (ℓ1 MCDA) is a state-of-the-art tool based exclusively on ℓ1-norm to identify the major components of a multivariate data set with irregularly positioned “spokes” and “clutters”. In this paper, we develop an algorithmic framework of ℓ1 MCDA for treating radial data clouds without the assumption of symmetry. This two-phase algorithm first locates the central point of the data by a pre-selection procedure to screen out candidate points with sufficient data points in the vicinity followed by solving an ℓ1-norm discrete minimization problem. It then calculates the major directions and median radii in those directions via a two-level median fitting process. Extensive computational experiments have been conducted on n-dimensional data sets of various configurations randomly generated from light-tailed and heavy-tailed distributions with possibly artificial outliers to support the accuracy and robustness of the proposed method.

Keywords

• ℓ1-norm
• Principal component analysis
• multivariate statistics