Three and Higher Dimensional Spaces

Based on the recent development of two dimensional `1 major component detection and analysis (`1 MCDA), we develop a scalable `1 MCDA in the n-dimensional space to identify the major directions of star-shaped heavy-tailed statistical distributions with irregularly positioned “spokes” and “clutters”. In order to achieve robustness and efficiency, the proposed `1 MCDA in n-dimensional space adopts a two-level median fit process in a local neighbor of a given direction in each iteration. Computational results indicate that in terms of accuracy `1 MCDA is competitive with two well-known PCAs when there is only one major direction in the data, and `1 MCDA can further determine multiple major directions of the n-dimensional data from superimposed Gaussians or heavy-tailed distributions without and with patterned artificial outliers. With the ability to recover complex spoke structures with heavy-tailed noise and clutter in the data, `1 MCDA has potential to generate better semantics than other methods.