Construction algorithm of principal curves in the sense of limit

Authors:
Lianwei Zhao;Yanchang Zhao;Siwei Luo;Chao Shao
Affiliations:
School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China;Faculty of Information Technology, University of Technology, Sydney, Australia;School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China;School of Computer and Information Technology, Beijing Jiaotong University, Beijing, China
Venue:
AI'05 Proceedings of the 18th Australian Joint conference on Advances in Artificial Intelligence
Year:
2005

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Abstract

Principal curves have been defined as self-consistent, smooth, one-dimensional curves which pass through the middle of a multidimensional data set. They are nonlinear generalization of the first Principal Component. In this paper, we take a new approach by defining principal curves as continuous curves based on the local tangent space in the sense of limit. It is proved that this new principal curves not only satisfy the self-consistent property, but also are the unique existence for any given open covering. Based on the new definition, a new practical algorithm for constructing principal curves is given. And the convergence properties of this algorithm are analyzed. The new construction algorithm of principal curves is illustrated on some simulated data sets.