Another Look at Principal Curves and Surfaces

By Pedro Delicado. 2001

132 citations recorded by the source corpus.

Principal curves have been defined as smooth curves passing through the “middle” of a multidimensional data set. They are nonlinear generalizations of the first principal component, a characterization of which is the basis of the definition of principal curves. We establish a new characterization of the first principal component and base our new definition of a principal curve on this property. We introduce the notion of principal oriented points and we prove the existence of principal curves passing through these points. We extend the definition of principal curves to multivariate data sets and propose an algorithm to find them. The new notions lead us to generalize the definition of total variance. Successive principal curves are recursively defined from this generalization. The new methods are illustrated on simulated and real data sets.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/835660fcd8b86e908a5c6a1be015efc223166793