Geometric Properties of Principal Curves in the Plane

Abstract

Principal curves were introduced to formalize the notion of “a curve passing through the middle of a dataset”. Vaguely speaking, a curve is said to pass through the middle of a dataset if every point on the curve is the average of the observations projecting onto it. This idea can be made precise by defining principal curves for probability densities. Principal curves can be regarded as a generalization of linear principal components—if a principal curve happens to be a straight line, then it is a principal component. In this paper we study principal curves in the plane. We show that principal curves are solutions of a differential equation. By solving this differential equation, we find principal curves for uniform densities on rectangles and annuii. There are oscillating solutions besides the obvious straight and circular ones, indicating that principal curves in general will not be unique. If a density has several principal curves, they have to cross, a property somewhat analogous to the orthogonality of principal components. Finally, we study principal curves for spherical and elliptical distributions.