On the Dynamical System of Principal Curves in $\mathbb R^d$

Abstract

Principal curves are natural generalizations of principal lines arising as first principal components in the Principal Component Analysis. They can be characterized-from a stochastic point of view-as so-called self-consistent curves based on the conditional expectation andfrom the variational-calculus point of view-as saddle points of the expected difference of a random variable and its projection onto some curve, where the current curve acts as argument of the energy functional. Beyond that, Stützle (1993,1996) showed that planar curves can by computed as solutions of a system of ordinary differential equations. The aim of this paper is to generalize this characterization of principal curves to R d with d ≥ 3. Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in R 3 .