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

TL;DR

This paper generalizes the dynamical system characterization of principal curves from the plane to higher dimensions, providing a system of ODEs and examples in three-dimensional space.

Contribution

It extends the differential equation framework for principal curves to $\

Findings

Derived a dynamical system for principal curves in $\

Provided examples of principal curves in $\

Generalized planar results to $\

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 and from 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, Duchamp and St\"utzle (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 $\mathbb R^d$ with $d \ge 3$. Having derived such a dynamical system, we provide several examples for principal curves related to uniform distribution on certain domains in $\mathbb R^3$.