标题： 关于$\mathbb R^d$中主曲线的动力系统 显示英文标题

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

摘要： 主曲线是主成分分析中第一主成分作为主线条的自然推广。它们可以从随机的观点被表征为基于条件期望的所谓自洽曲线，也可以从变分法的观点被表征为随机变量与其在某条曲线上的投影之间的期望差的鞍点，其中当前曲线作为能量泛函的参数。除此之外，Duchamp 和 Stützle（1993,1996）表明平面曲线可以通过求解常微分方程组来计算。本文的目的是将主曲线的这种表征推广到$\mathbb R^d$中的$d \ge 3$。在推导出这样的动力系统后，我们提供了与$\mathbb R^3$中某些区域上的均匀分布相关的主曲线的几个例子。 显示英文摘要

摘要： 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 $\mathbb R^3$.