On the Dynamical System of Principal Curves in $\mathbb R^d$
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910865446404096 |
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| author | Beinert, Robert Bërdëllima, Arian Gräf, Manuel Steidl, Gabriele |
| author_facet | Beinert, Robert Bërdëllima, Arian Gräf, Manuel Steidl, Gabriele |
| contents | 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 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 $\mathbb R^d$ 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 $\mathbb R^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_00227 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the Dynamical System of Principal Curves in $\mathbb R^d$ Beinert, Robert Bërdëllima, Arian Gräf, Manuel Steidl, Gabriele Dynamical Systems 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 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 $\mathbb R^d$ 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 $\mathbb R^3$. |
| title | On the Dynamical System of Principal Curves in $\mathbb R^d$ |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2108.00227 |