Journal ArticleDOI
Singular-value decomposition and embedding dimension
AI Mees,PE Rapp,LS Jennings +2 more
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TLDR
It is shown that a method which seems promising at first sight, estimating the rank of the matrix of embedded data, is unfortunately not useful in general, and can be avoided by a careful application of singular-value decomposition.Abstract:
Data from dynamical experiments are often studied with use of results due to Shaw et al. and to Takens, which generate points in a space of relatively high dimension by embedding measurements which are typically one dimensional. A number of questions arise from this, the most obvious being how should one choose the dimension of the embedding space. In this paper we show that a method which seems promising at first sight, estimating the rank of the matrix of embedded data, is unfortunately not useful in general. Previous encouraging results have almost certainly been due to numerical problems which can, in part, be avoided by a careful application of singular-value decomposition. We show that this process does not give useful dynamical information, though it is often useful in noise control.read more
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Journal ArticleDOI
The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview
Gaëtan Kerschen,Gaëtan Kerschen,Gaëtan Kerschen,Jean Claude Golinval,Alexander F. Vakakis,Alexander F. Vakakis,Lawrence A. Bergman +6 more
TL;DR: In this article, a different approach is adopted, and proper orthogonal decomposition is considered, and modes extracted from the decomposition may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant but unexpected structure hidden in the data.
Journal ArticleDOI
Proper orthogonal decomposition and its applications—part i: theory
TL;DR: The equivalence of the matrices for processing, the objective functions, the optimal basis vectors, the mean-square errors, and the asymptotic connections of the three POD methods are demonstrated and proved when the methods are used to handle the POD of discrete random vectors.