Principal Points and Self-Consistent Points of Elliptical Distributions

Tarpey, Thaddeus; Li, Luning; Flury, Bernard D.

doi:10.1214/aos/1176324457

Cited by 64 publications

(37 citation statements)

References 12 publications

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“…A vector quantizer is constrained to have at most k points, whereas a principal curve has a constrained length. This connection is further illustrated by a recent work of Tarpey et al [17] who define points y I Y F F F Y y k to be self-consistent if…”

Section: Remark (Connection With Vector Quantization)mentioning

confidence: 87%

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Learning and design of principal curves

Kégl

Krzyżak

Linder

et al. 2000

IEEE Trans. Pattern Anal. Machine Intell.

292

277

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AbstractÐPrincipal curves have been defined as ªself-consistentº smooth curves which pass through the ªmiddleº of a d-dimensional probability distribution or data cloud. They give a summary of the data and also serve as an efficient feature extraction tool. We take a new approach by defining principal curves as continuous curves of a given length which minimize the expected squared distance between the curve and points of the space randomly chosen according to a given distribution. The new definition makes it possible to theoretically analyze principal curve learning from training data and it also leads to a new practical construction. Our theoretical learning scheme chooses a curve from a class of polygonal lines with k segments and with a given total length to minimize the average squared distance over n training points drawn independently. Convergence properties of this learning scheme are analyzed and a practical version of this theoretical algorithm is implemented. In each iteration of the algorithm, a new vertex is added to the polygonal line and the positions of the vertices are updated so that they minimize a penalized squared distance criterion. Simulation results demonstrate that the new algorithm compares favorably with previous methods, both in terms of performance and computational complexity, and is more robust to varying data models.

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Section: Remark (Connection With Vector Quantization)mentioning

confidence: 87%

“…Thus, our principal curves correspond to optimal vector quantizers (ªprincipal pointsº by the terminology of [17]), while the HS principal curves correspond to self-consistent points.…”

Section: Remark (Connection With Vector Quantization)mentioning

confidence: 99%

Learning and design of principal curves

Kégl

Krzyżak

Linder

et al. 2000

IEEE Trans. Pattern Anal. Machine Intell.

292

277

View full text Add to dashboard Cite

show abstract

“…Otherwise, continue by letting A 2 denote the set of conditional means of the elements in A 1 over their respective domains of attraction, and so on, until convergence is reached. Similar ideas are used as well in the computation of semiparametric estimators of principal points (Flury, 1993), which are based on the k-means algorithm but restricted to follow certain patterns of principal points as suggested by the theory of principal points for elliptical distributions (Tarpey, Li and Flury, 1995).…”

Section: Discussionmentioning

confidence: 99%

“…The main result of this section, Theorem 4.1, is called the principal subspace theorem. It appeared originally in Tarpey, Li and Flury (1995) for the special case of selfconsistent approximations whose support consists of k distinct points. We show the theorem for random vectors X such that, for any orthogonal matrix A, the conditional mean of any subset of variables in A X, given the remaining variables, is linear.…”

Section: The Principal Subspace Theorem and Linear Principal Componentsmentioning

confidence: 99%

Self-consistency: a fundamental concept in statistics

Tarpey¹,

Flury²

1996

Statist. Sci.

Self Cite

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Abstract. The term "self-consistency" was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if ‫ޅ‬ X Y = Y almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give examples, show relationships between the various methods, discuss a related notion of self-consistent estimators and suggest directions for future research.

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“…In fact, principal points are applied to various problems, including product design and quality control problems, such as finding an optimal set of designs for masks (Flury, 1993), selective assembly for two parts in the manufacturing field (Mease et al, 2004;Mease and Nair, 2006;Matsuura, 2011), functional data analysis (Tarpey and Kinateder, 2003;Shimizu and Mizuta, 2007;Shimizu and Mizuta, 2008), and classification analysis of the placebo effect (Tarpey, et al, 2010). The properties of principal points have been widely studied in the literature (e.g., Tarpey and Flury, 1995;Zoppè, 1995;Yamamoto and Shinozaki, 2000;Gu and Mathew, 2001;Kurata, 2008;Bali and Boente, 2009;Matsuura and Kurata, 2011;Matsuura and Kurata, 2014), and the parametric estimation of principal points of some continuous distributions has also been discussed (e.g., Flury, 1993;Tarpey, 1997;Stampfer and Stadlober, 2002;Tarpey, 2007).…”

Section: Introductionmentioning

confidence: 99%

Comparison of model selection methods for the estimation of principal points for a multivariate binary distribution

Yamashita

Matsuura

Suzuki

2015

TQS

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Abstract:Recently, a parametric estimation method for principal points for a multivariate binary distribution using a log-linear model has been proposed, and Akaike information criterion (AIC) has been applied to model selection for log-linear model. This paper compares three model selection methods based on AIC, Bayesian information criterion (BIC), and the likelihood ratio test (LRT) for estimating principal points for a multivariate binary distribution. The performances of the model selection methods are shown through numerical simulation studies

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Principal Points and Self-Consistent Points of Elliptical Distributions

Cited by 64 publications

References 12 publications

Learning and design of principal curves

Learning and design of principal curves

Self-consistency: a fundamental concept in statistics

Comparison of model selection methods for the estimation of principal points for a multivariate binary distribution

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