A nonlinear PCA based on manifold approximation

Girard, Stáephane

doi:10.1007/s001800000025

Cited by 14 publications

(13 citation statements)

References 20 publications

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“…More recent research includes contributions to techniques for functional PCA (see for example Brumback and Rice (1998), Cardot (2000), Cardot et al. (2000), Girard (2000), James et al. (2000), Boente and Fraiman (2000) and He et al .…”

Section: Introductionmentioning

confidence: 99%

On Properties of Functional Principal Components Analysis

Hall

Hosseini‐Nasab

2005

Journal of the Royal Statistical Society Series B: Statistical Methodology

345

281

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Functional data analysis is intrinsically infinite dimensional; functional principal component analysis reduces dimension to a finite level, and points to the most significant components of the data. However, although this technique is often discussed, its properties are not as well understood as they might be. We show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results. Our approach quantifies the errors that arise through statistical approximation, in successive terms of orders "n"-super- - 1/2, "n"-super- - 1, "n"-super- - 3/2, …, where "n" denotes sample size. The expansions show how spacings among eigenvalues impact on statistical performance. The term of size "n"-super- - 1/2 illustrates first-order properties and leads directly to limit theory which describes the dominant effect of spacings. Thus, for example, spacings are seen to have an immediate, first-order effect on properties of eigenfunction estimators, but only a second-order effect on eigenvalue estimators. Our results can be used to explore properties of existing methods, and also to suggest new techniques. In particular, we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues, and also for individual eigenvalues and eigenvectors. Copyright 2006 Royal Statistical Society.

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Section: Introductionmentioning

confidence: 99%

On Properties of Functional Principal Components Analysis

Hall

Hosseini‐Nasab

2005

Journal of the Royal Statistical Society Series B: Statistical Methodology

345

281

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“…), it appears that the "optimal" axis maximizes the mean distance between the projected points. An attractive feature of the index (16) is that its maximization benefits from an explicit solution in terms of the eigenvectors of the empirical covariance matrix V k−1 defined in (13). Friedman et al [15,13], and more recently Hall [21], proposed an index to find clusters or use deviation from the normality measures to reveal more complex structures of the scatter-plot.…”

Section: Computation Of Principal Directionsmentioning

confidence: 99%

“…The matrix M k−1 = (m k−1 i,j ) is a first order contiguity matrix, whose value is 1 when R k−1 j is the nearest neighbor of R k−1 i , 0 otherwise. The upper part of (17) is proportional to the usual projected variance, see (16). The lower part is the distance between the projection of points which are nearest neighbor in R p .…”

Section: Computation Of Principal Directionsmentioning

confidence: 99%

Auto-Associative Models, Nonlinear Principal Component Analysis, Manifolds and Projection Pursuit

Girard

Iovleff

2008

Lecture Notes in Computational Science and Enginee

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Summary. Auto-associative models have been introduced as a new tool for building nonlinear Principal component analysis (PCA) methods. Such models rely on successive approximations of a dataset by manifolds of increasing dimensions. In this chapter, we propose a precise theoretical comparison between PCA and autoassociative models. We also highlight the links between auto-associative models, projection pursuit algorithms, and some neural network approaches. Numerical results are presented on simulated and real datasets. IntroductionPrincipal component analysis (PCA) is a well-known method for extracting linear structures from high-dimensional datasets. It computes the subspace best approaching the dataset from the Euclidean point of view. This method benefits from efficient implementations based either on solving an eigenvalue problem or on iterative algorithms. We refer to [27] for details. In a similar fashion, multi-dimensional scaling [3,35,44] addresses the problem of finding the linear subspace best preserving the pairwise distances. More recently, new algorithms have been proposed to compute low dimensional embeddings of high dimensional data. For instance, Isomap [46], LLE (Locally linear embedding) [42] and CDA (Curvilinear distance analysis) [9] aim at reproducing in the projection space the structure of the initial local neighborhood. These methods are mainly dedicated to visualization purposes. They cannot produce an analytic form of the transformation function, making it difficult to map new points into the dimensionality-reduced space. Besides, since they rely on local properties of pairwise distances, these methods are sensitive to noise and outliers. We refer to [38] for a comparison between Isomap and CDA and to [48] for a comparison between some features of LLE and Isomap.

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“…In more recent times the arguments have commonly been pressed into service in functional data analysis. For relatively recent contributions of this type, see, for example, Bosq (2000), Cardot (2000), Cardot et al (2000, 2003), Girard (2000), James et al (2000), Kneip and Utikal (2001), Boente and Fraiman (2000), He et al (2003), Yamanishi and Tanaka (2005), Hall et al (2006), Yao and Lee (2006), Cardot (2007), Cai and Hall (2006) and Hall and Hosseini‐Nasab (2006).…”

Section: Introductionmentioning

confidence: 99%

Ordering and Selecting Components in Multivariate or Functional Data Linear Prediction

Hall

Yang

2010

Journal of the Royal Statistical Society Series B: Statistical Methodology

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The problem of component choice in regression-based prediction has a long history. The main cases where important choices must be made are functional data analysis, and problems in which the explanatory variables are relatively high dimensional vectors. Indeed, principal component analysis has become the basis for methods for functional linear regression. In this context the number of components can also be interpreted as a smoothing parameter, and so the viewpoint is a little different from that for standard linear regression. However, arguments for and against conventional component choice methods are relevant to both settings and have received significant recent attention. We give a theoretical argument, which is applicable in a wide variety of settings, justifying the conventional approach. Although our result is of minimax type, it is not asymptotic in nature; it holds for each sample size. Motivated by the insight that is gained from this analysis, we give theoretical and numerical justification for cross-validation choice of the number of components that is used for prediction. In particular we show that cross-validation leads to asymptotic minimization of mean summed squared error, in settings which include functional data analysis. Copyright (c) 2010 Royal Statistical Society.

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A nonlinear PCA based on manifold approximation

Cited by 14 publications

References 20 publications

On Properties of Functional Principal Components Analysis

On Properties of Functional Principal Components Analysis

Auto-Associative Models, Nonlinear Principal Component Analysis, Manifolds and Projection Pursuit

Ordering and Selecting Components in Multivariate or Functional Data Linear Prediction

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