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Tensor Decompositions and Applications
 SIAM REVIEW
, 2009
"... This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal proce ..."
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Cited by 228 (14 self)
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This survey provides an overview of higherorder tensor decompositions, their applications, and available software. A tensor is a multidimensional or N way array. Decompositions of higherorder tensors (i.e., N way arrays with N â¥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, etc. Two particular tensor decompositions can be considered to be higherorder extensions of the matrix singular value decompo
sition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rankone tensors, and the Tucker decomposition is a higherorder form of principal components analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The Nway Toolbox and Tensor Toolbox, both for MATLAB, and the Multilinear Engine are examples of software packages for working with tensors.
Evaluating the use of exploratory factor analysis in psychological research
 Psychological Methods
, 1999
"... Despite the widespread use of exploratory factor analysis in psychological research, researchers often make questionable decisions when conducting these analyses. This article reviews the major design and analytical decisions that must be made when conducting a factor analysis and notes that each of ..."
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Cited by 65 (0 self)
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Despite the widespread use of exploratory factor analysis in psychological research, researchers often make questionable decisions when conducting these analyses. This article reviews the major design and analytical decisions that must be made when conducting a factor analysis and notes that each of these decisions has important consequences for the obtained results. Recommendations that have been made in the methodological literature are discussed. Analyses of 3 existing empirical data sets are used to illustrate how questionable decisions in conducting factor analyses can yield problematic results. The article presents a survey of 2 prominent journals that suggests that researchers routinely conduct analyses using such questionable methods. The implications of these practices for psychological research are discussed, and the reasons for current practices are reviewed. Since its initial development nearly a century ago (Spearman, 1904, 1927), exploratory factor analysis (EFA) has been one of the most widely used statistical procedures in psychological research. Despite this
Optimal Solutions for Sparse Principal Component Analysis
"... Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applica ..."
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Cited by 41 (8 self)
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Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n 3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n 3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.
A review of dimension reduction techniques
, 1997
"... The problem of dimension reduction is introduced as a way to overcome the curse of the dimensionality when dealing with vector data in highdimensional spaces and as a modelling tool for such data. It is defined as the search for a lowdimensional manifold that embeds the highdimensional data. A cl ..."
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Cited by 30 (4 self)
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The problem of dimension reduction is introduced as a way to overcome the curse of the dimensionality when dealing with vector data in highdimensional spaces and as a modelling tool for such data. It is defined as the search for a lowdimensional manifold that embeds the highdimensional data. A classification of dimension reduction problems is proposed. A survey of several techniques for dimension reduction is given, including principal component analysis, projection pursuit and projection pursuit regression, principal curves and methods based on topologically continuous maps, such as Kohonen’s maps or the generalised topographic mapping. Neural network implementations for several of these techniques are also reviewed, such as the projection pursuit learning network and the BCM neuron with an objective function. Several appendices complement the mathematical treatment of the main text.
Nonlinear multivariate and time series analysis by neural network methods
 Reviews of Geophysics
, 2004
"... [1] Methods in multivariate statistical analysis are essential for working with large amounts of geophysical data, data from observational arrays, from satellites, or from numerical model output. In classical multivariate statistical analysis, there is a hierarchy of methods, starting with linear re ..."
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Cited by 26 (13 self)
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[1] Methods in multivariate statistical analysis are essential for working with large amounts of geophysical data, data from observational arrays, from satellites, or from numerical model output. In classical multivariate statistical analysis, there is a hierarchy of methods, starting with linear regression at the base, followed by principal component analysis (PCA) and finally canonical correlation analysis (CCA). A multivariate time series method, the singular spectrum analysis (SSA), has been a fruitful extension of the PCA technique. The common drawback of these classical methods is that only linear structures can be correctly extracted from the data. Since the late 1980s, neural network methods have become popular for performing nonlinear regression and classification. More recently, neural network methods have been extended to perform nonlinear PCA (NLPCA), nonlinear CCA (NLCCA), and nonlinear SSA (NLSSA). This paper presents a unified view of the NLPCA, NLCCA, and NLSSA techniques and their applications to various data sets of the atmosphere and the ocean (especially for the El NiñoSouthern Oscillation and the stratospheric quasibiennial oscillation). These data sets reveal that the linear methods are often too simplistic to describe realworld systems, with a tendency to scatter a single oscillatory phenomenon into numerous unphysical modes or higher harmonics, which can be largely alleviated in the new nonlinear paradigm. INDEX TERMS: 3299
Principal Component Analysis
 (IN PRESS, 2010). WILEY INTERDISCIPLINARY REVIEWS: COMPUTATIONAL STATISTICS, 2
, 2010
"... Principal component analysis (pca) is a multivariate technique that analyzes a data table in which observations are described by several intercorrelated quantitative dependent variables. Its goal is to extract the important information from the table, to represent it as a set of new orthogonal var ..."
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Cited by 25 (5 self)
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Principal component analysis (pca) is a multivariate technique that analyzes a data table in which observations are described by several intercorrelated quantitative dependent variables. Its goal is to extract the important information from the table, to represent it as a set of new orthogonal variables called principal components, and to display the pattern of similarity of the observations and of the variables as points in maps. The quality of the pca model can be evaluated using crossvalidation techniques such as the bootstrap and the jackknife. Pca can be generalized as correspondence analysis (ca) in order to handle qualitative variables and as multiple factor analysis (mfa) in order to handle heterogenous sets of variables. Mathematically, pca depends upon the eigendecomposition of positive semidefinite matrices and upon the singular value decomposition (svd) of rectangular matrices.
Individual differences in chromatic (red/green) contrast sensitivity are constrained by the relative number of L versus Mcones in the eye
, 2002
"... ..."
ThreeDimensional Projection Pursuit
 J. Royal Statistical Society, Series C
, 1995
"... This article discusses various aspects of projection pursuit into three dimensions. The aim of projection pursuit is to find interesting linear combinations of variables in a multivariate data set. The precise definition of "interesting" is given later but clusters and other forms of nonlinear stru ..."
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Cited by 10 (0 self)
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This article discusses various aspects of projection pursuit into three dimensions. The aim of projection pursuit is to find interesting linear combinations of variables in a multivariate data set. The precise definition of "interesting" is given later but clusters and other forms of nonlinear structure are interesting. One and twodimensional projection pursuit have been dealt with extensively in the literature and some excellent software implementations are available. The benefit of projection into threedimensions is that more complex structures can be identified than with lowerdimensional projections. Projection pursuit into three dimensions is particularly attractive for two further perceptual reasons. Firstly, colours naturally correspond to 3vectors, for example through the RGB representation. Secondly, point clouds and other objects in three dimensions can be investigated on computer screens. For example through spinning 3D plots, which are immediately comprehensible because of our 3D intuition. These reasons are important when applying 3D projection pursuit to multispectral images (colour) and multivariate data sets (intuition). Section 2 briefly describes projection pursuit and includes details on projection indices and the process of sphering. Section 3 explains that we have chosen to extend Jones and Sibson's (1987) wellknown moments index into three dimensions because of its computational efficiency. The formulae for the moments index were analytically computed by the computer algebra package REDUCE (see Section 3.3). Section 3 also addresses the differentiation and optimization of the moments index, examines how outliers can be treated to provide better projection solutions and discusses how optimal projections can be rotated to give solutions that a...
Dimensionality reduction of electropalatographic data using latent variable models
 Speech Communication
, 1998
"... We consider the problem of obtaining a reduced dimension representation of electropalatographic (EPG) data. An unsupervised learning approach based on latent variable modelling is adopted, in which an underlying lower dimension representation is inferred directly from the data. Several latent variab ..."
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Cited by 8 (3 self)
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We consider the problem of obtaining a reduced dimension representation of electropalatographic (EPG) data. An unsupervised learning approach based on latent variable modelling is adopted, in which an underlying lower dimension representation is inferred directly from the data. Several latent variable models are investigated, including factor analysis and the generative topographic mapping (GTM). Experiments were carried out using a subset of the EURACCOR database, and the results indicate that these automatic methods capture important, adaptive structure in the EPG data. Nonlinear latent variable modelling clearly outperforms the investigated linear models in terms of loglikelihood and reconstruction error and suggests a substantially smaller intrinsic dimensionality for the EPG data than that claimed by previous studies. A twodimensional representation is produced with applications to speech therapy, language learning and articulatory dynamics.