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Restricted Canonical Correlations
, 1993
"... Given a pdimensional random variable yell and a qdimensional random variabley(2), the first canonical correlation leads to finding Q * E Rp and (3 * E R q which maximizes the correlation between Q'y(l) and (3'y(2). However, in many practical situations (e.g. educational testing problems, neural n ..."
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Given a pdimensional random variable yell and a qdimensional random variabley(2), the first canonical correlation leads to finding Q * E Rp and (3 * E R q which maximizes the correlation between Q'y(l) and (3'y(2). However, in many practical situations (e.g. educational testing problems, neural networks), some natural restrictions on the coefficients Q and (3 may arise which should be incorporated in the maximization procedure. The maximum correlation subject to these constraints is referred to as restricted canonical correlations. In this work the solution is obtained to the problem under the nonnegativity restriction on Q and (3. The analysis is extended"to more general form of inequality constraints, and also when the restrictions are present only on some of the coefficients. Also discussed are restricted versions of some other multivariate methods. This includes principal component analysis and different modifications of canonical correlation analysis. Some properties of restricted canonical correlations including its bounds have been studied. Since the restricted canonical correlation depends only on the covariance matrix, its sample version can naturally be obtained from sample covariance matrix. The asymptotic normality of this sample restricted canonical correlation is proved under reasonably mild conditions. The study of resampling methods becomes necessary because the asymptotic variance involves usually unknown fourth order moments of the population. The effectiveness of the jackknife method is shown in this case as well as for the usual canonical correlations. Bootstrapping also works out in both these cases. These theoretical results have been supplemented by simulation studies which compare the performances of the two resampling methods in the case of finite samples.
Biplots in reducedrank regression
 Biom. J
, 1994
"... SUMMARY Regression problems with a number of related response variables are typically analyzed by separate multiple regressions. This paper shows how these regressions can be visualized jointly in a biplot based on reducedrank regression. Reducedrank regression combines multiple regression and pri ..."
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SUMMARY Regression problems with a number of related response variables are typically analyzed by separate multiple regressions. This paper shows how these regressions can be visualized jointly in a biplot based on reducedrank regression. Reducedrank regression combines multiple regression and principal components analysis and can therefore be carried out with standard statistical packages. The proposed biplot highlights the major aspects of the regressions by displaying the leastsquares approximation of fitted values, regression coefficients and associated tratio's. The utility and interpretation of the reducedrank regression biplot is demonstrated with an example using public health data that were previously analyzed by separate multiple regressions.
Multivariate analysis of spatial patterns: a unified approach to local and global structures
 ENVIRONMENTAL AND ECOLOGICAL STATISTICS
, 1995
"... We propose a new approach to the multivariate analysis of data sets with known sampling site spatial positions. A betweensites neighbouring relationship must be derived from site positions and this relationship is introduced into the multivariate analyses through neighbouring weights (number of n ..."
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We propose a new approach to the multivariate analysis of data sets with known sampling site spatial positions. A betweensites neighbouring relationship must be derived from site positions and this relationship is introduced into the multivariate analyses through neighbouring weights (number of neighbours at each site) and through the matrix of the neighbouring graph. Eigenvector analysis methods (e.g., principal component analysis, correspondence analysis) can then be used to detect total, local and global structures. The introduction of the Dcentring (centring with respect to the neighbouring weights) allows us to write a total variance decomposition into local and global components, and to propose a unified view of several methods. After a brief review of the matrix approach to this problem, we present the results obtained on both simulated and real data sets, showing how spatial structure can be detected and analysed. Freely available computer programs to perform computations and graphical displays are proposed.
Interpreting canonical correlation analysis through biplots of structural correlations and weights
 Psychometrika
, 1990
"... This paper extends the biplot technique to canonical correlation analysis and redundancy analysis, The plot of structure correlations is shown to be optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression ..."
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This paper extends the biplot technique to canonical correlation analysis and redundancy analysis, The plot of structure correlations is shown to be optimal for displaying the pairwise correlations between the variables of the one set and those of the second. The link between multivariate regression and canonical correlation analysis/redundancy analysis is exploited for producing an optimal biplot that displays a matrix of regression coefficients. This plot can be made from the canonical weights of the predictors and the structure correlations of the criterion variables. An example is used to show how the proposed biptots may be interpreted. Key words: biplot, canonical correlation analysis, canonical weight, interbattery factor analysis, partial analysis, redundancy analysis, regression coefficient, reduced rank regression, structure correlations.
Partial least square modeling in research on educational achievement. Pp
 Reflections on educational achievement; Papers in honour of T. Neville Postlethwaite
, 1995
"... This paper contains a discussion of partial least squares (PLS) path modeling with latent constructs as a general method for research on educational achievement. To the extent that such research requires the analysis of comparatively large and complex models under mild supplementary assumptions, PLS ..."
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This paper contains a discussion of partial least squares (PLS) path modeling with latent constructs as a general method for research on educational achievement. To the extent that such research requires the analysis of comparatively large and complex models under mild supplementary assumptions, PLS is an extremely flexible and powerful tool for statistical model building. The formal specification, estimation, and evaluation of PLS models is described with special emphasis on the features that distinguish PLS from other methods for path analysis. This specifically concerns distributionfree least squares estimation and distributionfree model evaluation using jackknife techniques. 1
NeuroImage 56 (2011) 455–475 Contents lists available at ScienceDirect
"... journal homepage: www.elsevier.com/locate/ynimg ..."
Statistics for Industry and Technology, 235–246 c ○ 2004 Birkhäuser Verlag Basel/Switzerland Robust Redundancy Analysis by Alternating Regression
"... Abstract. Given two groups of variables redundancy analysis searches for linear combinations of variables in one group that maximizes the variance of the other group that is explained by the linear combination. The method is important as an alternative to canonical correlation analysis, and can be s ..."
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Abstract. Given two groups of variables redundancy analysis searches for linear combinations of variables in one group that maximizes the variance of the other group that is explained by the linear combination. The method is important as an alternative to canonical correlation analysis, and can be seen as an alternative to multivariate regression when there are collinearity problems in the dependent set of variables. Principal component analysis is itself a special case of redundancy analysis. In this work we propose a new robust method to estimate the redundancy analysis parameters based on alternating regressions. These estimators are compared with the classical estimator as well as other robust estimators based on robust covariance matrices. The behavior of the proposed estimators is also investigated under large contamination by the analysis of the empirical breakdown point.
3D Face Shape Approximation from Intensities Using Partial Least Squares
"... In this paper, we apply Partial Least Squares (PLS) regression to predict 3D face shape from a single image. PLS describes the relationship between independent (intensity images) and dependent (3D shape) variables by seeking directions in the space of the independent variables that are associated wi ..."
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In this paper, we apply Partial Least Squares (PLS) regression to predict 3D face shape from a single image. PLS describes the relationship between independent (intensity images) and dependent (3D shape) variables by seeking directions in the space of the independent variables that are associated with high variations in the dependent variables. We exploit this idea to construct statistical models of intensity and 3D shape that express strongly linked variations in both spaces. The outcome of this decomposition is the construction of two different models which express coupled variations in 3D shape and intensity. Using the intensity model, a set of parameters is obtained from outoftraining intensity examples. These intensity parameters can then be used directly in the 3D shape model to approximate facial shape. Experiments show that prediction is achieved with reasonable accuracy. 1.