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The Proximity of an Individual to a Population With Applications in Discriminant Analysis
, 1995
"... : We develop a proximity function between an individual and a population from a distance between multivariate observations. We study some properties of this construction and apply it to a distancebased discrimination rule, which contains the classic linear discriminant function as a particular ..."
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Cited by 18 (10 self)
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: We develop a proximity function between an individual and a population from a distance between multivariate observations. We study some properties of this construction and apply it to a distancebased discrimination rule, which contains the classic linear discriminant function as a particular case. Additionally, this rule can be used advantageously for categorical or mixed variables, or in problems where a probabilistic model is not well determined. This approach is illustrated and compared with other classic procedures using four real data sets. Keywords: Categorical and mixed data; Distances between observations; Multidimensional scaling; Discrimination; Classification rules. AMS Subject Classification: 62H30 The authors thank M.Abrahamowicz, J. C. Gower and M. Greenacre for their helpful comments, and W. J. Krzanowski for providing us with a data set and his quadratic location model program. Work supported in part by CGYCIT grant PB930784. Authors' address: Departam...
A family of matrices, the discretized Brownian Bridge and distancebased regression
, 1997
"... : The investigation of a distancebased regression model, using a onedimensional set of equally spaced points as regressor values, and p jx \Gamma yj as a distance function, leads to the study of a family of matrices which is closely related to a discrete analog of the Brownian Bridge stochasti ..."
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Cited by 4 (1 self)
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: The investigation of a distancebased regression model, using a onedimensional set of equally spaced points as regressor values, and p jx \Gamma yj as a distance function, leads to the study of a family of matrices which is closely related to a discrete analog of the Brownian Bridge stochastic process. We describe its eigenstructure and several properties, recovering in particular wellknown results on tridiagonal Toeplitz matrices and related topics. Keywords: Distancebased regression; Centrosymmetric matrices, Orthogonal polynomials. AMS Subject classification: 62H25, 62J02 1 Introduction The distancebased regression model (Cuadras 1989; Cuadras and Arenas 1990; Cuadras et al. 1996) is an extension of the ordinary linear model which can be applied to qualitative or, in general, to mixed continuous and discrete explanatory variables, provided that a distance ffi can be defined on the set of values of these variables. A brief description of the method is as follows: Assum...
Weighted continuous metric scaling
 Girko (Eds.), Multidimensional Statistical Analysis and Theory of Random Matrices
, 1996
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Metric Scaling Graphical Representation of Categorical Data
 Penn State University
, 1995
"... : Metric Scaling is a wellknown method to represent a finite set with respect to a given Euclidean distance matrix. Several methods to represent rows and columns of a twoway contingency table are available: Correspondence Analysis, Dual Scaling, Canonical Coordinates, etc. We show that metric s ..."
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Cited by 1 (1 self)
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: Metric Scaling is a wellknown method to represent a finite set with respect to a given Euclidean distance matrix. Several methods to represent rows and columns of a twoway contingency table are available: Correspondence Analysis, Dual Scaling, Canonical Coordinates, etc. We show that metric scaling provides a similar representation by using Hellinger or Rao distances together with Gower's addapoint formula and discuss its relationship with the other approaches. The present approach suggests an alternative to Multiple Correspondence Analysis for multivariate categorical data. Keywords: Categorical data; Correspondence Analysis; Distances between observations; Multidimensional scaling; Biplot. AMS Subject Classification: 62H25, 62H20, 6209. 1 Introduction The statistical methodology dealing with categorical data currently has an increasing interest. Under the name Correspondence Analysis (CA), the data analyst recognizes a method of graphical representation of categorical ...
Implementing pls for distancebased regression: computational issues
 Computational Statistics
"... Distancebased regression allows for a neat implementation of the Partial Least Squares recurrence. In this paper we address practical issues arising when dealing with moderately large datasets (n ~ 104) such as those typical of automobile insurance premium calculations. ..."
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Cited by 1 (1 self)
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Distancebased regression allows for a neat implementation of the Partial Least Squares recurrence. In this paper we address practical issues arising when dealing with moderately large datasets (n ~ 104) such as those typical of automobile insurance premium calculations.
Increasing The Correlations With The Response Variable May Not Increase The Coefficient Of Determination: A Pca Interpretation
 Trends in Probability and Statistics. Vol 3. Multivariate Statistics and Matrices in Statistics
, 1995
"... INTRODUCTION Tiit (1984) studied several regression models for which the multiple correlation coefficient and the regression parameters can be formally computed. One of these models deals with the regression of a response variable y on k equicorrelated variables x 1 ; : : : ; x k ; i.e, the correla ..."
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Cited by 1 (1 self)
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INTRODUCTION Tiit (1984) studied several regression models for which the multiple correlation coefficient and the regression parameters can be formally computed. One of these models deals with the regression of a response variable y on k equicorrelated variables x 1 ; : : : ; x k ; i.e, the correlation matrix between the explanatory variables is R = (1 \Gamma c)I+ cJ; (1) where I is the k\Thetak identity matrix, J is the k\Thetak matrix of ones and \Gamma(k \Gamma 1) \Gamma1 ! c ! 1. It is supposed in this model that the vector of correlations between y and x 1 ; : : : ; x<F4
The other uses of Multidimensional Scaling in Statistics
"... : Multidimensional Scaling is a useful tool to represent a finite set in an appropriate graphical display. But MDS can do much more in statistics, classification and data analysis. It is shown in this contribution that MDS and related methods based on distances provide techniques and solutions to a ..."
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: Multidimensional Scaling is a useful tool to represent a finite set in an appropriate graphical display. But MDS can do much more in statistics, classification and data analysis. It is shown in this contribution that MDS and related methods based on distances provide techniques and solutions to a wide field of topics: distance based regression with mixed variables and nonlinear regression; MDS interpretation of ridge regression; MDS representation of Hoeffding's maximum correlations; representing parametric estimable functions and comparing models in MANOVA; examining principal dimensions of a tree scaled in a Euclidean configuration; MDS representation of a statistical model; distance based approach in discrimination and classification; MDS representation of a random variable. All these applications show that MDS opens the way to new applications of this method and helps to give us a better understanding of the structure of data. 1 Introduction The earliest books on multivariate an...
The Importance of Geometry in Multivariate Analysis and some Applications
"... Geometrical concepts, including distance functions between observations, geometric variabilities and proximity functions, are used to develop some new aspects of multivariate analysis. These include the influence of principal components in comparing populations, the detection of atypical observa ..."
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Geometrical concepts, including distance functions between observations, geometric variabilities and proximity functions, are used to develop some new aspects of multivariate analysis. These include the influence of principal components in comparing populations, the detection of atypical observations in discrimination with mixed variables, and the construction of orthogonal expansions for a continuous random variable. Some illustrations are given using two wellknown data sets. KEYWORDS: Mahalanobis distance; Rao's score test; Principal components; Simpson's paradox; Typicality in discrimination; Orthogonal expansions; Goodnessoffit. 1 Introduction Multivariate Analysis is mainly based on results proceeding from three mathematical areas: matrix calculus, distribution theory and metric geometry. This last subject is fundamental in methods such as multidimensional scaling and correspondence analysis, where the notion of distance function plays a basic role. Following the ut...