e searching, zooming, selection of points, and display of data values on factor maps. The user interface is simple and homogeneous among all the programs; this contributes to making the use of ADE-4 very easy for nonspecialists in statistics, data analysis or computer science. Keywords: Multivariate analysis, principal component analysis, correspondence analysis, instrumental variables, canonical correspondence analysis, partial least squares regression, coinertia analysis, graphics, multivariate graphics, interactive graphics, Macintosh, HyperCard, Windows 95 1. Introduction ADE-4 is a multivariate analysis and graphical display software for Apple Macintosh and Windows 95 microcomputers. It is made up of several stand-alone applications, called modules, that feature a wide range of multivariate analysis methods, from simple one-table analysis to three-way table analysis and two-table coupling methods. It also provides many possibilitie

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Correlation and covariance matrices provide the basis for all classical multivariate techniques.

Canonical correspondence analysis (CCA) is introduced as a multivariate extension of weighted averaging ordination, which is a simple method for arranging species along environmental variables. CCA constructs those linear combinations of environmental variables, along which the distributions of the species are max-imally separated. The eigenvalues produced by CCA measure this separation. As its name suggests, CCA is also a correspondence analysis technique, but one in which the ordination axes are constrained to be linear combinations of environmental variables. The ordination diagram generated by CCA visualizes not only a pattern of community variation (as in standard ordination) but also the main features of the distributions of species along the environmental variables. Applications demonstrate that CCA can be used both for detecting species-environment relations, and for investigating specific questions about the response of species to environmental variables. Questions in community ecology that have typically been studied by 'indirect ' gradient analysis (i.e. ordination followed by external interpretation of the axes) can now be answered more directly by CCA.

Every teacher of linear algebra should be familiar with the matrix singular value decomposition (or SVD). It has interesting and attractive algebraic properties, and conveys important geometrical and theoretical insights about linear transformations. The close connection between the SVD and the well known theory of diagonalization for symmetric matrices makes the topic immediately accessible to linear algebra teachers, and indeed, a natural extension of what these teachers already know. At the same time, the SVD has fundamental importance in several different applications of linear algebra. Strang was aware of these facts when he introduced the SVD in his now classical text [22, page 142], observing...it is not nearly as famous as it should be. Golub and Van Loan ascribe a central significance to the SVD in their definitive explication of numerical matrix methods [8, page xiv] stating...perhaps the most recurring theme in the book is the practical and theoretical value of [the SVD]. Additional evidence of the significance of the SVD is its central role in a number of papers in recent years in Mathematics Magazine and The American Mathematical Monthly (for example [2, 3, 17, 23]). Although it is probably not feasible to include the SVD in the first linear algebra course, it definitely deserves a place in more advanced undergraduate courses, particularly those with a numerical or applied emphasis. My primary goals in this article are to bring the topic to the attention of a broad audience,

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In many fields of engineering science we have to deal with multivariate numerical data. In order to choose the technique that is best suited to a given task, it is necessary to get an insight into the data and to "understand" them. Much information allowing the understanding of multivariate data, that is the description of its global structure, the presence and shape of clusters or outliers, can be gained through data visualization. Multivariate data visualization can be realized through a reduction of the data dimensionality, which is often performed by mathematical and statistical tools that are well known. Such tools are Principal Components Analysis or Multidimensional Scaling. Artificial neural networks have developed and found applications mainly in the last two decades, and they are now considered as a mature field of research. This thesis investigates the use of existing algorithms as applied to multivariate data visualization. First an overview of existing neural and statistical techniques applied to data visualization is presented. Then a comparison is made between two chosen algorithms from the point of view of multivariate data visualization. The chosen neural network algorithm is Kohonen's Self-Organizing Maps, and the statistical technique is Multidimensional Scaling. The advantages and drawbacks from the theoretical and practical viewpoints of both approaches are put into light. The preservation of data topology involved by those two mapping techniques is discussed. The multidimensional scaling method was analyzed in details, the importance of each parameter was determined, and the technique was implemented in metric and non-metric versions. Improvements to the algorithm were proposed in order to increase the performance of the mapping process. A graphic...

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 reduced-rank regression. Reduced-rank 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 least-squares approximation of fitted values, regression coefficients and associated t-ratio's. The utility and interpretation of the reduced-rank regression biplot is demonstrated with an example using public health data that were previously analyzed by separate multiple regressions.

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 analy-sis, partial analysis, redundancy analysis, regression coefficient, reduced rank regression, struc-ture correlations.

la France was one of the foundation studies of modern social science. Guerry assembled data on crimes, suicides, literacy and other “moral statistics, ” and used tables and maps to analyze a variety of social issues in perhaps the first comprehensive study relating such variables. Indeed, the Essai may be considered the book that launched modern empirical social science, for the questions raised and the methods Guerry developed to try to answer them. Guerry’s data consist of a large number of variables recorded for each of the départments of France in the 1820–1830s and therefore involve both multivariate and geographical aspects. In addition to historical interest, these data provide the opportunity to ask how modern methods of statistics, graphics, thematic cartography and geovisualization can shed further light on the questions he raised. We present a variety of methods attempting to address Guerry’s challenge for multivariate spatial statistics.