Power transformations in correspondence analysis

Power transformations of positive data tables, prior to applying the correspondence analysis algorithm, are shown to open up a family of methods with direct connections to the analysis of log-ratios. Two variations of this idea are illustrated. The first approach is simply to power the original data and perform a correspondence analysis this method is shown to converge to unweighted log-ratio analysis as the power parameter tends to zero. The second approach is to apply the power transformation to thecontingency ratios, that is the values in the table relative to expected values based on the marginals this method converges to weighted log-ratio analysis, or the spectral map. Two applications are described: first, a matrix of population genetic data which is inherently two-dimensional, and second, a larger cross-tabulation with higher dimensionality, from a linguistic analysis of several books.

A IMPORTÂNCIA DO BRINQUEDO TERAPÊUTICO NO CUIDAR DA CRIANÇA HOSPITALIZADA

Este trabalho enquadra-se no âmbito da Monografia do Curso de Complemento de Licenciatura em Enfermagem e tem como objetivo analisar a literatura especializada do tema em questão e ao mesmo tempo realçar a importância do brinquedo terapêutico como forma de cuidar, agindo eficazmente na relação do profissional de saúde e a criança hospitalizada, diminuindo o medo, o sofrimento e a fragilidade por que esta poderá passar durante a hospitalização. O cuidar da criança hospitalizada através do brinquedo terapêutico é uma forma humana de cuidar em enfermagem enquanto profissão. A palavra cuidada, usada no singular, designa a atenção positiva e construtiva prestada a alguém, com o objetivo de fazer algo por esse alguém ou com ele (Hesbeen, 2001:16).Tendo em conta o conceito actual de saúde proposto pela OMS, bem como as políticas de saúde para atenção à criança hospitalizada, a autonomia da enfermagem numa visão diferente na prestação de cuidados de enfermagem, faz-se necessário proporcionar, no contexto hospitalar, um ambiente estruturado para as brincadeiras, que possibilite uma continuidade satisfatória no curso do desenvolvimento da criança. Brincar é primordial para a criança, esteja ela sadia ou doente, inclusive se, por uma circunstância de maior gravidade, precisa ser hospitalizada. Assim, os objectivos da enfermagem, no que diz respeito ao cuidado à criança, devem estar direcionados para facilitar o comportamento de adaptação desta frente a uma situação agressiva para ela. Sendo o brinquedo uma forma de humanizar a assistência de enfermagem pediátrica, o mesmo é essencial e indispensável ao cuidado da criança hospitalizada, tal como pretendemos demonstrar com este trabalho.

Contribution biplots

In order to interpret the biplot it is necessary to know which points usually variables are the ones that are important contributors to the solution, and this information is available separately as part of the biplot s numerical results. We propose a new scaling of the display, called the contribution biplot, which incorporates this diagnostic directly into the graphical display, showing visually the important contributors and thus facilitating the biplot interpretation and often simplifying the graphical representation considerably. The contribution biplot can be applied to a wide variety of analyses such as correspondence analysis, principal component analysis, log-ratio analysis and the graphical results of a discriminant analysis/MANOVA, in fact to any method based on the singular-value decomposition. In the contribution biplot one set of points, usually the rows of the data matrix, optimally represent the spatial positions of the cases or sample units, according to some distance measure that usually incorporates some form of standardization unless all data are comparable in scale. The other set of points, usually the columns, is represented by vectors that are related to their contributions to the low-dimensional solution. A fringe benefit is that usually only one common scale for row and column points is needed on the principal axes, thus avoiding the problem of enlarging or contracting the scale of one set of points to make the biplot legible. Furthermore, this version of the biplot also solves the problem in correspondence analysis of low-frequency categories that are located on the periphery of the map, giving the false impression that they are important, when they are in fact contributing minimally to the solution.

A total order in [0,1] defined through a 'next' operator

A `next' operator, s, is built on the set R1=(0,1]-{ 1-1/e} defining a partial order that, with the help of the axiom of choice, can be extended to a total order in R1. Besides, the orbits {sn(a)}nare all dense in R1 and are constituted by elements of the samearithmetical character: if a is an algebraic irrational of degreek all the elements in a's orbit are algebraic of degree k; if a istranscendental, all are transcendental. Moreover, the asymptoticdistribution function of the sequence formed by the elements in anyof the half-orbits is a continuous, strictly increasing, singularfunction very similar to the well-known Minkowski's ?(×) function.

Suivi du patient oncologique, garant de la permanence de soi [Follow-up of the cancer patient, maintaining self-identity]

L'individu confronté au diagnostic de cancer subit un bouleversement brutal de ses repères et de ses habitudes. La maladie représente une menace pour son équilibre de vie et sa stabilité sociale. Sa capacité à faire face et à opérer différents remaniements dans sa façon d'être au monde et d'envisager l'avenir est en partie déterminée par ses ressources personnelles. Toutefois, le soutien émotionnel peut également représenter un moyen privilégié de donner du sens à cette expérience singulière qu'est la maladie. La reconstruction narrative dans un cadre soutenant, caractérisé par une écoute bienveillante, offre au patient la possibilité de reconnaître sa souffrance comme partie intégrante de lui-même. Un récit de vie qui intègre la maladie lui permet de se réapproprier son histoire. Cette démarche nécessite de la part du thérapeute une disponibilité psychique et temporelle et la capacité de soutenir le patient dans un processus de liaison à travers les différentes étapes de la maladie. The individual facing the diagnosis of cancer is subjected to abrupt changes with regard to his inner world, his life, habits and social relationships. The patient's capacity to cope, to integrate changes in the way of living and to face the future is determined by his personal resources. However, psychological support may also be an important mean to search for and find sense to the singular experience of the illness. The narrative reconstruction within a supportive setting provides the patient a possibility to recognise his sufferance as an integral part of himself. A life narrative, which integrates the illness, allows the patient to re-appropriate his history again. Such a therapeutic project necessitates from the therapist a psychological and temporal disponibility and a capacity to create links all along the different stages of the disease.

Computation of multiple correspondence analysis, with code in R

The generalization of simple correspondence analysis, for two categorical variables, to multiple correspondence analysis where they may be three or more variables, is not straighforward, both from a mathematical and computational point of view. In this paper we detail the exact computational steps involved in performing a multiple correspondence analysis, including the special aspects of adjusting the principal inertias to correct the percentages of inertia, supplementary points and subset analysis. Furthermore, we give the algorithm for joint correspondence analysis where the cross-tabulations of all unique pairs of variables are analysed jointly. The code in the R language for every step of the computations is given, as well as the results of each computation.

Distributional equivalence and subcompositional coherence in the analysis of contingency tables, ratio-scale measurements and compositional data

We consider two fundamental properties in the analysis of two-way tables of positive data: the principle of distributional equivalence, one of the cornerstones of correspondence analysis of contingency tables, and the principle of subcompositional coherence, which forms the basis of compositional data analysis. For an analysis to be subcompositionally coherent, it suffices to analyse the ratios of the data values. The usual approach to dimension reduction in compositional data analysis is to perform principal component analysis on the logarithms of ratios, but this method does not obey the principle of distributional equivalence. We show that by introducing weights for the rows and columns, the method achieves this desirable property. This weighted log-ratio analysis is theoretically equivalent to spectral mapping , a multivariate method developed almost 30 years ago for displaying ratio-scale data from biological activity spectra. The close relationship between spectral mapping and correspondence analysis is also explained, as well as their connection with association modelling. The weighted log-ratio methodology is applied here to frequency data in linguistics and to chemical compositional data in archaeology.

Weighted metric multidimensional scaling

This paper establishes a general framework for metric scaling of any distance measure between individuals based on a rectangular individuals-by-variables data matrix. The method allows visualization of both individuals and variables as well as preserving all the good properties of principal axis methods such as principal components and correspondence analysis, based on the singular-value decomposition, including the decomposition of variance into components along principal axes which provide the numerical diagnostics known as contributions. The idea is inspired from the chi-square distance in correspondence analysis which weights each coordinate by an amount calculated from the margins of the data table. In weighted metric multidimensional scaling (WMDS) we allow these weights to be unknown parameters which are estimated from the data to maximize the fit to the original distances. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing a matrix and displaying its rows and columns in biplots.

Biplots of fuzzy coded data

A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well-known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure of fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data.

Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing oneor more parameters in their definition. Methods that can be linked in this way arecorrespondence analysis, unweighted or weighted logratio analysis (the latter alsoknown as "spectral mapping"), nonsymmetric correspondence analysis, principalcomponent analysis (with and without logarithmic transformation of the data) andmultidimensional scaling. In this presentation I will show how several of thesemethods, which are frequently used in compositional data analysis, may be linkedthrough parametrizations such as power transformations, linear transformations andconvex linear combinations. Since the methods of interest here all lead to visual mapsof data, a "movie" can be made where where the linking parameter is allowed to vary insmall steps: the results are recalculated "frame by frame" and one can see the smoothchange from one method to another. Several of these "movies" will be shown, giving adeeper insight into the similarities and differences between these methods.

A new light on Minkowski's? $(x)$ function

The well--known Minkowski's? $(x)$ function is presented as the asymptotic distribution function of an enumeration of the rationals in (0,1] based on their continued fraction representation. Besides, the singularity of ?$(x)$ is clearly proved in two ways: by exhibiting a set of measure one in which ?ï$(x)$ = 0; and again by actually finding a set of measure one which is mapped onto a set of measure zero and viceversa. These sets are described by means of metrical properties of different systems for real number representation.

Analysis of matched matrices

We consider the joint visualization of two matrices which have common rowsand columns, for example multivariate data observed at two time pointsor split accord-ing to a dichotomous variable. Methods of interest includeprincipal components analysis for interval-scaled data, or correspondenceanalysis for frequency data or ratio-scaled variables on commensuratescales. A simple result in matrix algebra shows that by setting up thematrices in a particular block format, matrix sum and difference componentscan be visualized. The case when we have more than two matrices is alsodiscussed and the methodology is applied to data from the InternationalSocial Survey Program.

From correspondence analysis to multiple and joint correspondence analysis

The generalization of simple (two-variable) correspondence analysis to more than two categorical variables, commonly referred to as multiple correspondence analysis, is neither obvious nor well-defined. We present two alternative ways of generalizing correspondence analysis, one based on the quantification of the variables and intercorrelation relationships, and the other based on the geometric ideas of simple correspondence analysis. We propose a version of multiple correspondence analysis, with adjusted principal inertias, as the method of choice for the geometric definition, since it contains simple correspondence analysis as an exact special case, which is not the situation of the standard generalizations. We also clarify the issue of supplementary point representation and the properties of joint correspondence analysis, a method that visualizes all two-way relationships between the variables. The methodology is illustrated using data on attitudes to science from the International Social Survey Program on Environment in 1993.

Report CR-A-219-B - Cabo Verde Estudo do Sector Agrícola

A. A Singularidade de Cabo Verde Cabo Verde e unica entre a familia de na90es. ~ uma republica de ilhas (Fig. 1.1) suficientemente distanciadas umas das outras e do continente africano (600 km) para dissuadir visitas imprevistas ou fortuitas. Seu isolamento serviu para fortalecer sua singular cultura crioula, porem a cllsta de contactos frequentes com influencias e inforrna90es externas. Cabo Verde e muito pequeno. Sua popula9ao de 300.000 habitantes, que mal chegaria para tornar uma cidade auto-suficiente num complexo industrial, esta dividida entre nove ilhas. A capital, Praia, conta somente com 23.000 habitantes, e e, essencialmente, urna pequena e simpatica cidade onde a maior parte dos vercursos pode ser feita a pe e onde encontros casuais entre funcionarios do governo sao provavelmente tao irnportantes quanto os formais meios de comunica9ao pelo telefone ou por escrito. Cabo Verde e tambem pequeno ern termos de area territorial, embora ocupe urna area relativamente grande de ocea~o. Sua superficie total e pouco mais de 4.000 km2 , porem suas nove ilhas principais estao espalhadas por uma POr9aO grande, quadrangular, do Atlantico medindo aproximadamente 240 qUilometros de urn lado. A superficie total (nao incluindo partes da plataforma continental que se estende para alem do litoral externo das ilhas) e comparavel as republicas de Togo ou Sri Lanka, ou cerca de metade da Guatemala. A nao ser pelo servi90 de barcas entre Fogo e Brava, e entre Sao Vicente e Santo Antao, as ilhas estao demasiado distantes entre si para urn conveniente transporte maritimo. Avioes sao, portanto, essenciais as comunica90es e trafego entre as ilhas, embora para transac90es lentas e volurnosas 0 transporte maritimo, seja por embarca90es a vela ou a motor, e sempre urna alternativa devido aos ventos constantes e condi90es atmosfericas geralmente boas.

Biplots of compositional data

The singular value decomposition and its interpretation as alinear biplot has proved to be a powerful tool for analysing many formsof multivariate data. Here we adapt biplot methodology to the specifficcase of compositional data consisting of positive vectors each of whichis constrained to have unit sum. These relative variation biplots haveproperties relating to special features of compositional data: the studyof ratios, subcompositions and models of compositional relationships. Themethodology is demonstrated on a data set consisting of six-part colourcompositions in 22 abstract paintings, showing how the singular valuedecomposition can achieve an accurate biplot of the colour ratios and howpossible models interrelating the colours can be diagnosed.