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.

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.

Optimization of multiclass queueing networks with changeover times via the achievable region approach: Part I, the single-station case

We address the performance optimization problem in a single-stationmulticlass queueing network with changeover times by means of theachievable region approach. This approach seeks to obtainperformance bounds and scheduling policies from the solution of amathematical program over a relaxation of the system's performanceregion. Relaxed formulations (including linear, convex, nonconvexand positive semidefinite constraints) of this region are developedby formulating equilibrium relations satisfied by the system, withthe help of Palm calculus. Our contributions include: (1) newconstraints formulating equilibrium relations on server dynamics;(2) a flow conservation interpretation of the constraintspreviously derived by the potential function method; (3) newpositive semidefinite constraints; (4) new work decomposition lawsfor single-station multiclass queueing networks, which yield newconvex constraints; (5) a unified buffer occupancy method ofperformance analysis obtained from the constraints; (6) heuristicscheduling policies from the solution of the relaxations.

Socio-economic inequalities in health in Catalonia

In this paper we measure the degree of income related inequality in mental health as measured by the GHQ instrument and general health as measured by the EQOL-5D instrument for the Catalan population. We find that income is the main contributor to inequality, although the share of inequality in mental health that can be explained by income is much greater than the corresponding share of inequality in general health. We also find that the variation in demographic structure reduces income related inequality in mental health but increases income related inequality in general health. The regional variations in both instruments for health are striking, with the Barcelona districts faring relatively bad with respect to the rest of geographical areas and Lleida being the health region where, all else held equal, the population reports the greatest level of health. A big share of inequality in the two health measures, but specially mental health, is due to the favourable position in both health and income of those who enjoy an indefinite contract with respect to the rest of individuals. We also find that risky working conditions affect both health measures and are able to explain an important share of socio-economic inequality.

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.

Optimization of multiclass queueing networks with changeover times via the achievable region method: Part II, the multi-station case

We address the problem of scheduling a multi-station multiclassqueueing network (MQNET) with server changeover times to minimizesteady-state mean job holding costs. We present new lower boundson the best achievable cost that emerge as the values ofmathematical programming problems (linear, semidefinite, andconvex) over relaxed formulations of the system's achievableperformance region. The constraints on achievable performancedefining these formulations are obtained by formulatingsystem's equilibrium relations. Our contributions include: (1) aflow conservation interpretation and closed formulae for theconstraints previously derived by the potential function method;(2) new work decomposition laws for MQNETs; (3) new constraints(linear, convex, and semidefinite) on the performance region offirst and second moments of queue lengths for MQNETs; (4) a fastbound for a MQNET with N customer classes computed in N steps; (5)two heuristic scheduling policies: a priority-index policy, anda policy extracted from the solution of a linear programmingrelaxation.

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.

Correspondence analysis of two transition tables

The case of two transition tables is considered, that is two squareasymmetric matrices of frequencies where the rows and columns of thematrices are the same objects observed at three different timepoints. Different ways of visualizing the tables, either separatelyor jointly, are examined. We generalize an existing idea where asquare matrix is descomposed into symmetric and skew-symmetric partsto two matrices, leading to a decomposition into four components: (1)average symmetric, (2) average skew-symmetric, (3) symmetricdifference from average, and (4) skew-symmetric difference fromaverage. The method is illustrated with an artificial example and anexample using real data from a study of changing values over threegenerations.

Reflections on gains and losses: A 2x2x7 experiment

We test whether risk attitudes change when losses instead of gains areinvolved. The study of gain-loss asymmetries has been largely confinedto reflected choices, where all the money amounts of a positiveprospect are multiplied by minus one. We define the decomposition reflection = translation + probability switch, and experimentally findboth a translation effect (risk attraction becomes more frequent whengains are translated into losses) and a probability switch effect (riskattraction becomes more frequent when the probability of the best outcomedecreases). Surprisingly, the switch effect is somewhat stronger than thetranslation effect, negating a conventional reflection effect when onestarts with choices between gains with a low probability of the bestoutcome. We conclude by arguing that, while both the translation effectand the switch effect contradict the expected utility hypothesis, thetranslation effect implies a deeper violation of standard preference theory.

Weighted Euclidean biplots

We construct a weighted Euclidean distance that approximates any distance or dissimilarity measure between individuals that is based on a rectangular cases-by-variables data matrix. In contrast to regular multidimensional scaling methods for dissimilarity data, the method leads to biplots of individuals and variables while preserving all the good properties of dimension-reduction methods that are based on the singular-value decomposition. The main benefits are the decomposition of variance into components along principal axes, which provide the numerical diagnostics known as contributions, and the estimation of nonnegative weights for each variable. The idea is inspired by the distance functions used in correspondence analysis and in principal component analysis of standardized data, where the normalizations inherent in the distances can be considered as differential weighting of the variables. In weighted Euclidean biplots we allow these weights to be unknown parameters, which are estimated from the data to maximize the fit to the chosen distances or dissimilarities. These weights are estimated using a majorization algorithm. Once this extra weight-estimation step is accomplished, the procedure follows the classical path in decomposing the matrix and displaying its rows and columns in biplots.

A Small Baseline DIFSAR Approach for Investigating Deformations on Full Resolution SAR Interferograms

This paper presents a differential synthetic apertureradar (SAR) interferometry (DIFSAR) approach for investigatingdeformation phenomena on full-resolution DIFSAR interferograms.In particular, our algorithm extends the capabilityof the small-baseline subset (SBAS) technique that relies onsmall-baseline DIFSAR interferograms only and is mainly focusedon investigating large-scale deformations with spatial resolutionsof about 100 100 m. The proposed technique is implemented byusing two different sets of data generated at low (multilook data)and full (single-look data) spatial resolution, respectively. Theformer is used to identify and estimate, via the conventional SBAStechnique, large spatial scale deformation patterns, topographicerrors in the available digital elevation model, and possibleatmospheric phase artifacts; the latter allows us to detect, onthe full-resolution residual phase components, structures highlycoherent over time (buildings, rocks, lava, structures, etc.), as wellas their height and displacements. In particular, the estimation ofthe temporal evolution of these local deformations is easily implementedby applying the singular value decomposition technique.The proposed algorithm has been tested with data acquired by theEuropean Remote Sensing satellites relative to the Campania area(Italy) and validated by using geodetic measurements.

Efecto de los usos del suelo en la emisión de dióxido de carbono del suelo a la atmósfera en un agroecosistema semiárido del Valle del Ebro

Los cambios en los usos del suelo han contribuido de manera importante al incremento de gases de efecto invernadero en la atmósfera, especialmente de dióxido de carbono, aumentando sus emisiones desde 1970 en un 80%. Estos cambios causan la alteración de los suelos provocando un impacto sobre el ciclo del carbono, aumentando las tasas de descomposición de la fracción orgánica creando así un flujo de CO2 a la atmosfera. Entre las recomendaciones del Panel Intergubernamental de expertos sobre el Cambio Climático (IPCC, en inglés), y contemplado en el Protocolo de Kyoto, se encuentra el proceso de secuestro de carbono en suelos, que implica la eliminación del CO2 atmosférico por parte de las plantas y su almacenamiento como materia orgánica del suelo. Para poder favorecer dicho proceso, en un determinado tipo de ecosistema, es fundamental conocer cuáles son los factores que gobiernan la respiración del suelo y el impacto que tienen los diferentes usos en la emisión de CO2. En el presente trabajo se han estudiado 4 usos del suelo representativos del secano aragonés: un monocultivo de cebada en siembra directa (NT), un suelo abandonado labrado (AC), un suelo abandonado no alterado (AU) y un suelo forestal (FR) con el objetivo de conocer sus tasas de respiración, la influencia de diferentes parámetros edáficos en ellas, y proponer cambios en el uso del suelo que ayuden a mitigar estas emisiones. Además, se ha dedicado un apartado para conocer cómo influyen diferentes técnicas de fertilización nitrogenada (mineral y orgánica) en la respiración de un monocultivo de cebada en siembra directa. En cuanto a los usos, los resultados obtenidos tanto in situ como en laboratorio muestran una mayor respiración en AC, siendo los valores más bajos los de NT y FR. Una de las principales conclusiones es que la supresión del laboreo y del periodo de barbecho largo, así como la conversión de tierras abandonadas y marginales a cultivos y zonas forestales se presentan en este tipo de ecosistemas como prácticas de secuestro de carbono. En el estudio de aplicación de fertilizantes, no se observó ningún cambio en la respiración del suelo después de la aplicación de nitrógeno mineral. En cambio, el suelo fertilizado con purín sí que mostró picos de emisión durante las siguientes horas a la incorporación de éste, debido fundamentalmente a su alto contenido de carbono lábil.