Lecture Notes on Compositional Data Analysis

These notes have been prepared as support to a short course on compositional data analysis. Their aim is to transmit the basic concepts and skills for simple applications, thus setting the premises for more advanced projects

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

CoDa-dendrogram: A new exploratory too

The use of orthonormal coordinates in the simplex and, particularly, balance coordinates, has suggested the use of a dendrogram for the exploratory analysis of compositional data. The dendrogram is based on a sequential binary partition of a compositional vector into groups of parts. At each step of a partition, one group of parts is divided into two new groups, and a balancing axis in the simplex between both groups is defined. The set of balancing axes constitutes an orthonormal basis, and the projections of the sample on them are orthogonal coordinates. They can be represented in a dendrogram-like graph showing: (a) the way of grouping parts of the compositional vector; (b) the explanatory role of each subcomposition generated in the partition process; (c) the decomposition of the total variance into balance components associated with each binary partition; (d) a box-plot of each balance. This representation is useful to help the interpretation of balance coordinates; to identify which are the most explanatory coordinates; and to describe the whole sample in a single diagram independently of the number of parts of the sample

Statistical Analysis of Eco-Toxicologic Data of Mammals from a Polluted Area

Several eco-toxicological studies have shown that insectivorous mammals, due to their feeding habits, easily accumulate high amounts of pollutants in relation to other mammal species. To assess the bio-accumulation levels of toxic metals and their in°uence on essential metals, we quantified the concentration of 19 elements (Ca, K, Fe, B, P, S, Na, Al, Zn, Ba, Rb, Sr, Cu, Mn, Hg, Cd, Mo, Cr and Pb) in bones of 105 greater white-toothed shrews (Crocidura russula) from a polluted (Ebro Delta) and a control (Medas Islands) area. Since chemical contents of a bio-indicator are mainly compositional data, conventional statistical analyses currently used in eco-toxicology can give misleading results. Therefore, to improve the interpretation of the data obtained, we used statistical techniques for compositional data analysis to define groups of metals and to evaluate the relationships between them, from an inter-population viewpoint. Hypothesis testing on the adequate balance-coordinates allow us to confirm intuition based hypothesis and some previous results. The main statistical goal was to test equal means of balance-coordinates for the two defined populations. After checking normality, one-way ANOVA or Mann-Whitney tests were carried out for the inter-group balances

Compositional analysis of bivariate discrete probabilities

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table has n rows and m columns and all probabilities are non-null. This kind of table can be seen as an element in the simplex of n · m parts. In this context, the marginals are identified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclidean elements of the Aitchison geometry of the simplex can also be translated into the table of probabilities: subspaces, orthogonal projections, distances. Two important questions are addressed: a) given a table of probabilities, which is the nearest independent table to the initial one? b) which is the largest orthogonal projection of a row onto a column? or, equivalently, which is the information in a row explained by a column, thus explaining the interaction? To answer these questions three orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independent two-way tables and fully dependent tables representing row-column interaction. An important result is that the nearest independent table is the product of the two (row and column)-wise geometric marginal tables. A corollary is that, in an independent table, the geometric marginals conform with the traditional (arithmetic) marginals. These decompositions can be compared with standard log-linear models. Key words: balance, compositional data, simplex, Aitchison geometry, composition, orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure, contingency table

Compositional data and Simpson's paradox

Simpson's paradox, also known as amalgamation or aggregation paradox, appears when dealing with proportions. Proportions are by construction parts of a whole, which can be interpreted as compositions assuming they only carry relative information. The Aitchison inner product space structure of the simplex, the sample space of compositions, explains the appearance of the paradox, given that amalgamation is a nonlinear operation within that structure. Here we propose to use balances, which are specific elements of this structure, to analyse situations where the paradox might appear. With the proposed approach we obtain that the centre of the tables analysed is a natural way to compare them, which avoids by construction the possibility of a paradox. Key words: Aitchison geometry, geometric mean, orthogonal projection

Quantifying rock fabrics: a test of autocorrelation of the spatial distribution of cristals

A novel test of spatial independence of the distribution of crystals or phases in rocks based on compositional statistics is introduced. It improves and generalizes the common joins-count statistics known from map analysis in geographic information systems. Assigning phases independently to objects in RD is modelled by a single-trial multinomial random function Z(x), where the probabilities of phases add to one and are explicitly modelled as compositions in the K-part simplex SK. Thus, apparent inconsistencies of the tests based on the conventional joins{count statistics and their possibly contradictory interpretations are avoided. In practical applications we assume that the probabilities of phases do not depend on the location but are identical everywhere in the domain of de nition. Thus, the model involves the sum of r independent identical multinomial distributed 1-trial random variables which is an r-trial multinomial distributed random variable. The probabilities of the distribution of the r counts can be considered as a composition in the Q-part simplex SQ. They span the so called Hardy-Weinberg manifold H that is proved to be a K-1-affine subspace of SQ. This is a generalisation of the well-known Hardy-Weinberg law of genetics. If the assignment of phases accounts for some kind of spatial dependence, then the r-trial probabilities do not remain on H. This suggests the use of the Aitchison distance between observed probabilities to H to test dependence. Moreover, when there is a spatial uctuation of the multinomial probabilities, the observed r-trial probabilities move on H. This shift can be used as to check for these uctuations. A practical procedure and an algorithm to perform the test have been developed. Some cases applied to simulated and real data are presented. Key words: Spatial distribution of crystals in rocks, spatial distribution of phases, joins-count statistics, multinomial distribution, Hardy-Weinberg law, Hardy-Weinberg manifold, Aitchison geometry

Compositional evolution with mass transfer in closed systems

Evolution of compositions in time, space, temperature or other covariates is frequent in practice. For instance, the radioactive decomposition of a sample changes its composition with time. Some of the involved isotopes decompose into other isotopes of the sample, thus producing a transfer of mass from some components to other ones, but preserving the total mass present in the system. This evolution is traditionally modelled as a system of ordinary di erential equations of the mass of each component. However, this kind of evolution can be decomposed into a compositional change, expressed in terms of simplicial derivatives, and a mass evolution (constant in this example). A rst result is that the simplicial system of di erential equations is non-linear, despite of some subcompositions behaving linearly. The goal is to study the characteristics of such simplicial systems of di erential equa- tions such as linearity and stability. This is performed extracting the compositional dif ferential equations from the mass equations. Then, simplicial derivatives are expressed in coordinates of the simplex, thus reducing the problem to the standard theory of systems of di erential equations, including stability. The characterisation of stability of these non-linear systems relays on the linearisation of the system of di erential equations at the stationary point, if any. The eigenvelues of the linearised matrix and the associated behaviour of the orbits are the main tools. For a three component system, these orbits can be plotted both in coordinates of the simplex or in a ternary diagram. A characterisation of processes with transfer of mass in closed systems in terms of stability is thus concluded. Two examples are presented for illustration, one of them is a radioactive decay

Studies in stoichiometry with compositional data

Geochemical data that is derived from the whole or partial analysis of various geologic materials represent a composition of mineralogies or solute species. Minerals are composed of structured relationships between cations and anions which, through atomic and molecular forces, keep the elements bound in specific configurations. The chemical compositions of minerals have specific relationships that are governed by these molecular controls. In the case of olivine, there is a well-defined relationship between Mn-Fe-Mg with Si. Balances between the principal elements defining olivine composition and other significant constituents in the composition (Al, Ti) have been defined, resulting in a near-linear relationship between the logarithmic relative proportion of Si versus (MgMnFe) and Mg versus (MnFe), which is typically described but poorly illustrated in the simplex. The present contribution corresponds to ongoing research, which attempts to relate stoichiometry and geochemical data using compositional geometry. We describe here the approach by which stoichiometric relationships based on mineralogical constraints can be accounted for in the space of simplicial coordinates using olivines as an example. Further examples for other mineral types (plagioclases and more complex minerals such as clays) are needed. Issues that remain to be dealt with include the reduction of a bulk chemical composition of a rock comprised of several minerals from which appropriate balances can be used to describe the composition in a realistic mineralogical framework. The overall objective of our research is to answer the question: In the cases where the mineralogy is unknown, are there suitable proxies that can be substituted? Kew words: Aitchison geometry, balances, mineral composition, oxides

Publiradio.net: innovación docente e investigación sobre publicidad radiofónica en el marco del EEES

La adaptación del sistema universitario catalán y español al Espacio Europeo de Educación Superior (EEES) está favoreciendo la aparición de nuevas metodologías docentes en respuesta a los requerimientos del llamado European Credit Transfer System (ECTS), una modalidad de crédito cuya principal característica consiste, como ya es ampliamente conocido, en focalizar la atención en el estudiante y, especialmente, en generar un proceso interactivo en el que el alumno se convierta en responsable activo de la adquisición de conocimientos. En el ámbito de las Ciencias de la Comunicación, una de las iniciativas más destacables de aproximación al EEES comienza en el año 2004, cuando el Departament d'Universitats, Investigació i Societat de la Informació de la Generalitat de Catalunya (DURSI) eligió a varios centros catalanes para que aplicaran de manera experimental el nuevo sistema común de créditos derivados de la Declaración de Bolonia. La Universidad Autónoma de Barcelona (UAB) fue uno de ellos, y ésta, a su vez, decidió poner en práctica la prueba piloto en la licenciatura de Publicidad y Relaciones Públicas. Es a partir de ese momento cuando los profesores del Departament de Comunicació Audiovisual i Publicitat de la UAB que forman el grupo de investigación Publiradio2 deciden poner en marcha un proyecto que, en el marco de las asignaturas de creatividad y de radio publicitaria, respondiera a las exigencias de la cultura docente en la que se inspira el crédito ECTS. Nace así Publiradio.net, un aplicativo on-line para la creación de publicidad radiofónica destinado, también, a facilitar y potenciar el aprendizaje autónomo entre los estudiantes, a fomentar, en sintonía con las apreciaciones de Aguaded, la utilización de 'técnicas y procedimientos que permitan al alumnado su análisis y uso como lenguajes propios' y a impulsar y facilitar a los estudiantes la investigación en publicidad radiofónica

Efecto de bajos niveles de control de la comprensión en el arraigo de ideas lamarckianas

La investigación didáctica ha revelado las destrezas cognitivas y metacognitivas como excelentes predictores del éxito académico frente a otros factores cognitivos e instruccionales (Campanario y Otero, 2000; Baker, 1994). Cuando un sujeto resuelve preguntas o lee con otro objetivo, debe ser capaz de comprender, utilizar información y reflexionar a partir de la información obtenida. Es en este último aspecto, cuando entra en juego el módulo de control, para revisar la representación y evaluar la ausencia de incoherencias temáticas y de contradicciones con su conocimiento previo. En síntesis, la comprensión del tema de Evolución Biológica y por ende cualquier texto científico, requiere tanto de habilidades de Comprensión Lectora para poder construir una buena representación semántica del texto (la Base del Texto), como de estrategias que Controlen el proceso de Comprensión por el que se vincula esta representación textual con el conocimiento previo para formar la representación Modelo de la Situación. Esta construcción requiere la vinculación vía inferencia de la información de la base del texto con la información de la MLP. Si se tiene un nivel alto de Control de la Comprensión, el número y fuerza de las conexiones realizadas vía inferencia, en esa construcción se optimiza. Asimismo se detectan inconsistencias (incoherencias) y se intentan superar. Dada la importancia del control de la comprensión, resulta interesante examinar la situación actual de dichas estrategias en temáticas particulares, tales como la evolución biológica y comprobar hasta que punto se replican los resultados obtenidos en investigaciones anteriores que indican que los niveles bajos en control de la comprensión se asocian en mayor grado con persistencia en ideas lamarckianas tras un tratamiento instruccional

Elaboración de software para la supervisión in situ de ejercicios prácticos de análisis de estructuras con enunciado personalizado

Se expone la experiencia de los autores en el desarrollo de las clases prácticas de análisis de estructuras, en la asignatura ESTRUCTURAS II de la ETS de Arquitectura de la Universidad de Granada, consistente en el desarrollo de ejercicios prácticos semanales con enunciados personalizados para cada alumno, previa elaboración de software que permite la supervisión y corrección in situ de los mismos. El objetivo de la metodología docente que se presenta, es la mejora de los tradicionalmente muy bajos índices de rendimiento académico, en las asignaturas de Análisis de Estructuras, en las escuelas de ingeniería y arquitectura, propiciando la participación activa del alumnado en el desarrollo de las clases mediante el planteamiento y resolución de ejercicios prácticos semanales de enunciado personalizado

La Presencia de la didáctica en los grados de educación: nuevo escenario para la formación de profesores

Interesados por la calidad de la formación profesional del futuro docente, coincidimos, al comenzar a trabajar en la elaboración de los planes de estudios, en la preocupación por verificar un soporte eminentemente didáctico tras ella. La gestión didáctica define uno de los papeles fundamentales de la profesión docente como ya lo señaló Escolano Benito (1996: 44-46) al denominarla la ingeniería de la instrucción

Evaluación de la utilización de un modelo inanimado para la extracción de sangre por parte de los estudiantes de patología general de veterinaria

Debido a las restricciones que impone la legislación de experimentación y bienestar animal referente al uso de animales vivos en las prácticas veterinarias, durante el curso 2009-2010 se desarrolló la fabricación de un modelo inanimado para la obtención de muestras de sangre en las prácticas de Patología General y Propedéutica Clínica. En el curso académico 2010-11 se ha puesto en funcionamiento estos modelos en las prácticas de Patología General y se ha evaluado el grado de satisfacción de los estudiantes. Los resultados han sido altamente satisfactorios, tanto para los estudiantes como para el profesorado que ha impartido dichas sesiones prácticas

Diseño, implantación y análisis de los resultados de un sistema de evaluación integral o 360º. Una propuesta alternativa en el proceso de evaluación de las competencias de trabajo en equipo

La Comunicación se enmarca dentro del análisis y efectividad que la evaluación 360 grados tiene en el desarrollo de las competencias de trabajo en equipo por parte el alumnado de la asignatura Análisis de Mercado Turístico. En la comunicación se detallan las fases de aplicación del sistema de evaluación así como los resultados obtenidos en la ejecución de la evaluación integral o 360º. En último lugar se presentan las conclusiones de la investigación, las cuales permiten plantear una metodología o sistema de evaluación novedoso y renovado en el que se mejoran y complementan los métodos de valoración de competencias así como los mecanismos formativos adscritos a los mismos