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

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

Statistical treatment of grain-size curves and empirical distributions: densities as compositions?

The preceding two editions of CoDaWork included talks on the possible consideration of densities as infinite compositions: Egozcue and D´ıaz-Barrero (2003) extended the Euclidean structure of the simplex to a Hilbert space structure of the set of densities within a bounded interval, and van den Boogaart (2005) generalized this to the set of densities bounded by an arbitrary reference density. From the many variations of the Hilbert structures available, we work with three cases. For bounded variables, a basis derived from Legendre polynomials is used. For variables with a lower bound, we standardize them with respect to an exponential distribution and express their densities as coordinates in a basis derived from Laguerre polynomials. Finally, for unbounded variables, a normal distribution is used as reference, and coordinates are obtained with respect to a Hermite-polynomials-based basis. To get the coordinates, several approaches can be considered. A numerical accuracy problem occurs if one estimates the coordinates directly by using discretized scalar products. Thus we propose to use a weighted linear regression approach, where all k- order polynomials are used as predictand variables and weights are proportional to the reference density. Finally, for the case of 2-order Hermite polinomials (normal reference) and 1-order Laguerre polinomials (exponential), one can also derive the coordinates from their relationships to the classical mean and variance. Apart of these theoretical issues, this contribution focuses on the application of this theory to two main problems in sedimentary geology: the comparison of several grain size distributions, and the comparison among different rocks of the empirical distribution of a property measured on a batch of individual grains from the same rock or sediment, like their composition

Revisiting the compositional data. Some fundamental questions and new prospects in Archaeometry and Archaeology

In this paper we examine the problem of compositional data from a different starting point. Chemical compositional data, as used in provenance studies on archaeological materials, will be approached from the measurement theory. The results will show, in a very intuitive way that chemical data can only be treated by using the approach developed for compositional data. It will be shown that compositional data analysis is a particular case in projective geometry, when the projective coordinates are in the positive orthant, and they have the properties of logarithmic interval metrics. Moreover, it will be shown that this approach can be extended to a very large number of applications, including shape analysis. This will be exemplified with a case study in architecture of Early Christian churches dated back to the 5th-7th centuries AD

Modelling of Mercury’s surface composition and remote detection from the orbit with the BepiColombo Mercury Planetary Orbiter

It can be assumed that the composition of Mercury’s thin gas envelope (exosphere) is related to the composition of the planets crustal materials. If this relationship is true, then inferences regarding the bulk chemistry of the planet might be made from a thorough exospheric study. The most vexing of all unsolved problems is the uncertainty in the source of each component. Historically, it has been believed that H and He come primarily from the solar wind, while Na and K originate from volatilized materials partitioned between Mercury’s crust and meteoritic impactors. The processes that eject atoms and molecules into the exosphere of Mercury are generally considered to be thermal vaporization, photonstimulated desorption (PSD), impact vaporization, and ion sputtering. Each of these processes has its own temporal and spatial dependence. The exosphere is strongly influenced by Mercury’s highly elliptical orbit and rapid orbital speed. As a consequence the surface undergoes large fluctuations in temperature and experiences differences of insolation with longitude. We will discuss these processes but focus more on the expected surface composition and solar wind particle sputtering which releases material like Ca and other elements from the surface minerals and discuss the relevance of composition modelling

Vertebrates Limb Geometry in the Simplex space

A novel metric comparison of the appendicular skeleton (fore and hind limb) of different vertebrates using the Compositional Data Analysis (CDA) methodological approach it’s presented. 355 specimens belonging in various taxa of Dinosauria (Sauropodomorpha, Theropoda, Ornithischia and Aves) and Mammalia (Prothotheria, Metatheria and Eutheria) were analyzed with CDA. A special focus has been put on Sauropodomorpha dinosaurs and the Aitchinson distance has been used as a measure of disparity in limb elements proportions to infer some aspects of functional morphology

Robust Factor Analysis for Compositional Data

Factor analysis as frequent technique for multivariate data inspection is widely used also for compositional data analysis. The usual way is to use a centered logratio (clr) transformation to obtain the random vector y of dimension D. The factor model is then y = Λf + e (1) with the factors f of dimension k < D, the error term e, and the loadings matrix Λ. Using the usual model assumptions (see, e.g., Basilevsky, 1994), the factor analysis model (1) can be written as Cov(y) = ΛΛT + ψ (2) where ψ = Cov(e) has a diagonal form. The diagonal elements of ψ as well as the loadings matrix Λ are estimated from an estimation of Cov(y). Given observed clr transformed data Y as realizations of the random vector y. Outliers or deviations from the idealized model assumptions of factor analysis can severely effect the parameter estimation. As a way out, robust estimation of the covariance matrix of Y will lead to robust estimates of Λ and ψ in (2), see Pison et al. (2003). Well known robust covariance estimators with good statistical properties, like the MCD or the S-estimators (see, e.g. Maronna et al., 2006), rely on a full-rank data matrix Y which is not the case for clr transformed data (see, e.g., Aitchison, 1986). The isometric logratio (ilr) transformation (Egozcue et al., 2003) solves this singularity problem. The data matrix Y is transformed to a matrix Z by using an orthonormal basis of lower dimension. Using the ilr transformed data, a robust covariance matrix C(Z) can be estimated. The result can be back-transformed to the clr space by C(Y ) = V C(Z)V T where the matrix V with orthonormal columns comes from the relation between the clr and the ilr transformation. Now the parameters in the model (2) can be estimated (Basilevsky, 1994) and the results have a direct interpretation since the links to the original variables are still preserved. The above procedure will be applied to data from geochemistry. Our special interest is on comparing the results with those of Reimann et al. (2002) for the Kola project data

Experimental design on the simplex

Optimum experimental designs depend on the design criterion, the model and the design region. The talk will consider the design of experiments for regression models in which there is a single response with the explanatory variables lying in a simplex. One example is experiments on various compositions of glass such as those considered by Martin, Bursnall, and Stillman (2001). Because of the highly symmetric nature of the simplex, the class of models that are of interest, typically Scheff´e polynomials (Scheff´e 1958) are rather different from those of standard regression analysis. The optimum designs are also rather different, inheriting a high degree of symmetry from the models. In the talk I will hope to discuss a variety of modes for such experiments. Then I will discuss constrained mixture experiments, when not all the simplex is available for experimentation. Other important aspects include mixture experiments with extra non-mixture factors and the blocking of mixture experiments. Much of the material is in Chapter 16 of Atkinson, Donev, and Tobias (2007). If time and my research allows, I would hope to finish with a few comments on design when the responses, rather than the explanatory variables, lie in a simplex. References Atkinson, A. C., A. N. Donev, and R. D. Tobias (2007). Optimum Experimental Designs, with SAS. Oxford: Oxford University Press. Martin, R. J., M. C. Bursnall, and E. C. Stillman (2001). Further results on optimal and efficient designs for constrained mixture experiments. In A. C. Atkinson, B. Bogacka, and A. Zhigljavsky (Eds.), Optimal Design 2000, pp. 225–239. Dordrecht: Kluwer. Scheff´e, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society, Ser. B 20, 344–360. 1

Habitat rural de Asturias.

Estudio del modo de estar distribuidos los lugares habitados en Asturias en el campo, los modos de vida, la agricultura tradicional, la parroquia, la propiedad de la tierra, la casería, la casa en las distintas zonas asturianas, los horreos y paneras, los molinos, los útiles de labranza, los productos más comunes como maíz, árboles frutales. los mercados como lugares de intercambio de bienes, etc. Finalmente se hace un recorrido sobre el panorama actual asturiano, la modernización de la maquinaria, la introducción de nuevos cultivos... y se sugieren algunas rutas tanto de lugares que conservan la estructura original como algunas más modernizadas.

La princesa y el dragón : el trabajo de Carlota.

Esta convocatoria forma parte de la campaña por una Educación no sexista que la Concejalía de la Mujer del Ayuntamiento de Gijón desarrolló en los centros de EGB

Averamientu a la Hestoria de la llingua asturiana. ' Acercamiento a la historia de la lengua asturiana'.

Se incluyen mapas ligüísticos de la península en distintas épocas

Itinerario en la ría de Villaviciosa.

Se incluyen en anexos una relación de plantas vasculares de la ría, otra de aves acuáticas y fichas para la determinación de plantas, de la actividad de las aves y de su identificación

El río: sistema biológico.

Se estudian los rios desde el punto de vista del flujo unidireccional. Se ha pretendido incluir los rudimentos básicos para comprender la complejidad del mundo acuático como un complicado conglomerado de relaciones y dependencias entre los organismos y el ambiente inanimado que los rodea. Trata de proporcionar los suficientes elementos que motiven a los lectores a profundizar en el estudio y análisis de la ecología de los rios, el respeto por su conservación y lucha por la recuperación de los que se han perdido. Consta de dos partes: 1.Descripción del ecosistema con la división clásica entre factores ambientales y comunidades biológicas, 2.Actividades que sirven para poner en práctica y contrastar alguno de los conceptos que se explican en la parte teórica. Se estudia la red fluvial, las características del cauce, medio, organismos acuáticos, sucesión longitudinal de poblaciones de animales, la contaminación y por último las prácticas.

Lagos de Covadonga.

Contiene fotografías explicativas y dibujos de vertebrados que se pueden observar en el parque

Arte Prerrománico Asturiano.

Guía didáctica sobre el prerrománico en Asturias dividiéndolo para su mejor estudio en tres etapas: 1.Estilo Alfonsí desde el 737 hasta el reinado de Alfonso II el Casto en 842; 2.Estilo Ramirense desde le reinado de Ramiro I en el 842 hasta el 866 reinando su hijo Ordoño I y 3 etapa desde el 866 al 916 reinando Alfonso III el Magno y años inmediatamente posteriores al traslado de la Corte de Oviedo a León. Se dan las características y luego se describen algunos de los monumentos prerrománicos más interesantes de cada uno de los estilos y para terminar se ofrece un vocabulario apoyado con dibujos de términos arquitectónicos utilizados en la guía.

Arte Románico en Asturias.

Ofrece una descripción del arte románico (entre los siglos X y XIII) en Asturias que presenta bastantes influencias del anterior arte prerrománico. El románico llega a Asturias con retraso y perdura más tiempo que en otros lugares. Se contemplan las características arquitectónicas románicas en Asturias y la decoración escultórica, dividiendo la región asturiana en zonas, destacando los edificios románicos más interesantes y fáciles de visitar de Oviedo y Siero, Villaviciosa, Teverga, Cangas de Onis, Avilés y Cabo Peñas y, por último, Oriente y Occidente. Se describen las características de esos edificios ofreciendo fotografías de los mismos y al final de la guía unos dibujos esquemáticos explicativos de las partes de que se componen: una iglesia románica asturiana, un arco de triunfo y abside (interior), una portada románica y una iglesia románica rural asturiana.