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

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

Hyperbolic Geometry

Exercises, exam questions and solutions for a fourth year hyperbolic geometry course. Diagrams for the questions are all together in the support.zip file, as .eps files

Teaching and learning geometry : issues and methods in mathematical education.

Recurso para la evaluación de la enseñanza y el aprendizaje de la geometría en la enseñanza secundaria desde la perspectiva de los nuevos docentes y de los que tienen más experiencia. Está diseñado para ampliar y profundizar el conocimiento de la materia y ofrecer consejos prácticos e ideas para el aula en el contexto de la práctica y la investigación actual. Hace especial hincapié en: comprender las ideas fundamentales del currículo de geometría; el aprendizaje de la geometría de manera efectiva; la investigación y la práctica actual; las ideas erróneas y los errores; el razonamiento de la geometría; la solución de problemas; el papel de la tecnología en el aprendizaje de la geometría.

Ontological beliefs and their impact on teaching elementary geometry. 'Creencias ontológicas y su impacto en la enseñanza de la geometría elemental'.

Resumen basado en el de la publicación

Non-compositional forms and the continuity assumption. 'La hipótesis de la continuidad y las formas no composicionales'.

Monográfico con el título: 'The debate on language acquisitions: constructivism versus innatism'. Resumen basado en el de la publicación

Reasoning by contradiction in dynamic geometry

Resumen basado en el de la publicación

Measurement Quality in Indicators of Compositions: a Compositional Multitrait-Multimethod Approach

Compositional data, also called multiplicative ipsative data, are common in survey research instruments in areas such as time use, budget expenditure and social networks. Compositional data are usually expressed as proportions of a total, whose sum can only be 1. Owing to their constrained nature, statistical analysis in general, and estimation of measurement quality with a confirmatory factor analysis model for multitrait-multimethod (MTMM) designs in particular are challenging tasks. Compositional data are highly non-normal, as they range within the 0-1 interval. One component can only increase if some other(s) decrease, which results in spurious negative correlations among components which cannot be accounted for by the MTMM model parameters. In this article we show how researchers can use the correlated uniqueness model for MTMM designs in order to evaluate measurement quality of compositional indicators. We suggest using the additive log ratio transformation of the data, discuss several approaches to deal with zero components and explain how the interpretation of MTMM designs di ers from the application to standard unconstrained data. We show an illustration of the method on data of social network composition expressed in percentages of partner, family, friends and other members in which we conclude that the faceto-face collection mode is generally superior to the telephone mode, although primacy e ects are higher in the face-to-face mode. Compositions of strong ties (such as partner) are measured with higher quality than those of weaker ties (such as other network members)

Compositional effects on productivity, labour cost and export adjustments. Bruegel Policy Contribution 2012/11, June 2012

Sectoral shifts, such as shrinkage of low labour productivity and the low-wage construction sector, can lead to apparent increased aggregate average labour productivity and average wages, especially when capital intensity differs across sectors. For 11 main sectors and 13 manufacturing sub-sectors, we quantify the compositional effects on productivity, wages and unit labour costs (ULCs) based and real effective exchange rates (REER), for 24 EU countries. Compositional effects are greatest in Ireland, where the pharmaceutical sector drives the growth of output and productivity, but other sectors have suffered greatly and have not yet recovered. Our new ULC-REER measurements, which are free from compositional effects, correlate well with export performance. Among the countries facing the most severe external adjustment challenges, Lithuania, Portugal and Ireland have been the most successful based on five indicators, and Latvia, Estonia and Greece the least successful. There is evidence of downward wage flexibility in some countries, but wage cuts have corrected just a small fraction of pre-crisis wage rises and came with massive reductions in employment even in the business sector excluding construction and real estate, highlighting the difficulty of adjusting wages downward.

The effect of flow field & geometry on the dynamic contact angle

A number of recent experiments suggest that, at a given wetting speed, the dynamic contact angle formed by an advancing liquid-gas interface with a solid substrate depends on the flow field and geometry near the moving contact line. In the present work, this effect is investigated in the framework of an earlier developed theory that was based on the fact that dynamic wetting is, by its very name, a process of formation of a new liquid-solid interface (newly “wetted” solid surface) and hence should be considered not as a singular problem but as a particular case from a general class of flows with forming or/and disappearing interfaces. The results demonstrate that, in the flow configuration of curtain coating, where a liquid sheet (“curtain”) impinges onto a moving solid substrate, the actual dynamic contact angle indeed depends not only on the wetting speed and material constants of the contacting media, as in the so-called slip models, but also on the inlet velocity of the curtain, its height, and the angle between the falling curtain and the solid surface. In other words, for the same wetting speed the dynamic contact angle can be varied by manipulating the flow field and geometry near the moving contact line. The obtained results have important experimental implications: given that the dynamic contact angle is determined by the values of the surface tensions at the contact line and hence depends on the distributions of the surface parameters along the interfaces, which can be influenced by the flow field, one can use the overall flow conditions and the contact angle as a macroscopic multiparametric signal-response pair that probes the dynamics of the liquid-solid interface. This approach would allow one to investigate experimentally such properties of the interface as, for example, its equation of state and the rheological properties involved in the interface’s response to an external torque, and would help to measure its parameters, such as the coefficient of sliding friction, the surface-tension relaxation time, and so on.