http://hdl.handle.net/10256/643 2013-05-18T21:38:29Z 2013-05-18T21:38:29Z Ongaro, Andrea Migliorati, Sonia Monti, Gianna Serafina http://hdl.handle.net/10256/726 2012-11-30T09:15:22Z 2008-05-29T00:00:00Z

A new distribution on the simplex containing the Dirichlet family Ongaro, Andrea; Migliorati, Sonia; Monti, Gianna Serafina Daunis i Estadella, Josep; Martín Fernández, Josep Antoni The Dirichlet family owes its privileged status within simplex distributions to easyness of interpretation and good mathematical properties. In particular, we recall fundamental properties for the analysis of compositional data such as closure under amalgamation and subcomposition. From a probabilistic point of view, it is characterised (uniquely) by a variety of independence relationships which makes it indisputably the reference model for expressing the non trivial idea of substantial independence for compositions. Indeed, its well known inadequacy as a general model for compositional data stems from such an independence structure together with the poorness of its parametrisation. In this paper a new class of distributions (called Flexible Dirichlet) capable of handling various dependence structures and containing the Dirichlet as a special case is presented. The new model exhibits a considerably richer parametrisation which, for example, allows to model the means and (part of) the variance-covariance matrix separately. Moreover, such a model preserves some good mathematical properties of the Dirichlet, i.e. closure under amalgamation and subcomposition with new parameters simply related to the parent composition parameters. Furthermore, the joint and conditional distributions of subcompositions and relative totals can be expressed as simple mixtures of two Flexible Dirichlet distributions. The basis generating the Flexible Dirichlet, though keeping compositional invariance, shows a dependence structure which allows various forms of partitional dependence to be contemplated by the model (e.g. non-neutrality, subcompositional dependence and subcompositional non-invariance), independence cases being identified by suitable parameter configurations. In particular, within this model substantial independence among subsets of components of the composition naturally occurs when the subsets have a Dirichlet distribution

2008-05-29T00:00:00Z Jarauta Bragulat, Eusebio Egozcue, Juan José http://hdl.handle.net/10256/725 2012-06-28T12:30:36Z 2008-05-29T00:00:00Z

Compositional evolution with mass transfer in closed systems Jarauta Bragulat, Eusebio; Egozcue, Juan José Daunis i Estadella, Josep; Martín Fernández, Josep Antoni 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

2008-05-29T00:00:00Z Chacón, J.E. Martín Fernández, Josep Antoni Mateu i Figueras, Glòria http://hdl.handle.net/10256/724 2012-11-19T08:55:48Z 2008-05-29T00:00:00Z

A comparison of the alr and ilr transformations for kernel density estimation of compositional data Chacón, J.E.; Martín Fernández, Josep Antoni; Mateu i Figueras, Glòria Daunis i Estadella, Josep; Martín Fernández, Josep Antoni In a seminal paper, Aitchison and Lauder (1985) introduced classical kernel density estimation techniques in the context of compositional data analysis. Indeed, they gave two options for the choice of the kernel to be used in the kernel estimator. One of these kernels is based on the use the alr transformation on the simplex SD jointly with the normal distribution on RD-1. However, these authors themselves recognized that this method has some deficiencies. A method for overcoming these dificulties based on recent developments for compositional data analysis and multivariate kernel estimation theory, combining the ilr transformation with the use of the normal density with a full bandwidth matrix, was recently proposed in Martín-Fernández, Chacón and Mateu- Figueras (2006). Here we present an extensive simulation study that compares both methods in practice, thus exploring the finite-sample behaviour of both estimators

2008-05-29T00:00:00Z Bergman, Jakob http://hdl.handle.net/10256/723 2012-06-28T12:30:36Z 2008-05-29T00:00:00Z

Compositional Time Series: An Application Bergman, Jakob Daunis i Estadella, Josep; Martín Fernández, Josep Antoni The composition of the labour force is an important economic factor for a country. Often the changes in proportions of different groups are of interest. I this paper we study a monthly compositional time series from the Swedish Labour Force Survey from 1994 to 2005. Three models are studied: the ILR-transformed series, the ILR-transformation of the compositional differenced series of order 1, and the ILRtransformation of the compositional differenced series of order 12. For each of the three models a VAR-model is fitted based on the data 1994-2003. We predict the time series 15 steps ahead and calculate 95 % prediction regions. The predictions of the three models are compared with actual values using MAD and MSE and the prediction regions are compared graphically in a ternary time series plot. We conclude that the first, and simplest, model possesses the best predictive power of the three models

2008-05-29T00:00:00Z Aguilar, Lucía Barceló i Vidal, Carles http://hdl.handle.net/10256/722 2012-06-28T12:30:36Z 2008-05-29T00:00:00Z

Multivariate ARIMA Compositional Time Series Analysis Aguilar, Lucía; Barceló i Vidal, Carles Daunis i Estadella, Josep; Martín Fernández, Josep Antoni A compositional time series is obtained when a compositional data vector is observed at different points in time. Inherently, then, a compositional time series is a multivariate time series with important constraints on the variables observed at any instance in time. Although this type of data frequently occurs in situations of real practical interest, a trawl through the statistical literature reveals that research in the field is very much in its infancy and that many theoretical and empirical issues still remain to be addressed. Any appropriate statistical methodology for the analysis of compositional time series must take into account the constraints which are not allowed for by the usual statistical techniques available for analysing multivariate time series. One general approach to analyzing compositional time series consists in the application of an initial transform to break the positive and unit sum constraints, followed by the analysis of the transformed time series using multivariate ARIMA models. In this paper we discuss the use of the additive log-ratio, centred log-ratio and isometric log-ratio transforms. We also present results from an empirical study designed to explore how the selection of the initial transform affects subsequent multivariate ARIMA modelling as well as the quality of the forecasts

2008-05-29T00:00:00Z