Session 5: Natural constraints in coda
http://hdl.handle.net/10256/643
Mon, 16 Sep 2024 15:03:22 GMT2024-09-16T15:03:22ZA new distribution on the simplex containing the Dirichlet family
http://hdl.handle.net/10256/726
A new distribution on the simplex containing the Dirichlet family
Ongaro, Andrea; Migliorati, Sonia; Monti, Gianna Serafina
Daunis-i-Estadella, Pepus; 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
Thu, 29 May 2008 00:00:00 GMThttp://hdl.handle.net/10256/7262008-05-29T00:00:00ZCompositional evolution with mass transfer in closed systems
http://hdl.handle.net/10256/725
Compositional evolution with mass transfer in closed systems
Jarauta Bragulat, Eusebio; Egozcue, Juan José
Daunis-i-Estadella, Pepus; 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
Thu, 29 May 2008 00:00:00 GMThttp://hdl.handle.net/10256/7252008-05-29T00:00:00ZA comparison of the alr and ilr transformations for kernel density estimation of compositional data
http://hdl.handle.net/10256/724
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, Pepus; 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
Thu, 29 May 2008 00:00:00 GMThttp://hdl.handle.net/10256/7242008-05-29T00:00:00ZCompositional Time Series: An Application
http://hdl.handle.net/10256/723
Compositional Time Series: An Application
Bergman, Jakob
Daunis-i-Estadella, Pepus; 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
Thu, 29 May 2008 00:00:00 GMThttp://hdl.handle.net/10256/7232008-05-29T00:00:00Z