CODAWORK’08http://hdl.handle.net/10256/6182017-12-16T03:25:58Z2017-12-16T03:25:58ZExperimental design on the simplexAtkinson, A.C.http://hdl.handle.net/10256/7532013-07-17T09:56:48Z2008-05-30T00:00:00ZExperimental design on the simplex
Atkinson, A.C.
Martín Fernández, Josep Antoni; Daunis i Estadella, Josep
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.
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2008-05-30T00:00:00ZRobust Factor Analysis for Compositional DataFilzmoser, PeterHron, KarelReimann, ClemensGarrett, Robert G.http://hdl.handle.net/10256/7522012-06-28T12:30:36Z2008-05-30T00:00:00ZRobust Factor Analysis for Compositional Data
Filzmoser, Peter; Hron, Karel; Reimann, Clemens; Garrett, Robert G.
Daunis i Estadella, Josep; Martín Fernández, Josep Antoni
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
2008-05-30T00:00:00ZVertebrates Limb Geometry in the Simplex spaceDaunis i Estadella, JosepMateu i Figueras, GlòriaThió i Fernández de Henestrosa, SantiagoRodrigues, L.http://hdl.handle.net/10256/7512012-06-28T12:30:36Z2008-05-30T00:00:00ZVertebrates Limb Geometry in the Simplex space
Daunis i Estadella, Josep; Mateu i Figueras, Glòria; Thió i Fernández de Henestrosa, Santiago; Rodrigues, L.
Daunis i Estadella, Josep; Martín Fernández, Josep Antoni
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
2008-05-30T00:00:00ZModelling of Mercury’s surface composition and remote detection from the orbit with the BepiColombo Mercury Planetary OrbiterLammer, HelmutWurz, PeterMartín Fernández, Josep AntoniLichtenegger, Herbert I.M.Khodachenko, Maxim L.http://hdl.handle.net/10256/7502012-11-19T08:56:48Z2008-05-30T00:00:00ZModelling of Mercury’s surface composition and remote detection from the orbit with the BepiColombo Mercury Planetary Orbiter
Lammer, Helmut; Wurz, Peter; Martín Fernández, Josep Antoni; Lichtenegger, Herbert I.M.; Khodachenko, Maxim L.
Daunis i Estadella, Josep; Martín Fernández, Josep Antoni
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
2008-05-30T00:00:00Z