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Daunis i Estadella, Josep
MartÃn FernÃ¡ndez, Josep Antoni 

Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Tolosana Delgado, Raimon
Boogaart, K. Gerald van den Mikes, TÃ¼nde Eynatten, Hilmar von 

2008 May 30  
The preceding two editions of CoDaWork included talks on the possible consideration
of densities as infinite compositions: Egozcue and DÂ´Ä±azBarrero (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 Hermitepolynomialsbased 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 2order Hermite polinomials (normal reference)
and 1order 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 Geologische Vereinigung; Institut dâ€™EstadÃstica de Catalunya; International Association for Mathematical Geology; CÃ tedra LluÃs SantalÃ³ dâ€™Aplicacions de la MatemÃ tica; Generalitat de Catalunya, Departament dâ€™InnovaciÃ³, Universitats i Recerca; Ministerio de EducaciÃ³n y Ciencia; Ingenio 2010. 

application/pdf  
Tolosana Delgado, R. et al. â€™Statistical treatment of grainsize curves and empirical distributions: densities as compositions?â€™ a CODAWORKâ€™08. Girona: La Universitat, 2008 [consulta: 15 maig 2008]. Necessita Adobe Acrobat. Disponible a Internet a: http://hdl.handle.net/10256/748  
http://hdl.handle.net/10256/748  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Tots els drets reservats  
Probabilitats
EstadÃstica matemÃ tica 

Statistical treatment of grainsize curves and empirical distributions: densities as compositions?  
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