Item
DaunisiEstadella, Pepus
MartÃn FernÃ¡ndez, Josep Antoni 

Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Egozcue, Juan JosÃ©
DÃaz Barrero, JosÃ© Luis PawlowskyGlahn, Vera 

2008 May 28  
A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table has
n rows and m columns and all probabilities are nonnull. This kind of table can be
seen as an element in the simplex of n Â· m parts. In this context, the marginals are
identified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclidean
elements of the Aitchison geometry of the simplex can also be translated into the table
of probabilities: subspaces, orthogonal projections, distances.
Two important questions are addressed: a) given a table of probabilities, which is
the nearest independent table to the initial one? b) which is the largest orthogonal
projection of a row onto a column? or, equivalently, which is the information in a
row explained by a column, thus explaining the interaction? To answer these questions
three orthogonal decompositions are presented: (1) by columns and a rowwise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independent
twoway tables and fully dependent tables representing rowcolumn interaction. An
important result is that the nearest independent table is the product of the two (row
and column)wise geometric marginal tables. A corollary is that, in an independent
table, the geometric marginals conform with the traditional (arithmetic) marginals.
These decompositions can be compared with standard loglinear models.
Key words: balance, compositional data, simplex, Aitchison geometry, composition,
orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure,
contingency table 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  
Egozcue, J.J.; DÃaz Barrero, J.L.; Pawlowsky Glahn, V. â€™Compositional analysis of bivariate discrete probabilitiesâ€™ a CODAWORKâ€™08. Girona: La Universitat, 2008 [consulta: 12 maig 2008]. Necessita Adobe Acrobat. Disponible a Internet a: http://hdl.handle.net/10256/717  
http://hdl.handle.net/10256/717  
eng  
Universitat de Girona. Departament dâ€™InformÃ tica i MatemÃ tica Aplicada  
Tots els drets reservats  
Aitchison, Geometria dâ€™
Probabilitats 

Compositional analysis of bivariate discrete probabilities  
info:eurepo/semantics/conferenceObject  
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