Compositional analysis of bivariate discrete probabilities

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 non-null. This kind of table can be seen as an element in the simplex of n · 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 row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independent two-way tables and fully dependent tables representing row-column 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 log-linear models. Key words: balance, compositional data, simplex, Aitchison geometry, composition, orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure, contingency table

Using appearance and context for outdoor scene object classification

We propose a probabilistic object classifier for outdoor scene analysis as a first step in solving the problem of scene context generation. The method begins with a top-down control, which uses the previously learned models (appearance and absolute location) to obtain an initial pixel-level classification. This information provides us the core of objects, which is used to acquire a more accurate object model. Therefore, their growing by specific active regions allows us to obtain an accurate recognition of known regions. Next, a stage of general segmentation provides the segmentation of unknown regions by a bottom-strategy. Finally, the last stage tries to perform a region fusion of known and unknown segmented objects. The result is both a segmentation of the image and a recognition of each segment as a given object class or as an unknown segmented object. Furthermore, experimental results are shown and evaluated to prove the validity of our proposal