Mixing compositions and other scales

Theory of compositional data analysis is often focused on the composition only. However in practical applications we often treat a composition together with covariableswith some other scale. This contribution systematically gathers and develop statistical tools for this situation. For instance, for the graphical display of the dependenceof a composition with a categorical variable, a colored set of ternary diagrams mightbe a good idea for a first look at the data, but it will fast hide important aspects ifthe composition has many parts, or it takes extreme values. On the other hand colored scatterplots of ilr components could not be very instructive for the analyst, if theconventional, black-box ilr is used.Thinking on terms of the Euclidean structure of the simplex, we suggest to set upappropriate projections, which on one side show the compositional geometry and on theother side are still comprehensible by a non-expert analyst, readable for all locations andscales of the data. This is e.g. done by defining special balance displays with carefully-selected axes. Following this idea, we need to systematically ask how to display, explore,describe, and test the relation to complementary or explanatory data of categorical, real,ratio or again compositional scales.This contribution shows that it is sufficient to use some basic concepts and very fewadvanced tools from multivariate statistics (principal covariances, multivariate linearmodels, trellis or parallel plots, etc.) to build appropriate procedures for all these combinations of scales. This has some fundamental implications in their software implementation, and how might they be taught to analysts not already experts in multivariateanalysis

CoDa-dendrogram: A new exploratory too

The use of orthonormal coordinates in the simplex and, particularly, balance coordinates, has suggested the use of a dendrogram for the exploratory analysis of compositional data. The dendrogram is based on a sequential binary partition of a compositional vector into groups of parts. At each step of a partition, one group of parts isdivided into two new groups, and a balancing axis in the simplex between both groupsis defined. The set of balancing axes constitutes an orthonormal basis, and the projections of the sample on them are orthogonal coordinates. They can be represented in adendrogram-like graph showing: (a) the way of grouping parts of the compositional vector; (b) the explanatory role of each subcomposition generated in the partition process;(c) the decomposition of the total variance into balance components associated witheach binary partition; (d) a box-plot of each balance. This representation is useful tohelp the interpretation of balance coordinates; to identify which are the most explanatory coordinates; and to describe the whole sample in a single diagram independentlyof the number of parts of the sample

Scene Classification Using a Hybrid Generative/Discriminative Approach

We investigate whether dimensionality reduction using a latent generative model is beneficial for the task of weakly supervised scene classification. In detail, we are given a set of labeled images of scenes (for example, coast, forest, city, river, etc.), and our objective is to classify a new image into one of these categories. Our approach consists of first discovering latent ";topics"; using probabilistic Latent Semantic Analysis (pLSA), a generative model from the statistical text literature here applied to a bag of visual words representation for each image, and subsequently, training a multiway classifier on the topic distribution vector for each image. We compare this approach to that of representing each image by a bag of visual words vector directly and training a multiway classifier on these vectors. To this end, we introduce a novel vocabulary using dense color SIFT descriptors and then investigate the classification performance under changes in the size of the visual vocabulary, the number of latent topics learned, and the type of discriminative classifier used (k-nearest neighbor or SVM). We achieve superior classification performance to recent publications that have used a bag of visual word representation, in all cases, using the authors' own data sets and testing protocols. We also investigate the gain in adding spatial information. We show applications to image retrieval with relevance feedback and to scene classification in videos

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 hasn rows and m columns and all probabilities are non-null. This kind of table can beseen as an element in the simplex of n · m parts. In this context, the marginals areidentified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclideanelements of the Aitchison geometry of the simplex can also be translated into the tableof probabilities: subspaces, orthogonal projections, distances.Two important questions are addressed: a) given a table of probabilities, which isthe nearest independent table to the initial one? b) which is the largest orthogonalprojection of a row onto a column? or, equivalently, which is the information in arow explained by a column, thus explaining the interaction? To answer these questionsthree orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independenttwo-way tables and fully dependent tables representing row-column interaction. Animportant result is that the nearest independent table is the product of the two (rowand column)-wise geometric marginal tables. A corollary is that, in an independenttable, 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

Folner sequences and finite operators

This article analyzes Folner sequences of projections for bounded linear operators and their relationship to the class of finite operators introduced by Williams in the 70ies. We prove that each essentially hyponormal operator has a proper Folner sequence (i.e. a Folner sequence of projections strongly converging to 1). In particular, any quasinormal, any subnormal, any hyponormal and any essentially normal operator has a proper Folner sequence. Moreover, we show that an operator is finite if and only if it has a proper Folner sequence or if it has a non-trivial finite dimensional reducing subspace. We also analyze the structure of operators which have no Folner sequence and give examples of them. For this analysis we introduce the notion of strongly non-Folner operators, which are far from finite block reducible operators, in some uniform sense, and show that this class coincides with the class of non-finite operators.

Discovering similarities in the time use patterns of the Spanish Autonomous Communities by Fuzzy techniques

In 2000 the European Statistical Office published the guidelines for developing theHarmonized European Time Use Surveys system. Under such a unified framework,the first Time Use Survey of national scope was conducted in Spain during 2002–03. The aim of these surveys is to understand human behavior and the lifestyle ofpeople. Time allocation data are of compositional nature in origin, that is, they aresubject to non-negativity and constant-sum constraints. Thus, standard multivariatetechniques cannot be directly applied to analyze them. The goal of this work is toidentify homogeneous Spanish Autonomous Communities with regard to the typicalactivity pattern of their respective populations. To this end, fuzzy clustering approachis followed. Rather than the hard partitioning of classical clustering, where objects areallocated to only a single group, fuzzy method identify overlapping groups of objectsby allowing them to belong to more than one group. Concretely, the probabilistic fuzzyc-means algorithm is conveniently adapted to deal with the Spanish Time Use Surveymicrodata. As a result, a map distinguishing Autonomous Communities with similaractivity pattern is drawn.Key words: Time use data, Fuzzy clustering; FCM; simplex space; Aitchison distance

Bandwidth allocation based on real time calculations using the convolution approach

This paper focuses on one of the methods for bandwidth allocation in an ATM network: the convolution approach. The convolution approach permits an accurate study of the system load in statistical terms by accumulated calculations, since probabilistic results of the bandwidth allocation can be obtained. Nevertheless, the convolution approach has a high cost in terms of calculation and storage requirements. This aspect makes real-time calculations difficult, so many authors do not consider this approach. With the aim of reducing the cost we propose to use the multinomial distribution function: the enhanced convolution approach (ECA). This permits direct computation of the associated probabilities of the instantaneous bandwidth requirements and makes a simple deconvolution process possible. The ECA is used in connection acceptance control, and some results are presented

On concentrators and related approximation constants

Pippenger [Pi77] showed the existence of (6m,4m,3m,6)-concentrator for each positive integer m using a probabilistic method. We generalize his approach and prove existence of (6m,4m,3m,5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44.5 (established by Kalton and Roberts in [KaRo83] to 39. We show a more direct connection of the latter problem to the Whitney type estimate for approximation of continuous functions on a cube in &b&R&/b&&sup&d&/sup& by linear functions, and improve the estimate of this Whitney constant from 802 (proved by Brudnyi and Kalton in [BrKa00] to 73.

Mapeo con robot móvil: caracterización y modelado

El projecte tracta de la fabricació d’un robot mòbil que sigui capaç de realitzar el mapeig del seu entorn, evitant els obstacles que es pugui trobar al transcurs del seu recorregut. Per tal d’obtenir els resultats desitjats, s’ha realitzat la caracterització dels seus components emprant algorismes probabilístics ,amb els que s’ha pogut determinar la seva eficiència. Finalment s’han realitzat els algorismes corresponents per graficar el mapeig del recorregut del robot i del seu entorn, aconseguint el objectiu plantejat.