Evidence functions: a compositional approach to information

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.Peer Reviewe

Polinomis i coeficients de reflexió

Els polinomis es solen representar o bé pels seus coeficients o bé pels seus zeros. Les dues representacions estan lligades per les fórmules de Cardano-Viète que expressen els coeficients com a funcions simètriques elementals dels zeros. La recursió descendent de Levinson defineix els coeficients de reflexió d'un polinomi. En aquest article es veu com es poden caracteritzar els polinomis en termes dels seus coeficients de reflexió, es donen resultats sobre polinomis autoreversos, que juguen un paper singular en aquesta representació, i es donen fórmules homòlogues a les de Cardano-Viète que relacionen zeros amb coeficients de reflexió. També es caracteritzen els polinomis de tipus Kakeya en termes de coeficients de reflexió, cosa que permet donar una demostració alternativa del teorema d'Eneström-Kakeya sobre la localització de zeros d'un polinomi. Molts d'aquests desenvolupaments estan relacionats amb teoria de control i anàlisi de senyals. En aquest context, els texts clàssics de localització de zeros són recursius. Hi ha casos singulars en els quals el procés recursiu queda aturat i s'ha de recórrer a tècniques de pertorbació per continuar-los. Aquestes tècniques sempre funcionen però no estan en general ben fonamentades. Aquí es prova que els polinomis no singulars són densos, amb la norma L2, al disc unitat, cosa que dóna base matemàtica a les tècniques de pertorbació.Polynomials can be represented by their coefficients or by their zeros. The link between these two representations is the Cardan–Vi`ete’s formulas that allow expressing coefficients as elementary symmetric functions in the zeros. Backward Levinson’s recursion defines reflection coefficients of a polynomial. These coefficients can be used to characterize polynomials. A complete classification of the set of all polynomials is obtained and two theorems on self-inversive polynomials are given. As a consequence of Levinson recursion, a counterpart of Cardan-Vi`ete’s formulas is presented. They express polynomial coefficients in terms of its reflection coefficients. Backward Levinson’s recursion for polynomials is used again to obtain the characterization of polynomials of Kakeya type by their reflection coefficients. This result leads to an alternative proof of Enestr¨om and Kakeya theorems on the location of zeros of polynomials. Most of these developments are related to control and signal analysis. In this framework, classical tests for locating zeroes of polynomials are recursive. There are singular cases in which such recursive tests are stopped and perturbation techniques should be applied to proceed. Perturbation techniques, although always successful, are not proven to be well-founded. The non-singular polynomials are proven to be dense in the set of all polynomials with respect to the L2-norm in the unit circle thus giving a mathematical foundation to perturbation techniques

Space-time compositional models: an introduction to simplicial partial differential operators

A function assigning a composition to space-time points is called a compositional or simplicial field. These fields can be analysed using the compositional analysis tools. A study of linear models for evolutionary compositions depending on one variable, usually time, was formulated by Egozcue and Jarauta-Bragulat (2014) in terms of the so-called simplicial linear differential equations. The foundations of differential and integral calculus for simplex-valued functions of one real variable, was presented by Egozcue, Jarauta-Bragulat and Díaz-Barrero (2011). In order to study compositions depending on space and/or time, reformulation and interpretation of traditional partial differential operators is required. These operators such as: partial derivatives, compositional gradient, directional derivative and divergence are of primary importance to state alternative models of processes as diffusion, advection and waves, from the compositional perspective. This kind of models, usually based on continuity of mass, circulation of a vector field along a curve and flux through surfaces, should be analyzed when compositional operators are used instead of the traditional gradient or divergence. This study is aimed at setting up the definitions, mathematical basis and interpretation of such operators.Peer ReviewedPostprint (author's final draft

Un estimador de error residual para el método de los elementos finitos

Los estimadores de error residuales se basan en resolver de manera aproximada la ecuación que caracteriza al error. En este artículo se presenta un estimador de error basado en la resolución de problemas locales mediante submallas que discretizan cada uno de los elementos. En cada uno de estos problemas elementales se imponen condiciones de contorno de Dirichlet. De esta manera se obtiene una primera estima que sólo tiene en cuenta información interior a los elementos. En esta primera fase no se considera la contribución al error asociada a los saltos de flujo a través de los lados de los elementos. En una segunda fase se incluye esta información. Sin embargo, a diferencia de otros estimadores, esto se lleva a cabo sin calcular los saltos y, por consiguiente, se evita tener que equilibrar los flujos de error. Esto se hace conservando la filosofía de la primera fase, es decir resolviendo problemas locales discretizados mediante submallas. Los subdominios asociados a estos problemas se solapan con los elementos y recubren sus lados. Esto último hace que esta segunda fase recoja el efecto de los saltos de flujo. En esta segunda fase, la estima se somete a restricciones adicionales que permiten que se pueda sumar al resultado de la primera fase. El estimador que se calcula a partir de la combinación de las dos fases proporciona buenos resultados en los ejemplos de aplicación, comparado con otros estimadores existentes.Peer Reviewe

Evidence information in Bayesian updating

Bayes theorem (discrete case) is taken as a paradigm of information acquisition. As men-tioned by Aitchison, Bayes formula can be identified with perturbation of a prior probability vector and a discrete likelihood function, both vectors being compositional. Considering prior, poste-rior and likelihood as elements of the simplex, a natural choice of distance between them is the Aitchison distance. Other geometrical features can also be considered using the Aitchison geom-etry. For instance, orthogonality in the simplex allows to think of orthogonal information, or the perturbation-difference to think of opposite information. The Aitchison norm provides a size of compositional vectors, and is thus a natural scalar measure of the information conveyed by the likelihood or captured by a prior or a posterior. It is called evidence information, or e-information for short. In order to support such e-information theory some principles of e-information are discussed. They essentially coincide with those of compositional data analysis. Also, a comparison of these principles of e-information with the axiomatic Shannon-information theory is performed. Shannon-information and developments thereof do not satisfy scale invariance and also violate subcomposi-tional coherence. In general, Shannon-information theory follows the philosophy of amalgamation when relating information given by an evidence-vector and some sub-vector, while the dimension reduction for the proposed e-information corresponds to orthogonal projections in the simplex. The result of this preliminary study is a set of properties of e-information that may constitute the basis of an axiomatic theory. A synthetic example is used to motivate the ideas and the subsequent discussion

Approaching predator-prey Lotka-Volterra equations by simplicial linear differential equations

Predator-prey Lotka-Volterra equations was one of the rst models reflecting interaction of different species and modeling evolution of respective populations. It considers a large population of hares (preys) which is depredated by an also large population of lynxes (predators). It proposes an increasing/decreasing law of the number of individuals in each population thus resulting in an apparently simple system of ordinary differential equations. However, the Lotka-Volterra equation, and most of its modi cations, is non-linear and its generalization to a larger number of species is not trivial. The present aim is to study approximations of the evolution of the proportion of species in the Lotka-Volterra equations using some simple model de ned in the simplex. Calculus in the simplex has been recently developed on the basis of the Aitchison geometry and the simplicial derivative. Evolution of proportions in time (or other parameters) can be represented as simplicial ordinary di erential equations from which the simpler models are the linear ones. Simplicial Linear Ordinary Di erential Equations are not able to model the evolution of the total mass of the population (total number of predators plus preys) but only the evolution of the proportions of the different species (ratio predators over preys). This way of analysis has been successful showing that the compositional growth of a population in the Malthusian exponential model and the Verhulst logistic model were exactly the same one: the rst order simplicial linear di erential equation with constant coeffcients whose solution is a compositional straight-line. This strategy of studying the total mass evolution and the compositional evolution separately is used to get a simplicial differential equation whose solutions approach suitably the compositional behavior of the Lotka-Volterra equations. This approach has additional virtues: it is linear and can be extended in an easy way to a number of species larger than two

Un estimador de error residual para el método de los elementos finitos

Los estimadores de error residuales se basan en resolver de manera aproximada la ecuación que caracteriza al error. En este artículo se presenta un estimador de error basado en la resolución de problemas locales mediante submallas que discretizan cada uno de los elementos. En cada uno de estos problemas elementales se imponen condiciones de contorno de Dirichlet. De esta manera se obtiene una primera estima que sólo tiene en cuenta información interior a los elementos. En esta primera fase no se considera la contribución al error asociada a los saltos de flujo a través de los lados de los elementos. En una segunda fase se incluye esta información. Sin embargo, a diferencia de otros estimadores, esto se lleva a cabo sin calcular los saltos y, por consiguiente, se evita tener que equilibrar los flujos de error. Esto se hace conservando la filosofía de la primera fase, es decir resolviendo problemas locales discretizados mediante submallas. Los subdominios asociados a estos problemas se solapan con los elementos y recubren sus lados. Esto último hace que esta segunda fase recoja el efecto de los saltos de flujo. En esta segunda fase, la estima se somete a restricciones adicionales que permiten que se pueda sumar al resultado de la primera fase. El estimador que se calcula a partir de la combinación de las dos fases proporciona buenos resultados en los ejemplos de aplicación, comparado con otros estimadores existentes.Peer Reviewe

Bayesian estimation of the orthogonal decomposition of a contingency table

In a multinomial sampling, contingency tables can be parametrized by probabilities of each cell. These probabilities constitute the joint probability function of two or more discrete random variables. These probability tables have been previously studied from a compositional point of view. The compositional analysis of probability tables ensures coherence when analysing sub-tables. The main results are: (1) given a probability table, the closest independent probability table is the product of their geometric marginals; (2) the probability table can be orthogonally decomposed into an independent table and an interaction table; (3) the departure of independence can be measured using simplicial deviance, which is the Aitchison square norm of the interaction table. In previous works, the analysis has been performed from a frequentist point of view. This contribution is aimed at providing a Bayesian assessment of the decomposition. The resulting model is a log-linear one, which parameters are the centered log-ratio transformations of the geometric marginals and the interaction table. Using a Dirichlet prior distribution of multinomial probabilities, the posterior distribution of multinomial probabilities is again a Dirichlet distribution. Simulation of this posterior allows to study the distribution of marginal and interaction parameters, checking the independence of the observed contingency table and cell interactions. The results corresponding to a two-way contingency table example are presented.Peer ReviewedPostprint (published version

Wave-height hazard analysis in Eastern Coast of Spain : Bayesian approach using generalized Pareto distribution

Standard practice of wave-height hazard analysis often pays little attention to the uncertainty of assessed return periods and occurrence probabilities. This fact favors the opinion that, when large events happen, the hazard assessment should change accordingly. However, uncertainty of the hazard estimates is normally able to hide the effect of those large events. This is illustrated using data from the Mediterranean coast of Spain, where the last years have been extremely disastrous. Thus, it is possible to compare the hazard assessment based on data previous to those years with the analysis including them. With our approach, no significant change is detected when the statistical uncertainty is taken into account. The hazard analysis is carried out with a standard model. Time-occurrence of events is assumed Poisson distributed. The wave-height of each event is modelled as a random variable which upper tail follows a Generalized Pareto Distribution (GPD). Moreover, wave-heights are assumed independent from event to event and also independent of their occurrence in time. A threshold for excesses is assessed empirically. The other three parameters (Poisson rate, shape and scale parameters of GPD) are jointly estimated using Bayes' theorem. Prior distribution accounts for physical features of ocean waves in the Mediterranean sea and experience with these phenomena. Posterior distribution of the parameters allows to obtain posterior distributions of other derived parameters like occurrence probabilities and return periods. Predictives are also available. Computations are carried out using the program BGPE v2.

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case, for example, with strictly positive data, or with compositional data, such as percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models that better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated.Peer Reviewe