Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Searching for consensus in ahp-group decision making. A bayesian perspective

This paper sets out to identify the initial positions of the different decision makers who intervene in a group decision making process with a reduced number of actors, and to establish possible consensus paths between these actors. As a methodological support, it employs one of the most widely-known multicriteria decision techniques, namely, the Analytic Hierarchy Process (AHP). Assuming that the judgements elicited by the decision makers follow the so-called multiplicative model (Crawford and Williams, 1985; Altuzarra et al., 1997; Laininen and Hämäläinen, 2003) with log-normal errors and unknown variance, a Bayesian approach is used in the estimation of the relative priorities of the alternatives being compared. These priorities, estimated by way of the median of the posterior distribution and normalised in a distributive manner (priorities add up to one), are a clear example of compositional data that will be used in the search for consensus between the actors involved in the resolution of the problem through the use of Multidimensional Scaling tools

Group preference structures in AHP–group decision making

This paper presents a procedure that allows us to determine the preference structures (PS) associated to each of the different groups of actors that can be identified in a group decision making problem with a large number of individuals. To that end, it makes use of the Analytic Hierarchy Process (AHP) (Saaty, 1980) as the technique to solve discrete multicriteria decision making problems. This technique permits the resolution of multicriteria, multienvironment and multiactor problems in which subjective aspects and uncertainty have been incorporated into the model, constructing ratio scales corresponding to the priorities relative to the elements being compared, normalised in a distributive manner (wi = 1). On the basis of the individuals’ priorities we identify different clusters for the decision makers and, for each of these, the associated preference structure using, to that end, tools analogous to those of Multidimensional Scaling. The resulting PS will be employed to extract knowledge for the subsequent negotiation processes and, should it be necessary, to determine the relative importance of the alternatives being compared using anyone of the existing procedures

Analysis of Pleistocene paleodrainage evolution in the Po Basin (Italy) by multivariate statistical techniques

In order to obtain a high-resolution Pleistocene stratigraphy, eleven continuously cored boreholes, 100 to 220m deep were drilled in the northern part of the Po Plain by Regione Lombardia in the last five years. Quantitative provenance analysis (QPA, Weltje and von Eynatten, 2004) of Pleistocene sands was carried out by using multivariate statistical analysis (principal component analysis, PCA, and similarity analysis) on an integrated data set, including high-resolution bulk petrography and heavy-mineral analyses on Pleistocene sands and of 250 major and minor modern rivers draining the southern flank of the Alps from West to East (Garzanti et al, 2004; 2006). Prior to the onset of major Alpine glaciations, metamorphic and quartzofeldspathic detritus from the Western and Central Alps was carried from the axial belt to the Po basin longitudinally parallel to the SouthAlpine belt by a trunk river (Vezzoli and Garzanti, 2008). This scenario rapidly changed during the marine isotope stage 22 (0.87 Ma), with the onset of the first major Pleistocene glaciation in the Alps (Muttoni et al, 2003). PCA and similarity analysis from core samples show that the longitudinal trunk river at this time was shifted southward by the rapid southward and westward progradation of transverse alluvial river systems fed from the Central and Southern Alps. Sediments were transported southward by braided river systems as well as glacial sediments transported by Alpine valley glaciers invaded the alluvial plain. Kew words: Detrital modes; Modern sands; Provenance; Principal Components Analysis; Similarity, Canberra Distance; palaeodrainage

Desenvolupament d'un sistema expert com a eina per a una millor gestió de la qualitat de les aigües fluvials

Avui en dia no es pot negar el fet que els humans són un component més de les conques fluvials i que la seva activitat afecta enormement la qualitat de les aigües. A nivell europeu, l'elevada densitat de població situada en les conques fluvials ha comportat un increment de la mala qualitat de les seves aigües fluvials. En les darreres dècades l'increment de les càrregues de nutrients en els sistemes aquàtics ha esdevingut un problema prioritari a solucionar per les administracions competents en matèria d'aigua. La gestió dels ecosistemes fluvials no és una tasca fàcil. Els gestors es troben amb què són sistemes molt complexos, donada l'estreta relació existent entre els ecosistemes fluvials i els ecosistemes terrestres que drenen. Addicionalment a la complexitat d'aquests sistemes es troba la dificultat associada de la gestió o control de les entrades de substàncies contaminants tant de fonts puntuals com difoses. Per totes aquestes raons la gestió de la qualitat de les aigües fluvials esdevé una tasca complexa que requereix un enfocament multidisciplinar. Per tal d'assolir aquest enfocament diverses eines han estat utilitzades, des de models matemàtics fins a sistemes experts i sistemes de suport a la decisió. Però, la major part dels esforços han estat encarats cap a la resolució de problemes de reduïda complexitat, fent que molts dels problemes ambientals complexos, com ara la gestió dels ecosistemes fluvials, no hagin estat vertaderament tractats. Per tant, es requereix l'aplicació d'eines que siguin de gran ajuda en els processos de presa de decisions i que incorporin un ampli coneixement heurístic i empíric: sistemes experts i sistemes de suport a la decisió. L'òptima gestió de la qualitat de l'aigua fluvial requereix una aproximació integrada i multidisciplinar, que pot ésser aconseguida amb una eina intel·ligent construïda sobre els conceptes i mètodes del raonament humà. La present tesi descriu la metodologia desenvolupada i aplicada per a la creació i construcció d'un Sistema Expert, així com el procés de desenvolupament d'aquest Sistema Expert, com el principal mòdul de raonament d'un Sistema de Suport a la Decisió Ambiental. L'objectiu principal de la present tesi ha estat el desenvolupament d'una eina d'ajuda en el procés de presa de decisions dels gestors de l'aigua en la gestió de trams fluvials alterats antròpicament per tal de millorar la qualitat de la seva aigua fluvial. Alhora, es mostra el funcionament de l'eina desenvolupada a través de dos casos d'estudi. Els resultats derivats del Sistema Expert desenvolupat, implementat i presentat en la present tesi mostren que aquests sistemes poden ésser eines útils per a millorar la gestió dels ecosistemes fluvials.