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

Mathematical Methods 4

Exercises, exams and solutions for a third year maths course.

Mathematical Methods 2

Exercises, exams and solutions for a first year maths course.

Mathematical Programming

Exam questions and solutions for a third year mathematical programming course.

Mathematical Population Biology

math3052 lecture notes (2008-9)

Dispute resolution and “environmental” provisions in the WTO: promising developments for environmental matters

En la actualidad, la política internacional tiende hacia la unificación económica y cultural (globalización), en la cual la totalidad de las naciones se encuentran vinculadas por interacciones comerciales. Para que los compromisos adquiridos en materia de protección ambiental sean respetados, el derecho ambiental debe dejar de ser considerado como una disciplina separada y buscar sinergias que le permitan convertirse en un actor importante en las relaciones comerciales. En el presente ensayo se argumenta que el comercio internacional y el derecho ambiental han coevolucionado en las últimas tres décadas, hasta llegar a un balance en el concepto de desarrollo sostenible.Este artículo explora esta coevolución, mientras propone que la Organización Mundial del Comercio (OMC) puede jugar un importante rol en el cumplimiento de los objetivos internacionales de protección ambiental si sus disposiciones “verdes” y su sistema de resolución de conflictos son utilizados para promoverlos, tal como está sucediendo. Para ilustrar este punto, se presenta un breve resumen de los tratados comerciales y ambientales suscritos desde los años setenta, seguido de una explicación de las disposiciones legales de la OMC que pueden ser consideradas como “verdes”, con el fin de hallar puntos de convergencia que puedan ser utilizados por los países miembros no solo para justificar la adopción de normativas domésticas de protección ambiental, sino también para instar a los otros signatarios en el cumplimiento de sus obligaciones en esta área.

Intervention strategies on teachers focused on creating meaningful educational environments for mathematical abilities development

This article describes an intervention process undertaken in a training program for preschool and first grade teachers from public schools in Cali, Colombia. The objective of this process is to provide a space for teachers to reflect on pedagogical practices which allow them to generate educational processes that foster children’s understanding of mathematical knowledge in the classroom. A set of support strategies was presented for helping teachers in the design, analysis and implementation of learning environments as meaningful educational spaces. Furthermore, participants engaged in an analysis of their own intervention modalities to identify which modalities facilitate the development of mathematical abilities in children. In order to ascertain the transformations in the teachers’ learning environments, the mathematical competences and cognitive processes underlying the activities proposed in the classroom, as well as teacher intervention modalities and the types of student participation in classroom activities were examined both before and after the intervention process. Transformations in the teachers’ conceptions about the children’s abilities and their own practices in teaching mathematics in the classroom were evidenced.

The Construction of a State Peace Policy and the Process of Conflict Resolution

The academic literature specialized on processes of conflict resolution has focused on how third party intervention and mediation can contribute to end violent conflict, however it has also ignored the potential role of the state in countries affected by internal war. This article calls for a better understanding of the state, not only as a source of contemporary conflict but as a potential advocate of conflict resolution processes. It suggests that it is necessary and possible to involve the state in more effective processes of conflict resolution through the implementation of a State Peace Policy. The central argument is that some of the critical elements that should guide the state action when confronting the destructive logic of war and violence can be effectively undertaken through a public policy focused on building peace. After suggesting a general definition of State Peace Policy and highlighting some of its main attributes, the article recognizes that this sort of policy can also orientate the unfinished process of state consolidation in Colombia and other countries affected by internal conflict. The role of the state in the process of conflict resolution is crucial; however there are some issues normally ignored within the policy-making process, they must be seriously taken into account in order to eliminate the underlying structures that perpetuate conflict and delay the consolidation of sustainable peace.-----La literatura académica especializada en procesos de resolución de conflictos se ha centrado en el estudio de cómo la intervención de terceras partes y la labor de mediación pueden contribuir a poner fin a los conflictos violentos, sin embargo, también ha ignorado el rol potencial del estado en aquellos países afectados por la guerra interna. Este artículo llama la atención sobre la necesidad de estudiar al estado no sólo como la fuente de los conflictos contemporáneos, sino como un colaborador potencial en procesos de resolución de conflictos. Se sugiere que es necesario y también posible involucrar al estado en este tipo de procesos de una manera más efectiva a través de la implementación de una Política de Estado centrada en la Paz. El argumento central es que algunos de los elementos claves que deben guiar la acción del estado cuando éste se enfrenta a la lógica destructiva de la guerra y la violencia, pueden ser efectivamente tratados a través de una política pública que de prioridad a la construcción de la paz. Luego de sugerir una definición general y destacar algunos de los principales atributos de ésta clase de Política de Estado, el artículo reconoce que ésta podría también orientar el proceso inacabado de consolidación del estado en Colombia y en otros países afectados por conflictos internos. El rol del estado en el proceso para resolver los conflictos es crucial, sin embargo hay algunos aspectos normalmente ignorados durante el proceso de formulación de políticas que deben ser tomados en cuenta seriamente para eliminar las estructuras que perpetúan el conflicto y que retrasan la consolidación de la paz sostenible.

Teaching and learning geometry : issues and methods in mathematical education.

Recurso para la evaluación de la enseñanza y el aprendizaje de la geometría en la enseñanza secundaria desde la perspectiva de los nuevos docentes y de los que tienen más experiencia. Está diseñado para ampliar y profundizar el conocimiento de la materia y ofrecer consejos prácticos e ideas para el aula en el contexto de la práctica y la investigación actual. Hace especial hincapié en: comprender las ideas fundamentales del currículo de geometría; el aprendizaje de la geometría de manera efectiva; la investigación y la práctica actual; las ideas erróneas y los errores; el razonamiento de la geometría; la solución de problemas; el papel de la tecnología en el aprendizaje de la geometría.

Mathematical Studies for the IB Diploma.

Este práctico libro de ejercicios cubre las necesidades básicas para conseguir el Diploma de Bachillerato Internacional en estudios de matemáticas. Incluye consejos de profesores y estudiantes para realizar un examen óptimo y obtener el mayor rendimiento posible. Los temas del libro son: álgebra y números (ecuaciones de segundo grado), lógica y probabilidad, funciones (funciones exponenciales y gráficos), geometría y trigonometría (la regla del seno y el área de un triángulo, la regla del coseno), estadísticas (clasificación de datos, tablas de frecuencia y polígonos), cálculo diferencial (ecuaciones de línea tangentes), matemáticas financieras.

Improving awareness about the meaning of the principle of mathematical induction. 'Fomentar la conciencia sobre el significado del principio de inducción matemática'.

Resumen basado en el de la publicación

Why children differ in their mathematical attainment at primary school?.'¿Por qué los niños difieren en su competencia matemática en la escuela primaria?'.

Resúmen tomado parcialmente de la publicación.- El artículo forma parte de un monográfico dedicado a Psicología de las Matemáticas