Dynamic graphics of parametrically linked multivariate methods used in compositional data analysis

Many multivariate methods that are apparently distinct can be linked by introducing one or more parameters in their definition. Methods that can be linked in this way are correspondence analysis, unweighted or weighted logratio analysis (the latter also known as "spectral mapping"), nonsymmetric correspondence analysis, principal component analysis (with and without logarithmic transformation of the data) and multidimensional scaling. In this presentation I will show how several of these methods, which are frequently used in compositional data analysis, may be linked through parametrizations such as power transformations, linear transformations and convex linear combinations. Since the methods of interest here all lead to visual maps of data, a "movie" can be made where where the linking parameter is allowed to vary in small steps: the results are recalculated "frame by frame" and one can see the smooth change from one method to another. Several of these "movies" will be shown, giving a deeper insight into the similarities and differences between these methods

Robust Factor Analysis for Compositional Data

Factor analysis as frequent technique for multivariate data inspection is widely used also for compositional data analysis. The usual way is to use a centered logratio (clr) transformation to obtain the random vector y of dimension D. The factor model is then y = Λf + e (1) with the factors f of dimension k < D, the error term e, and the loadings matrix Λ. Using the usual model assumptions (see, e.g., Basilevsky, 1994), the factor analysis model (1) can be written as Cov(y) = ΛΛT + ψ (2) where ψ = Cov(e) has a diagonal form. The diagonal elements of ψ as well as the loadings matrix Λ are estimated from an estimation of Cov(y). Given observed clr transformed data Y as realizations of the random vector y. Outliers or deviations from the idealized model assumptions of factor analysis can severely effect the parameter estimation. As a way out, robust estimation of the covariance matrix of Y will lead to robust estimates of Λ and ψ in (2), see Pison et al. (2003). Well known robust covariance estimators with good statistical properties, like the MCD or the S-estimators (see, e.g. Maronna et al., 2006), rely on a full-rank data matrix Y which is not the case for clr transformed data (see, e.g., Aitchison, 1986). The isometric logratio (ilr) transformation (Egozcue et al., 2003) solves this singularity problem. The data matrix Y is transformed to a matrix Z by using an orthonormal basis of lower dimension. Using the ilr transformed data, a robust covariance matrix C(Z) can be estimated. The result can be back-transformed to the clr space by C(Y ) = V C(Z)V T where the matrix V with orthonormal columns comes from the relation between the clr and the ilr transformation. Now the parameters in the model (2) can be estimated (Basilevsky, 1994) and the results have a direct interpretation since the links to the original variables are still preserved. The above procedure will be applied to data from geochemistry. Our special interest is on comparing the results with those of Reimann et al. (2002) for the Kola project data

Implementación del modelo hidrológico de Témez para la evaluación de recursos hídricos con GRASS GIS: fase superficial y subterránea

Se presenta la implementación del modelo hidrológico distribuido de Témez sobre GRASS GIS. Este modelo se utiliza para la evaluación de recursos hídricos en régimen natural con paso mensual y para la totalidad del territorio español, tal como aparece en el Libro Blanco del Agua en España. A partir de las variables de entrada, precipitación y evapotranspiración potencial y los parámetros hidrológicos, el modelo obtiene los mapas de los distintos almacenamientos, humedad en el suelo y volumen de acuífero, y de las variables de salida del ciclo hidrológico, evapotranspiración y escorrentía total, obtenida esta última como suma de la escorrentía superficial y subterránea. El objetivo final del trabajo es la implementación de los componentes superficiales y subterráneos en el modelo hidrológico, desarrollando para ello un programa que hace funcional en GRASS GIS el modelo matemático en que se basa la evaluación de recursos hídricos

ROC parameters in item and context recognition. 'Par??metros ROC en el reconocimiento de items y contextos'

Resumen tomado de la publicaci??n. Resumen tambi??n en ingl??s

Enhancing MPLS QoS routing algorithms by using the network protection degree paradigm

IP based networks still do not have the required degree of reliability required by new multimedia services, achieving such reliability will be crucial in the success or failure of the new Internet generation. Most of existing schemes for QoS routing do not take into consideration parameters concerning the quality of the protection, such as packet loss or restoration time. In this paper, we define a new paradigm to develop new protection strategies for building reliable MPLS networks, based on what we have called the network protection degree (NPD). This NPD consists of an a priori evaluation, the failure sensibility degree (FSD), which provides the failure probability and an a posteriori evaluation, the failure impact degree (FID), to determine the impact on the network in case of failure. Having mathematical formulated these components, we point out the most relevant components. Experimental results demonstrate the benefits of the utilization of the NPD, when used to enhance some current QoS routing algorithms to offer a certain degree of protection

Calibration of a Structured Light-Based Stereo Catadioptric Sensor

Catadioptric sensors are combinations of mirrors and lenses made in order to obtain a wide field of view. In this paper we propose a new sensor that has omnidirectional viewing ability and it also provides depth information about the nearby surrounding. The sensor is based on a conventional camera coupled with a laser emitter and two hyperbolic mirrors. Mathematical formulation and precise specifications of the intrinsic and extrinsic parameters of the sensor are discussed. Our approach overcomes limitations of the existing omni-directional sensors and eventually leads to reduced costs of production

Prediction of glucose excursions under uncertain parameters and food intake in intensive insulin therapy for type 1 diabetes mellitus

Considering the difficulty in the insulin dosage selection and the problem of hyper- and hypoglycaemia episodes in type 1 diabetes, dosage-aid systems appear as tremendously helpful for these patients. A model-based approach to this problem must unavoidably consider uncertainty sources such as the large intra-patient variability and food intake. This work addresses the prediction of glycaemia for a given insulin therapy face to parametric and input uncertainty, by means of modal interval analysis. As result, a band containing all possible glucose excursions suffered by the patient for the given uncertainty is obtained. From it, a safer prediction of possible hyper- and hypoglycaemia episodes can be calculated

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)

Una aproximación de la variable aleatoria a procesos de toma de decisión que implican condiciones de riesgo e incertidumbre

La variable aleatoria es una función matemática que permite asignar valores numéricos a cada uno de los posibles resultados obtenidos en un evento de naturaleza aleatoria. Si el número de estos resultados se puede contar, se tiene un conjunto discreto; por el contrario, cuando el número de resultados es infinito y no se puede contar, se tiene un conjunto continuo. El objetivo de la variable aleatoria es permitir adelantar estudios probabilísticos y estadísticos a partir del establecimiento de una asignación numérica a través de la cual se identifiquen cada uno de los resultados que pueden ser obtenidos en el desarrollo de un evento determinado. El valor esperado y la varianza son los parámetros por medio de los cuales es posible caracterizar el comportamiento de los datos reunidos en el desarrollo de una situación experimental; el valor esperado permite establecer el valor sobre el cual se centra la distribución de la probabilidad, mientras que la varianza proporciona información acerca de la manera como se distribuyen los datos obtenidos. Adicionalmente, las distribuciones de probabilidad son funciones numéricas asociadas a la variable aleatoria que describen la asignación de probabilidad para cada uno de los elementos del espacio muestral y se caracterizan por ser un conjunto de parámetros que establecen su comportamiento funcional, es decir, cada uno de los parámetros propios de la distribución suministra información del experimento aleatorio al que se asocia. El documento se cierra con una aproximación de la variable aleatoria a procesos de toma de decisión que implican condiciones de riesgo e incertidumbre.

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.

Option pricing in market models driven by telegraph processes with jumps

Esta tesis está dividida en dos partes: en la primera parte se presentan y estudian los procesos telegráficos, los procesos de Poisson con compensador telegráfico y los procesos telegráficos con saltos. El estudio presentado en esta primera parte incluye el cálculo de las distribuciones de cada proceso, las medias y varianzas, así como las funciones generadoras de momentos entre otras propiedades. Utilizando estas propiedades en la segunda parte se estudian los modelos de valoración de opciones basados en procesos telegráficos con saltos. En esta parte se da una descripción de cómo calcular las medidas neutrales al riesgo, se encuentra la condición de no arbitraje en este tipo de modelos y por último se calcula el precio de las opciones Europeas de compra y venta.

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.