Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

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

Comparing methods for dimensionality reduction when data are density functions

Functional Data Analysis (FDA) deals with samples where a whole function is observed for each individual. A particular case of FDA is when the observed functions are density functions, that are also an example of infinite dimensional compositional data. In this work we compare several methods for dimensionality reduction for this particular type of data: functional principal components analysis (PCA) with or without a previous data transformation and multidimensional scaling (MDS) for diferent inter-densities distances, one of them taking into account the compositional nature of density functions. The difeerent methods are applied to both artificial and real data (households income distributions)

An LMI approach to H∞ synchronization of second-order neutral master-slave systems

The H∞ synchronization problem of the master and slave structure of a second-order neutral master-slave systems with time-varying delays is presented in this paper. Delay-dependent sufficient conditions for the design of a delayed output-feedback control are given by Lyapunov-Krasovskii method in terms of a linear matrix inequality (LMI). A controller, which guarantees H∞ synchronization of the master and slave structure using some free weighting matrices, is then developed. A numerical example has been given to show the effectiveness of the method

Computing Distance Functions from Generalized Sources on Weighted Polyhedral Surfaces

We present algorithms for computing approximate distance functions and shortest paths from a generalized source (point, segment, polygonal chain or polygonal region) on a weighted non-convex polyhedral surface in which obstacles (represented by polygonal chains or polygons) are allowed. We also describe an algorithm for discretizing, by using graphics hardware capabilities, distance functions. Finally, we present algorithms for computing discrete k-order Voronoi diagrams

Negative Fukui functions: new insights based on electronegativity equalization

The occurrence of negative values for Fukui functions was studied through the electronegativity equalization method. Using algebraic relations between Fukui functions and different other conceptual DFT quantities on the one hand and the hardness matrix on the other hand, expressions were obtained for Fukui functions for several archetypical small molecules. Based on EEM calculations for large molecular sets, no negative Fukui functions were found

Critical thoughts on computing atom condensed Fukui functions

Different procedures to obtain atom condensed Fukui functions are described. It is shown how the resulting values may differ depending on the exact approach to atom condensed Fukui functions. The condensed Fukui function can be computed using either the fragment of molecular response approach or the response of molecular fragment approach. The two approaches are nonequivalent; only the latter approach corresponds in general with a population difference expression. The Mulliken approach does not depend on the approach taken but has some computational drawbacks. The different resulting expressions are tested for a wide set of molecules. In practice one must make seemingly arbitrary choices about how to compute condensed Fukui functions, which suggests questioning the role of these indicators in conceptual density-functional theory

Linear response functions for a vibrational configuration interaction state

Linear response functions are implemented for a vibrational configuration interaction state allowing accurate analytical calculations of pure vibrational contributions to dynamical polarizabilities. Sample calculations are presented for the pure vibrational contributions to the polarizabilities of water and formaldehyde. We discuss the convergence of the results with respect to various details of the vibrational wave function description as well as the potential and property surfaces. We also analyze the frequency dependence of the linear response function and the effect of accounting phenomenologically for the finite lifetime of the excited vibrational states. Finally, we compare the analytical response approach to a sum-over-states approach

New delay-dependent stability criteria for uncertain neutral systems with mixed time-varying delays and nonlinear perturbations

The problem of stability analysis for a class of neutral systems with mixed time-varying neutral, discrete and distributed delays and nonlinear parameter perturbations is addressed. By introducing a novel Lyapunov-Krasovskii functional and combining the descriptor model transformation, the Leibniz-Newton formula, some free-weighting matrices, and a suitable change of variables, new sufficient conditions are established for the stability of the considered system, which are neutral-delay-dependent, discrete-delay-range dependent, and distributeddelay-dependent. The conditions are presented in terms of linear matrix inequalities (LMIs) and can be efficiently solved using convex programming techniques. Two numerical examples are given to illustrate the efficiency of the proposed method

Estimation of age- and stage-specific Catalan breast cancer survival functions using US and Catalan survival data

During the last part of the 1990s the chance of surviving breast cancer increased. Changes in survival functions reflect a mixture of effects. Both, the introduction of adjuvant treatments and early screening with mammography played a role in the decline in mortality. Evaluating the contribution of these interventions using mathematical models requires survival functions before and after their introduction. Furthermore, required survival functions may be different by age groups and are related to disease stage at diagnosis. Sometimes detailed information is not available, as was the case for the region of Catalonia (Spain). Then one may derive the functions using information from other geographical areas. This work presents the methodology used to estimate age- and stage-specific Catalan breast cancer survival functions from scarce Catalan survival data by adapting the age- and stage-specific US functions. Methods: Cubic splines were used to smooth data and obtain continuous hazard rate functions. After, we fitted a Poisson model to derive hazard ratios. The model included time as a covariate. Then the hazard ratios were applied to US survival functions detailed by age and stage to obtain Catalan estimations. Results: We started estimating the hazard ratios for Catalonia versus the USA before and after the introduction of screening. The hazard ratios were then multiplied by the age- and stage-specific breast cancer hazard rates from the USA to obtain the Catalan hazard rates. We also compared breast cancer survival in Catalonia and the USA in two time periods, before cancer control interventions (USA 1975–79, Catalonia 1980–89) and after (USA and Catalonia 1990–2001). Survival in Catalonia in the 1980–89 period was worse than in the USA during 1975–79, but the differences disappeared in 1990–2001. Conclusion: Our results suggest that access to better treatments and quality of care contributed to large improvements in survival in Catalonia. On the other hand, we obtained detailed breast cancer survival functions that will be used for modeling the effect of screening and adjuvant treatments in Catalonia

Multi-objective optimization scheme for static and dynamic multicast flows

Muchas de las nuevas aplicaciones emergentes de Internet tales como TV sobre Internet, Radio sobre Internet,Video Streamming multi-punto, entre otras, necesitan los siguientes requerimientos de recursos: ancho de banda consumido, retardo extremo-a-extremo, tasa de paquetes perdidos, etc. Por lo anterior, es necesario formular una propuesta que especifique y provea para este tipo de aplicaciones los recursos necesarios para su buen funcionamiento. En esta tesis, proponemos un esquema de ingeniería de tráfico multi-objetivo a través del uso de diferentes árboles de distribución para muchos flujos multicast. En este caso, estamos usando la aproximación de múltiples caminos para cada nodo egreso y de esta forma obtener la aproximación de múltiples árboles y a través de esta forma crear diferentes árboles multicast. Sin embargo, nuestra propuesta resuelve la fracción de la división del tráfico a través de múltiples árboles. La propuesta puede ser aplicada en redes MPLS estableciendo rutas explícitas en eventos multicast. En primera instancia, el objetivo es combinar los siguientes objetivos ponderados dentro de una métrica agregada: máxima utilización de los enlaces, cantidad de saltos, el ancho de banda total consumido y el retardo total extremo-a-extremo. Nosotros hemos formulado esta función multi-objetivo (modelo MHDB-S) y los resultados obtenidos muestran que varios objetivos ponderados son reducidos y la máxima utilización de los enlaces es minimizada. El problema es NP-duro, por lo tanto, un algoritmo es propuesto para optimizar los diferentes objetivos. El comportamiento que obtuvimos usando este algoritmo es similar al que obtuvimos con el modelo. Normalmente, durante la transmisión multicast los nodos egresos pueden salir o entrar del árbol y por esta razón en esta tesis proponemos un esquema de ingeniería de tráfico multi-objetivo usando diferentes árboles para grupos multicast dinámicos. (en el cual los nodos egresos pueden cambiar durante el tiempo de vida de la conexión). Si un árbol multicast es recomputado desde el principio, esto podría consumir un tiempo considerable de CPU y además todas las comuicaciones que están usando el árbol multicast serán temporalmente interrumpida. Para aliviar estos inconvenientes, proponemos un modelo de optimización (modelo dinámico MHDB-D) que utilice los árboles multicast previamente computados (modelo estático MHDB-S) adicionando nuevos nodos egreso. Usando el método de la suma ponderada para resolver el modelo analítico, no necesariamente es correcto, porque es posible tener un espacio de solución no convexo y por esta razón algunas soluciones pueden no ser encontradas. Adicionalmente, otros tipos de objetivos fueron encontrados en diferentes trabajos de investigación. Por las razones mencionadas anteriormente, un nuevo modelo llamado GMM es propuesto y para dar solución a este problema un nuevo algoritmo usando Algoritmos Evolutivos Multi-Objetivos es propuesto. Este algoritmo esta inspirado por el algoritmo Strength Pareto Evolutionary Algorithm (SPEA). Para dar una solución al caso dinámico con este modelo generalizado, nosotros hemos propuesto un nuevo modelo dinámico y una solución computacional usando Breadth First Search (BFS) probabilístico. Finalmente, para evaluar nuestro esquema de optimización propuesto, ejecutamos diferentes pruebas y simulaciones. Las principales contribuciones de esta tesis son la taxonomía, los modelos de optimización multi-objetivo para los casos estático y dinámico en transmisiones multicast (MHDB-S y MHDB-D), los algoritmos para dar solución computacional a los modelos. Finalmente, los modelos generalizados también para los casos estático y dinámico (GMM y GMM Dinámico) y las propuestas computacionales para dar slución usando MOEA y BFS probabilístico.