Publication: A VARIATIONAL APPROACH TO A CUMULATIVE DISTRIBUTION FUNCTION ESTIMATION PROBLEM UNDER STOCHASTIC AMBIGUITY
Date
2022-10-30
Authors
URREA CASTILLO, FERNANDA PAZ
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Abstract
En esta tesis se propone un método para el problema de estimación de funciones de distribución de probabilidad bajo ambigüedad estocástica. La ambigüedad estocástica está representada por un conjunto de incertidumbre en el que se debe encontrar la función de distribución acumulada y el método propuesto considera a las funciones de distribución acumulada como un subconjunto de una clase más grande de funciones, el espacio de las funciones semicontinuas superior. Presentamos varios resultados para introducir las propiedades topológicas del espacio y también para justificar nuestra elección del espacio. Las herramientas utilizadas para desarrollar este método se basan en la teoría del análisis variacional. En particular, trabajamos sobre el espacio de las funciones semicontinuas superior dotado de la distancia de Attouch Wets para las que se propone una aproximación y en conjunto con el uso de epi-splines (funciones polinomiales a trozos) impulsan un esquema de aproximación lineal a nuestro enfoque. Implementamos un algoritmo para el caso bivariado que nos permite calcular soluciones al problema aproximado así como también incorporar información suave y condiciones de crecimiento al modelo. Enunciamos condiciones que garantizan la convergencia de los minimizadores cercanos de la sucesión de problemas aproximados en soluciones para el problema original y entregamos una clase de funciones que satisfacen aquella condición. Probamos nuestro algoritmo a modo de prueba de concepto con dos ejemplos distintos, donde analizamos varios parámetros y también realizamos experiencias numéricas para el problema de estimación de la posición de un vehículo submarino no tripulado dadas fuentes de información ruidosas.
In this thesis we propose a method to the problem of estimating cumulative distribution functions under stochastic ambiguity. The stochastic ambiguity is represented by a set of uncertainty in which the cumulative distribution function must be found and the proposed method considers cumulative distribution functions as a subset of a bigger class of functions, the space of upper semi-continuous functions. We present several results in order to introduce the topological properties of the space and also, to give justification of our choice. The tools used to develop this method rely on the theory of variational analysis. In particular, we work on the space of upper semicontinuous functions endowed with the Attouch Wets distance for which an approximation is proposed and altogether with the use of epi-splines (piecewise polynomial functions) drive a linear approximating scheme to our setting. We implemented an algorithm for the bivariate case that allowed us to compute solutions to the approximating problem as well as incorporating soft information and growing conditions to the model. We give conditions that guarantee the convergence of near-minimizers of the approximating sequence into solutions for the original problem and we hand out a class of functions that satisfy it. We test our algorithm as a proof of concept with two different examples, where we analyze various parameters and in addition, we performed numerical experiences for the problem of estimating the position of an unmanned underwater vehicle given noisy sources of information.
In this thesis we propose a method to the problem of estimating cumulative distribution functions under stochastic ambiguity. The stochastic ambiguity is represented by a set of uncertainty in which the cumulative distribution function must be found and the proposed method considers cumulative distribution functions as a subset of a bigger class of functions, the space of upper semi-continuous functions. We present several results in order to introduce the topological properties of the space and also, to give justification of our choice. The tools used to develop this method rely on the theory of variational analysis. In particular, we work on the space of upper semicontinuous functions endowed with the Attouch Wets distance for which an approximation is proposed and altogether with the use of epi-splines (piecewise polynomial functions) drive a linear approximating scheme to our setting. We implemented an algorithm for the bivariate case that allowed us to compute solutions to the approximating problem as well as incorporating soft information and growing conditions to the model. We give conditions that guarantee the convergence of near-minimizers of the approximating sequence into solutions for the original problem and we hand out a class of functions that satisfy it. We test our algorithm as a proof of concept with two different examples, where we analyze various parameters and in addition, we performed numerical experiences for the problem of estimating the position of an unmanned underwater vehicle given noisy sources of information.
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Keywords
CALCULO DE VARIACIONES , CONVERGENCIA , ANALISIS FUNCIONAL