Functional linear models

  1. Mingotti, Nicola
Supervised by:
  1. Rosa Elvira Lillo Rodríguez Director
  2. Juan José Romo Urroz Director

Defence university: Universidad Carlos III de Madrid

Fecha de defensa: 21 July 2015

Committee:
  1. Francisco Javier Prieto Fernández Chair
  2. Eva Senra Díaz Secretary
  3. Ana Justel Committee member

Type: Thesis

Abstract

This work aims at the exposition of two different results we have obtained in Functional Data Analysis. The first is a variable selection method in Functional Regression which is an adaptation of the well known Lasso technique. The second is a brand new Random Walk test for Functional Time Series. Being the results afferent to different areas of Functional Data Analysis, as well as of general Statistics, the introduction will be divided in three parts. Firstly we expose the fundamentals of Functional Data Analysis. Then we will recall some variable selection methods in ordinary Linear Regression. Finally we will review some basics of Time Series analysis and brie y review some existing Random Walk tests. These introductory sections will motivate our research putting it in a general framework. Since Functional Data Analysis can be seen as a data reduction method we will talk incidentally of Big Data and we will provide some comments on the current definition of it. All results of our research are supported by extensive computer simulations and in general, all of FDA is based on extensive computer deployment so some attention will be given to software and computation methods. The Lasso has been used in Functional Regression before this work, our contribution is twofold, we provide a reduction of Lasso in Functional Regression from a functional optimization problem to a numerical one via algebraic manipulations, no sampling is required. Then, we augment the Lasso with a post hoc analysis method which helps deciding which regressors have to be dropped, we called this augmented strategy The Shaked Lasso. About testing if a Functional Autoregressive Process can be considered a Random Walk, our proposed test, as far as we could establish, is the first one in literature.