Resumen de Statistical learning methods for functional data with applications to prediction, classification and outlier detection

Nicolás Jorge Hernández Banadik

• In the era of big data, Functional Data Analysis has become increasingly important in- sofar as it constitutes a powerful tool to tackle inference problems in statistics. In par- ticular in this thesis we have proposed several methods aimed to solve problems of prediction of time series, classification and outlier detection from a functional approach.

  The thesis is organized as follows: In Chapter 1 we introduce the concept of func- tional data and state the overview of the thesis. In Chapter 2 of this work we present the theoretical framework used to we develop the proposed methodologies.

  In Chapters 3 and 4 two new ordering mappings for functional data are proposed.

  The first is a Kernel depth measure, which satisfies the corresponding theoretical prop- erties, while the second is an entropy measure. In both cases we propose a parametric and non-parametric estimation method that allow us to define an order in the data set at hand. A natural application of these measures is the identification of atypical obser- vations (functions).

  In Chapter 5 we study the Functional Autoregressive Hilbertian model. We also propose a new family of basis functions for the estimation and prediction of the afore- mentioned model, which belong to a reproducing kernel Hilbert space. The properties of continuity obtained in this space allow us to construct confidence bands for the cor- responding predictions in a detracted time horizon.

  In order to boost different classification methods, in Chapter 6 we propose a diver- gence measure for functional data. This metric allows us to determine in which part of the domain two classes of functional present divergent behavior. This methodology is framed in the field of domain selection, and it is aimed to solve classification problems by means of the elimination of redundant information.

  Finally in Chapter 7 the general conclusions of this work and the future research lines are presented.