Rainbow plots, Bagplots and Boxplots for Functional Data

We propose new tools for visualizing large numbers of functional data in the form of smooth curves or surfaces. The proposed tools include functional versions of the bagplot and boxplot, and make use of the first two robust principal component scores, Tukey's data depth and highest density regions. By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with exiting methods for detecting outliers in functional data and show that our methods are better able to identify the outliers.Highest density regions, Robust principal component analysis, Kernel density estimation, Outlier detection, Tukey's halfspace depth

Density kernel depth for outlier detection in functional data

In this paper, we propose a novel approach to address the problem of functional outlier detection. Our method leverages a low-dimensional and stable representation of functions using Reproducing Kernel Hilbert Spaces (RKHS).We define a depth measure based on density kernels that satisfy desirable properties.We also address the challenges associated with estimating the density kernel depth. Throughout aMonte Carlo simulation we assess the performance of our functional depth measure in the outlier detection task under different scenarios. To illustrate the effectiveness of our method, we showcase the proposed method in action studying outliers in mortality rate curves.Este artículo se encuentra publicado en International Journal of Data Science and Analytics (Springer Nature)https://doi.org/10.1007/s41060-023-00420-

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

In the era of big data, Functional Data Analysis has become increasingly important insofar as it constitutes a powerful tool to tackle inference problems in statistics. In particular 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 functional 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 properties, 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 observations (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 aforementioned 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 corresponding predictions in a detracted time horizon. In order to boost different classification methods, in Chapter 6 we propose a divergence 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.Financial support received from the Spanish Ministry of Economy and Competitiveness ECO2015-66593-P and the UC3M PIF scholarship for doctoral studies.Programa de Doctorado en Economía de la Empresa y Métodos Cuantitativos por la Universidad Carlos III de MadridPresidente: Santiago Velilla Cerdán; Secretario: Kalliopi Mylona; Vocal: Luis Antonio Belanche Muño