3 research outputs found
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 EconomĂa de la Empresa y MĂ©todos Cuantitativos por la Universidad Carlos III de MadridPresidente: Santiago Velilla Cerdán; Secretario: Kalliopi Mylona; Vocal: Luis Antonio Belanche Muño
Kernel depth funcions for functional data
In the last years the concept of data depth has been increasingly used in Statistics as a center-outward ordering of sample points in multivariate data sets. Recently data depth has been extended to functional data. In this paper we propose new intrinsic functional data depths based on the representation of functional data on Reproducing Kernel Hilbert Spaces, and test its performance against a number of well known alternatives in the problem of functional outlier detection.The authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness ECO2015-66593-P
Entropy Measures for Stochastic Processes with Applications in Functional Anomaly Detection
We propose a definition of entropy for stochastic processes. We provide a reproducing kernel Hilbert space model to estimate entropy from a random sample of realizations of a stochastic process, namely functional data, and introduce two approaches to estimate minimum entropy sets. These sets are relevant to detect anomalous or outlier functional data. A numerical experiment illustrates the performance of the proposed method; in addition, we conduct an analysis of mortality rate curves as an interesting application in a real-data context to explore functional anomaly detection.The first and third authors acknowledge financial support from the Spanish Ministry
of Economy and Competitiveness ECO2015-66593-P. The Second author acknowledges CONICET
Argentina Project 20020150200110BA. The fourth author acknowledges the Spanish Ministry of
Economy and Competitiveness Projects GROMA(MTM2015-63710-P), PPI (RTC-2015-3580-7) and
UNIKO(RTC-2015-3521-7) and the “methaodos.org” research group at URJC