QAMML: probability distributions for functional data

Student thesis: Doc typesDoctor of Sciences

Abstract

Functional data analysis is the part of data analysis which is interested specifically by the functional data. Functional data arise naturally in the study of many phenomenons, and are any continuous phenomenon which can be measured for any values of a varying parameter, this parameter can be the time, but not in all the cases. The study of functional data could become mainstream due to the increasing interest for the data analysis of streaming data, which become ubiquitous in our networked world. But a specificity of functional data is that they belong to an infinite dimensional space, which increases the difficulty to define certain concepts, like the probability distribution of a functional random variable. However probability distributions are be valuable tools, because they can be seen as the Swiss army knife of the data analysis: they are used in many procedures: unsupervised classification by mixture decomposition, Bayesian supervised classification, regression functions, statistical inference… Then, there were a strong interest to develop probability distributions for functional data, and this is the cornerstone of this thesis: the definition, the construction and the use in the data analysis framework, of probability distributions directly defined in the infinite dimensional space of functional data. We define a new kind of probability distributions, called QAMML distributions, in mixing two close concepts: Archimedean copulas and Quasi-arithmetic means. QAMML distributions, directly defined in the infinite dimensional space of functional data, need also the definition of an adapted kind of density, and for this we use a directional differential called the Gateaux differential. Our approach is not only a theoretical one, because we use these new tools, in data analysis of functional and symbolic data, simply like a “plug-in” in two existing methods: the unsupervised classification by mixture decomposition and the Bayesian supervised classification. We use also the QAMML distributions to build functional confidence intervals.
Date of Award25 Sep 2009
Original languageEnglish
Awarding Institution
  • University of Namur
SupervisorMonique Fraiture (Supervisor), Jean-Marie Jacquet (President), Jean-Paul Leclercq (Jury), Marcel Remon (Jury), Edwin Diday (Jury) & I VAN KEYLEGOM (Jury)

Cite this

QAMML: probability distributions for functional data
Cuvelier, E. (Author). 25 Sep 2009

Student thesis: Doc typesDoctor of Sciences