Wavelet regression using a Lévy prior model
Author(s)
Missaoui, Badr
Type
Thesis
Abstract
This thesis is concerned with nonparametric regression and regularization. In
particular, wavelet regression using a Lévy prior model is investigated. The use of
this prior is motivated by the statistical properties, such as heavy-tails, common in
many datasets of interest, such as those in financial time series.
The Lévy process we propose captures the heavy tails of the wavelet coefficients
of an unknown function. We study the Besov regularity of the wavelet coefficients
and establish the connection between the parameters of the Lévy wavelet prior model
and Besov spaces. At first, we gave a necessary and sufficient condition such that
the realizations of the prior model fall into a certain class of Besov spaces. We show
that the tempered stable distribution preserves its functional form for different time
scales. We prove that this scaling behaviour can model the exponential-decay-across-scale
property of the wavelet coefficients without imposing any specified structure on
the coefficients’ energy.
We also introduce a Lévy wavelet mixture model to capture the sparseness of the
wavelet coefficients. We show that this sparse model exhibits a thresholding rule.
We also study the Lévy tempered stable prior model under a Bayesian framework.
For the prior specified, we gave a closed form to the posterior Lévy measure of the
wavelet coefficients and estimate the hyperparameters of the prior model in both a
simulation study and for the S&P 500 time series.
We focus on density estimation using a penalized likelihood approach. Primarily,
we study the wavelet Tsallis entropy and Fisher information and give closed-form
expressions for these measures when the wavelet coefficients are driven by a tempered
stable process. Then, we develop an entropic regularization based on the wavelet Tsallis entropy and show that the penalized maximum likelihood method improves
the convergence of the estimates.
particular, wavelet regression using a Lévy prior model is investigated. The use of
this prior is motivated by the statistical properties, such as heavy-tails, common in
many datasets of interest, such as those in financial time series.
The Lévy process we propose captures the heavy tails of the wavelet coefficients
of an unknown function. We study the Besov regularity of the wavelet coefficients
and establish the connection between the parameters of the Lévy wavelet prior model
and Besov spaces. At first, we gave a necessary and sufficient condition such that
the realizations of the prior model fall into a certain class of Besov spaces. We show
that the tempered stable distribution preserves its functional form for different time
scales. We prove that this scaling behaviour can model the exponential-decay-across-scale
property of the wavelet coefficients without imposing any specified structure on
the coefficients’ energy.
We also introduce a Lévy wavelet mixture model to capture the sparseness of the
wavelet coefficients. We show that this sparse model exhibits a thresholding rule.
We also study the Lévy tempered stable prior model under a Bayesian framework.
For the prior specified, we gave a closed form to the posterior Lévy measure of the
wavelet coefficients and estimate the hyperparameters of the prior model in both a
simulation study and for the S&P 500 time series.
We focus on density estimation using a penalized likelihood approach. Primarily,
we study the wavelet Tsallis entropy and Fisher information and give closed-form
expressions for these measures when the wavelet coefficients are driven by a tempered
stable process. Then, we develop an entropic regularization based on the wavelet Tsallis entropy and show that the penalized maximum likelihood method improves
the convergence of the estimates.
Date Issued
2013-01
Date Awarded
2013-06
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
McCoy, Emma
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)