BURA Collection:http://bura.brunel.ac.uk/handle/2438/2352017-06-09T09:52:46Z2017-06-09T09:52:46ZBoundary-domain integral equation systems for the stokes system with variable viscosity and diffusion equation in inhomogeneous mediaFresneda-Portillo, Carloshttp://bura.brunel.ac.uk/handle/2438/145212017-05-11T02:00:32Z2016-01-01T00:00:00ZTitle: Boundary-domain integral equation systems for the stokes system with variable viscosity and diffusion equation in inhomogeneous media
Authors: Fresneda-Portillo, Carlos
Abstract: The importance of the Stokes system stems from the fact that the Stokes system is the stationary linearised form of the Navier Stokes system [Te01, Chapter1]. This linearisation is allowed when neglecting the inertial terms at a low Reinolds numbers Re << 1. The Stokes system essentially models the behaviour of a non - turbulent viscous fluid. The mixed interior boundary value problem related to the compressible Stokes system is reduced to two different BDIES which are equivalent to the original boundary value problem. These
boundary-domain integral equation systems (BDIES) can be expressed in terms of surface and volume parametrix-based potential type operators whose properties are also analysed in appropriate Sobolev spaces. The invertibility and Fredholm properties related to the matrix operators that de ne the BDIES are
also presented. Furthermore, we also consider the mixed compressible Stokes system with variable
viscosity in unbounded domains. An analysis of the similarities and differences with regards to the bounded domain case is presented. Furthermore, we outline the mapping properties of the surface and volume parametrix-based potentials in weighted Sobolev spaces. Equivalence and invertibility results still hold under certain decay conditions on the variable coeffi cient The last part of the thesis refers to the mixed boundary value problem for the stationary heat transfer partial di erential equation with variable coe cient. This BVP is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed by Chkadua, Mikhailov and Natroshvili in the paper [CMN09].
Mapping properties of the potential type integral operators appearing in these equations are presented in appropriate Sobolev spaces. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed in both bounded and unbounded domains.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2016-01-01T00:00:00ZDiscrete Weibull regression model for count dataKalktawi, Hadeel Salehhttp://bura.brunel.ac.uk/handle/2438/144762017-04-28T07:44:32Z2017-01-01T00:00:00ZTitle: Discrete Weibull regression model for count data
Authors: Kalktawi, Hadeel Saleh
Abstract: Data can be collected in the form of counts in many situations. In other words, the number of deaths from an accident, the number of days until a machine stops working or the number of annual visitors to a city may all be considered as interesting variables for study. This study is motivated by two facts; first, the vital role of the continuous Weibull distribution in survival analyses and failure time studies. Hence, the discrete Weibull (DW) is introduced analogously to the continuous Weibull distribution, (see, Nakagawa and Osaki (1975) and Kulasekera (1994)). Second, researchers usually focus on modeling count data, which take only non-negative integer values as a function of other variables. Therefore, the DW, introduced by Nakagawa and Osaki (1975), is considered to investigate the relationship between count data and a set of covariates. Particularly, this DW is generalised by allowing one of its parameters to be a function of covariates. Although the Poisson regression can be considered as the most common model for count data, it is constrained by its equi-dispersion (the assumption of equal mean and variance). Thus, the negative binomial (NB) regression has become the most widely used method for count data regression. However, even though the NB can be suitable for the over-dispersion cases, it cannot be considered as the best choice for modeling the under-dispersed data. Hence, it is required to have some models that deal with the problem of under-dispersion, such as the generalized Poisson regression model (Efron (1986) and Famoye (1993)) and COM-Poisson regression (Sellers and Shmueli (2010) and Sáez-Castillo and Conde-Sánchez (2013)). Generally, all of these models can be considered as modifications and developments of Poisson models. However, this thesis develops a model based on a simple distribution with no modification. Thus, if the data are not following the dispersion system of Poisson or NB, the true structure generating this data should be detected. Applying a model that has the ability to handle different dispersions would be of great interest. Thus, in this study, the DW regression model is introduced. Besides the exibility of the DW to model under- and over-dispersion, it is a good model for inhomogeneous and highly skewed data, such as those with excessive zero counts, which are more disperse than Poisson. Although these data can be fitted well using some developed models, namely, the zero-inated and hurdle models, the DW demonstrates a good fit and has less complexity than these modifed models. However, there could be some cases when a special model that separates the probability of zeros from that of the other positive counts must be applied. Then, to cope with the problem of too many observed zeros, two modifications of the DW regression are developed, namely, zero-inated discrete Weibull (ZIDW) and hurdle discrete Weibull (HDW) models. Furthermore, this thesis considers another type of data, where the response count variable is censored from the right, which is observed in many experiments. Applying the standard models for these types of data without considering the censoring may yield misleading results. Thus, the censored discrete Weibull (CDW) model is employed for this case. On the other hand, this thesis introduces the median discrete Weibull (MDW) regression model for investigating the effect of covariates on the count response through the median which are more appropriate for the skewed nature of count data. In other words, the likelihood of the DW model is re-parameterized to explain the effect of the predictors directly on the median. Thus, in comparison with the generalized linear models (GLMs), MDW and GLMs both investigate the relations to a set of covariates via certain location measurements; however, GLMs consider the means, which is not the best way to represent skewed data. These DW regression models are investigated through simulation studies to illustrate their performance. In addition, they are applied to some real data sets and compared with the related count models, mainly Poisson and NB models. Overall, the DW models provide a good fit to the count data as an alternative to the NB models in the over-dispersion case and are much better fitting than the Poisson models. Additionally, contrary to the NB model, the DW can be applied for the under-dispersion case.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2017-01-01T00:00:00ZAcoustic scattering in circular cylindrical shells: a modal approach based on a generalised orthogonality relationPullen, Ryan Michaelhttp://bura.brunel.ac.uk/handle/2438/144672017-04-28T07:45:34Z2017-01-01T00:00:00ZTitle: Acoustic scattering in circular cylindrical shells: a modal approach based on a generalised orthogonality relation
Authors: Pullen, Ryan Michael
Abstract: During the past 60 years fluid-structure interaction in a wide range of three dimensional circular cylinder problems have been studied. Initial problems considered a rigid wall structure which were solved using impedance model comparisons. Soon after, further solution techniques were used, such as computer simulation, transfer matrix methods and finite element techniques. However such problems were only valid for low frequencies when compared with experiments, this was because that did not include higher order modes. The importance of higher order modes was then established and studies have since included these modes. More recently, mode matching methods have been used to find the amplitudes of waves in structures comprising two or more ducts. This has been done with using an orthogonality relation to find integrals which occur from the application this method. This methodology is demonstrated in as background information and is applied to prototype problems formed of rigid ducts. The rigid duct theory led to the consideration of elastic shells, of which several shell modelling equations were available from the vibration theory. In this thesis, the Donnell-Mustari equations of motion are used to model thin, elastic, fluid-loaded shells of circular cross-section. It is demonstrated that generalised orthogonality relations exist for such shells. Two such relations are found: one for shells subject to axisymmetric motion and one for shells subject to non-axisymmetric motion. These generalised orthogonality relations are new to the field of acoustics and are specific to shells modelled with the Donnell-Mustari equations of motion. The mode matching method is used to find the amplitudes of waves propagating in prototype problems and the generalised orthogonality relations are used to find integrals which occur through this method. Expressions for energy for all considered structure types are used to find the resulting energy for each prototype problem and results for equivalent problems are compared. In addition, verification of the resulting amplitudes is done by ensuring that the matching conditions are suitably satisfied. It is anticipated that the method will have application to the understanding and control of the vibration of cylindrical casings such as those enclosing turbo-machinery. Another application of the method would be the tuning of cylindrical casings, such as those featured on car exhaust systems or HVAC (heating, ventilation and air conditioning) systems.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2017-01-01T00:00:00ZNovel regularization models for dynamic and discrete response dataHamed, Haseli Mashhadihttp://bura.brunel.ac.uk/handle/2438/141352017-02-28T03:01:12Z2017-01-01T00:00:00ZTitle: Novel regularization models for dynamic and discrete response data
Authors: Hamed, Haseli Mashhadi
Abstract: Regularized regression models have gained popularity in recent years. The addition of a penalty term to the likelihood function allows parameter estimation where traditional methods fail, such as in the p » n case. The use of an l1 penalty in particular leads to simultaneous parameter estimation and variable selection, which is rather convenient in practice. Moreover, computationally efficient algorithms make these methods really attractive in many applications. This thesis is inspired by this literature and investigates the development of novel penalty functions and regression methods within this context. In particular, Chapter 2 deals with linear models for time-dependent response and explanatory variables. This is beyond the independent framework which is common to many of the developed regularized regression models. We propose to account for the time dependency in the data by explicitly adding autoregressive terms to the response variable together with an autoregressive process for the residuals. In addition, the use of a l1 penalized likelihood approach for parameter estimation leads to automatic order and variable selection and makes this method feasible for high-dimensional data. Theoretical properties of the estimators are provided and an extensive simulation study is performed. Finally, we show the application of the model on air pollution and stock market data and discuss its implementation in the R package DREGAR, which is freely available in CRAN. In Chapter 3, we develop a new penalty function. Despite all the advantages of the l1 penalty, this penalty is not differentiable at zero, and neither are the alternatives that are proposed in the literature. The only exception is the ridge penalty, which does not lead to variable selection. Motivated by this gap, and noting the advantages that a differentiable penalty can give, such as increased computational efficiency in some cases and the derivation of more accurate model selection criteria, we develop a new penalty function based on the error function. We study the theoretical properties of this function and of the estimators obtained in a regularized regression context. Finally, we perform a simulation study and we use the new penalty to analyse a diabetes and prostate cancer dataset. The new method is implemented in the R package DLASSO, that is freely available in CRAN. Finally, Chapter 4 deals with regression models for discrete response data, which is frequently collected in many application areas. In particular, we consider a discrete Weibull regression model that has recently been introduced in the literature. In this chapter, we propose the first Bayesian implementation of this model. We consider a general parametrization, where both parameters of the discrete Weibull distribution can be conditioned on the predictors, and show theoretically how, under a uniform noninformative
prior, the posterior distribution is proper with finite moments. In addition, we consider closely the case of Laplace priors for parameter shrinkage and variable selection. A simulation study and the analysis of four real datasets of medical records show the applicability of this approach to the analysis of count data. The method is implemented in the R package BDWreg, which is freely available in CRAN.
Description: This thesis was submitted for the award of Doctor of Philosophy and was awarded by Brunel University London2017-01-01T00:00:00Z