DSpace Collection:
http://theses.ncl.ac.uk/jspui/handle/10443/5374
2023-03-29T19:38:42ZSubjective Bayesian Methods in the Design and Analysis of Clinical Trials
http://theses.ncl.ac.uk/jspui/handle/10443/5732
Title: Subjective Bayesian Methods in the Design and Analysis of Clinical Trials
Authors: Williams, Cameron
Abstract: Assurance provides a Bayesian alternative to commonly used frequentist sample size calculation methods. As part of sample size calculations, an estimate of a treatment’s effect
size or a test’s accuracy is typically required. When using Bayesian methods, these unknown quantities can be represented with a prior distribution, rather than using a single
point estimate, allowing for more nuanced information about the unknown quantity to be
incorporated into the sample size calculation.
In this thesis, we first review common sample size calculation methods and elicitation
techniques. We consider the problem of aggregating expert prior beliefs to form a single
prior distribution, to be used in sample size calculations. Common methods of prior
distribution aggregation include mathematical methods, which use a mathematical rule
to combine priors, and behavioural methods, which provide experts with a framework to
assist them in creating an aggregate prior during a group discussion.
Though not a recent development, assurance is not commonly used in practice. We
provide a case study of a diagnostic study, investigating a novel diagnostic test for Motor
Neurone Disease, for which prior distributions are elicited and aggregated across experts,
and sample size calculations are conducted using both frequentist and assurance methods.
As a result of the requirements involved in using each method of aggregation, few
comparisons between behavioural and mathematical aggregation methods exist. In order
to make comparisons, we structured a series of elicitations as part of the case study. We
demonstrate how any method of aggregation outperforms individual experts, and that the
Sheffield Elicitation Framework and Classical Method perform best out of the aggregation
methods compared. We also demonstrate that all of the considered aggregation methods
perform better than a randomly selected individual expert.
In order to explore the behaviour of assurance, we provide a number of simulation
studies comparing assurance and power calculations. We investigate the sensitivity of
power and assurance to changes in input parameters, the effect of misrepresenting an effect
size, and the effect of using different prior distributions in the design and analysis stages
of assurance calculations. We consider these behaviours for both Normal and binomial
observations.
We use the resulting aggregated prior distributions for assurance and power calculations, to determine appropriate sample sizes within the case study and more generally. We
compare assurance calculations with different priors, analysis methods and target values
to further demonstrate differences between assurance and power, and their properties. We
demonstrate how the choice of model and prior distribution can have a large impact on
the final results of a sample size calculation.
Description: Ph. D. Thesis2022-01-01T00:00:00ZSubcritical Behaviour in Rotating Convection and Convectively-Driven Dynamos
http://theses.ncl.ac.uk/jspui/handle/10443/5720
Title: Subcritical Behaviour in Rotating Convection and Convectively-Driven Dynamos
Authors: Cooper, Robert George
Abstract: In planets and stars, convection is thought to be key for generating and maintaining largescale magnetic fields. Many planets possess a hydromagnetic dynamo driven by convective
motions, such as the geodynamo. However, a number of smaller planetary bodies, such
as Mars, show evidence of once possessing a dynamo that suddenly ceased to exist. One
suggested cause for the sudden cessation of the Martian dynamo is that it was operating in
a subcritical parameter regime, that is, the dynamo continued to exist when its controlling
parameter decreased below the critical value for linear onset, before eventually collapsing
towards the non-magnetic trivial state. This thesis aims to explore subcritical behaviour
in dynamo action and convection in order to better understand the dynamic processes
that affect planetary dynamos.
In the first part of this thesis, we focus upon the simpler problem of rotating convection in the absence of a magnetic field. In two-dimensional rotating convection, localised
states, known as ‘convectons’, have previously been observed for moderate rotation rates.
Convectons are associated with systematic shear flows which locally reduce the inhibiting
nature of rotation on convection, potentially promoting subcritical behaviour. We study
convectons in 2D Boussinesq convection in a rotating plane layer and perform parametric
surveys in both a fully-truncated model with restricted symmetries, and a model where the
full horizontal structure is allowed. We successfully obtain rotating convectons for rapid
rotation and explore their bifurcation structure, stability and key features. In particular,
we show that convectons are typically associated with a full local reduction in the effective
rotation.
In the second part of this thesis, we study dynamo action using 3D numerical simulations of planar Boussinesq convection at rapid rotation, focussing again on subcritical
behaviour. We first generate a large-scale magnetic field in the supercritical regime that
significantly influences convective motions. Subcritical solutions are then found by tracking this solution branch into the subcritical regime. Here the dynamo is sustained for
convective driving below the critical value for the linear onset of non-magnetic convection.
We show that increasing rotation leads to an extension of the subcritical range to an optimal value. At more rapid rotation, subcriticality is then hampered by the emergence of
a large-scale convective mode. The inability of the large-scale mode to sustain dynamo
action leads to an intermittent behaviour that appears to inhibit subcriticality. Finally, we
study the key parameter regimes at which subcritical dynamos exist, such as an optimal
magnetic Reynolds number.
Description: PhD Thesis2022-01-01T00:00:00ZA flat extension theorem for truncated matrix-valued multisequences
http://theses.ncl.ac.uk/jspui/handle/10443/5699
Title: A flat extension theorem for truncated matrix-valued multisequences
Authors: Trachana, Matina
Abstract: Given a truncated multisequence of p × p Hermitian matrices S := (Sγ1,...,γd
) (γ1,...,γd)∈Nd
0
0≤γ1+···+γd≤m
, the
truncated matrix-valued moment problem on R
d asks whether or not there exists a p×p positive
semidefinite matrix-valued measure T, with convergent moments of all orders, such that
Sγ1,...,γd =
Z
· · · Z
Rd
x
γ1
1
· · · x
γd
d
dT(x1, . . . , xd)
for all (γ1, . . . , γd) ∈ N
d
0 which satisfy 0 ≤
Pd
j=1 γj ≤ m. When such a measure exists we say
that T is a representing measure for S. We shall see that if m is even, then S has a minimal
representing measure (that is, Pκ
a=1
rank Qa is as small as possible) if and only if a block matrix
determined entirely by S has a rank-preserving positive extension. In this case, the support
of the representing measure has a connection with zeros (suitably interpreted) of a system of
matrix-valued polynomials which describe the rank-preserving extension. The proof of this
result relies on operator theory and certain results for ideals of multivariate matrix-valued
polynomials. Our result subsumes the celebrated flat extension theorem of Curto and Fialkow.
We shall pay particularly close attention to the bivariate quadratic matrix-valued moment
problem (that is, where d = 2 and m = 2).
Description: PhD Thesis2022-01-01T00:00:00ZRepresentation Theory of Non-graded, Non-restricted Modular Lie algebras
http://theses.ncl.ac.uk/jspui/handle/10443/5645
Title: Representation Theory of Non-graded, Non-restricted Modular Lie algebras
Authors: Guerra Ocampo, Horacio Z.
Abstract: We classify in a uni ed approach the simple restricted modules for the minimal p-envelope
of the non-graded, non-restricted Hamiltonian Lie algebra H.2I.1; 1/I ˆ.1// over an
algebraically closed eld k of characteristic p 5. We also give the restrictions of these
modules to a subalgebra isomorphic to the rst Witt Algebra, a result stated in [S. Herpel
and D. Stewart, Selecta Mathematica 22:2 (2016) 765–799] with an incomplete proof. We
end by completing the classi cation of the simple restricted modules over elds of all
characteristics by considering the characteristic 3 case separately
Description: Ph. D. Thesis.2021-01-01T00:00:00Z