Mathematical & Computer Sciences
http://hdl.handle.net/10399/20
Wed, 01 Jul 2020 00:12:26 GMT2020-07-01T00:12:26ZHigher gauge theory, self-dual strings and 6D superconformal field theory
http://hdl.handle.net/10399/4150
Higher gauge theory, self-dual strings and 6D superconformal field theory
Schmidt, Lennart
We present two explicit constructions in higher gauge theory of relevance to string
and M-theory: the non-abelian self-dual string and a six-dimensional (1,0) super
conformal ﬁeld theory.
We start by outlining higher gauge theory from the point of view of morphisms
of graded diﬀerential algebras and extend this to generalized higher gauge theory.
We discuss two models of the string Lie 2-algebra and give twisted versions of these
that are suitable for our non-abelian constructions.
We argue from analogy to monopoles that the string Lie 2-algebra is the relevant
higher gauge structure for the non-abelian generalization of the self-dual string. We
show that the twisted versions can be used to write down consistent non-abelian
self-dual string equations. Moreover, we give the elementary solution, which passes
the relevant consistency checks.
We also use this gauge structure to present an action for a six-dimensional super
conformal ﬁeld theory containing a non-abelian tensor multiplet based on ingredients
available in the literature. The resulting (1,0)-model contains the ﬁeld content of
the (2,0)-theory, allows for a self-dual three-form curvature and straightforwardly
reduces to a four-dimensional supersymmetric Yang–Mills theory. It can be regarded
as a stepping stone towards a potential construction of the (2,0)-theory.
Sat, 01 Dec 2018 00:00:00 GMThttp://hdl.handle.net/10399/41502018-12-01T00:00:00ZNew sampling and optimization methods for topic inference and text classification
http://hdl.handle.net/10399/4131
New sampling and optimization methods for topic inference and text classification
Khalifa, Osama
Topic modelling (TM) methods, such as latent Dirichlet allocation (LDA), are advanced statistical models which are used to uncover hidden thematic structures or topics in the unstructured text. In this context, a topic is a distribution over words, and a document is a distribution over topics. Topic models are usually unsupervised; however, supervised variants have been proposed, such as supervised LDA (SLDA) which can be used for text classiﬁcation. To evaluate a supervised topic model, one could measure its classiﬁcation accuracy. However, unsupervised topic model’s evaluation is not straightforward, and it is usually done by calculating metrics known as held-out perplexity and coherence. Held-out perplexity evaluates the model’s ability to generalize to unseen documents; coherence calculates a semantic distance between the words within each topic. This thesis explores ideas for enhancing the performance of TM, both supervised and unsupervised. Firstly, multi-objective topic modelling (MOEA-TM) is proposed, which uses a multi-objective evolutionary algorithm (MOEA) to optimize two objectives: coverage and coherence. MOEA-TM has two settings: ’start from scratch’ and ’start from an estimated topic model’. In the later, the held-out perplexity is added as another objective. In both settings, MOEA-TM achieves highly coherent topics. Further, a genetic algorithm is developed with LDA log-likelihood as a ﬁtness function. This algorithm can improve log-likelihood by up to 10%; however, perplexity scores slightly deteriorate due to over-ﬁtting. Hyperparameters play a signiﬁcant role in TM; thus, Gibbs-Newton (GN), which is an eﬃcient approach to learn a multivariate Pólya distribution parameter, is proposed. A closer look at the LDA model reveals that it comprises two multivariate Pólya distributions: one is used to model topics, whereas the other is used to model topics proportions in documents. Consequently, a better approach to learn multivariate Pólya distribution parameter may enhance TM. GN is benchmarked against Minka’s ﬁxed-point iteration approach, a slice sampling technique and the moments’ method. We ﬁnd that GN provides the same level of accuracy as Minka’s ﬁxed-point iteration method but in less time, and with better accuracy than the other approaches. Also, LDA-GN is proposed, which makes use of the GN method in topic modelling. This algorithm can achieve better perplexity scores than the original LDA on three corpora tested. Moreover, LDA-GN is tested on a supervised task using SLDA-GN, which is the SLDA model equipped with the GN method to learn its hyperparameters. SLDA-GN outperforms the original SLDA, which optimizes its hyperparameters using Minka’s ﬁxed point iteration method. Furthermore, LDA-GN is evaluated on a spam ﬁltering task using the Multi-corpus LDA (MC-LDA) model; where LDA-GN shows a more stable performance compared with the standard LDA. Finally, most topic models are based on the “Bag of Words” assumption, where a document word order is lost, and only frequency is preserved. We propose LDA-crr model, which represents word order as an observed variable. LDA-crr introduces only minor additional complexity to TM; thus, it can be applied readily to large corpora. LDA-crr is benchmarked against the original LDA using ﬁxed hyperparameters to isolate their inﬂuence. LDA-crr outperforms LDA in terms of perplexity and shows slightly more coherent topics when the number of topics increases. Also, LDA-crr is equipped with both the GN approach and the slice sampling technique in LDA-crrGN and LDA-crrGSS models respectively. LDA-crrGN shows a slightly better ability to generalize to unseen documents compared with LDA-GN on one corpus when the number of topics is high. However, in general, LDA-crrGSS shows better coherence scores compared with the LDA-GN and the original LDA. Furthermore, experiments to investigate LDA-crr performance in a classiﬁcation task were run; thus, SLDA is extended to incorporate word orders in the SLDA-crr model. The GN and the GSS techniques are used in the SLDA-crrGN and the SLDA-crrGSS models respectively to learn its parameters. Compared with the SLDA-GN and the original SLDA, the SLDA-crrGN shows better accuracy results in classifying unseen documents. This reveals that SLDA-crrGN can pick up more useful information from the training corpus which consequently helps the model to perform better.
Sat, 01 Dec 2018 00:00:00 GMThttp://hdl.handle.net/10399/41312018-12-01T00:00:00ZOn weak and strong convergence rate for the Heston stochastic volatility model
http://hdl.handle.net/10399/4106
On weak and strong convergence rate for the Heston stochastic volatility model
Zheng, Chao
The Heston stochastic volatility model is one of the most fundamental models in mathematical ﬁnance. Recently, many numerical schemes have been developed for the Heston model. However, in the literature, there is no weak or strong convergence rate obtained for the full parameter regime. In this PhD thesis, we shall focus on the numerical scheme that simulates the variance process exactly and applies the stochastic trapezoidal rule to approximate the time integral of the variance process in the SDE of the logarithmic asset process. Our goal is to obtain the weak and strong convergence rates of such a numerical scheme for the Heston model. The weak convergence rate is of traditional interest, because it is an important measure on how fast the bias of a numerical scheme decays. We prove that the numerical scheme we consider converges at rate two for the whole parameter regime, and the test function can be any polynomial of the logarithmic asset process. The rate is consistent with the standard rate of the stochastic trapezoidal rule, although the Lipschitz assumption is not satisﬁed. The strong convergence analysis is meaningful in the framework of Multi-level Monte Carlo (MLMC). The MLMC can be regarded as a variance reduction technique for numerical schemes on SDEs, as long as there is a MLMC estimator with a good strong convergence rate. We establish efﬁcient MLMC estimators, separately for the path-independent and path-dependent simulations. We are able to provide the strong convergence rates in both situations.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/10399/41062016-06-01T00:00:00ZStatistical modelling of the consistency of symptoms reported during hypoglycaemia for individual patients
http://hdl.handle.net/10399/4095
Statistical modelling of the consistency of symptoms reported during hypoglycaemia for individual patients
Zulkaﬂi, Hani Syahida
In this thesis, we use Bayesian methodology and Markov chain Monte Carlo tech-
niques to construct logistic-type latent variable statistical models for estimating the
consistency of hypoglycaemic symptoms experienced by individual diabetic patients.
Consistency in reporting experienced symptoms of hypoglycaemia is related to early
detection of symptoms and is therefore important for fast corrective action. Based
on a model developed by Zammit et al. (2011) we classify symptoms into diﬀerent
groups and consider between-groups variability. Our work also explores a number of
possible symptom-experiencing thresholds that can be used in the consistency model.
To evaluate the performance of each consistency model, we develop ideas based on
Bayesian latent residuals (Streftaris and Gibson, 2012) to check on the models’ ﬁt and
utilise posterior predictive checking methodology (Gelman et al., 1996 and Streftaris
et al., 2013) to assess relevant performance. The impact of using data from hypo-
glycaemic episodes occurring within 24 hours from an earlier episode is also explored
using various approaches, as previous work claims that such episodes might lead to di-
minished intensity of the episodes. Using generalised linear-type model methodology,
we investigate how various factors such as age, gender, type and duration of diabetes,
body mass index, retinopathy and others, or their interaction, can aﬀect patients’
consistency. Additionally, we develop a hierarchical model that is able to estimate
consistency and identify factors aﬀecting it in a single setting. Finally, we work on
determining the best sets of variables for a predictive model. For this purpose, we
use Gibbs variable selection and a stepwise regression procedure. Due to model un-
certainty, we apply Bayesian model averaging to a number of selected models given
by Gibbs variable selection.
Mon, 01 May 2017 00:00:00 GMThttp://hdl.handle.net/10399/40952017-05-01T00:00:00Z