MMathSci Thesis Talks

This semester’s MMathSci thesis talks will be held on Wednesday 30 Nov.  All are welcome
to come.  There is a mix of in person and online talks, between 10am-12pm and
2:30-4:30pm.  All relevant information is below.  I hope to see you there! 

Location: Carslaw AGR and https://uni-sydney.zoom.us/j/89458130430 

Schedule: 

10:00 - Mingxi Huang (online) 

10:30 - Tiancong Cheng (in person) 

11:00 - Shuyang Mo (online) 

11:30 - Richard Alessi (in person) 

2:30 - Muzhi Yu (online) 

3:00 - Paul Fortuin (in person) 

3:30 - Ruixuan Bi (online) 

4:00 - Zhoujiajing Wang (online) 

Talk Info: 

1.  Richard Alessi 

Title: Allocation strategies in binary response trials.  

Abstract: This thesis provides an examination of strategies to sequentially allocate
patients to treatments within the setting of a clinical trial with a binary outcome in
the presence of known covariates with the aim of making these techniques accessible in
business contexts with scenarios analogous to clinical trials..  We explore a range of
techniques through a simulation study with differing performance.  The allocation
strategy that best suits a situation is dependent on the relative importance of having a
high success rate and making reliable inference as well as the treatment effect size and
sample size.  

2.  Ruixuan Bi 

Title: On Theory and Applications of Time Series Models 

Abstract: Volatility is a statistical measure of the dispersion of returns for a given
stock or market index.  In time series literature, models which attempt to explain the
changes in conditional variance are generally known as conditional heteroscedastic
models.  In this study, fractionally integrated GARCH (FIGARCH) models are applied to
financial time series with long memory behaviour.  Our study also introduces stochastic
volatility (SV) models whose main advantage is to consider a random component adaptable
to abrupt changes.  The parameter estimation method based on the state-space methods is
proved and found to be accurate through a simulation study.  Following this, these
models are used to forecast the one-step-ahead volatility of Nasdaq 100 data.  The
results indicate the good performance of FIGARCH and SV models in estimating financial
market volatility.  

3.  Tiancong Cheng 

Title: Analysis and Applications of Bilinear Time Series models with Heteroscedastic
Errors 

Abstract: ARMA models are used in modeling linear time series.  However, there
are many time series in practice which are not linear.  In order to analyse certain time
series, Granger and Anderson (1978) proposed a family of bilinear time series models
with errors are a secquence of uncorrelated random variables with zero mean and constant
variance.  As many financel time series, the constant variance of the noise is too
restrictive and consider as heteroscedastic.  In order to model bilinear time series
with time dependent variance the generalize autoregressive conditional hterscedastic
(GARCH) can be used as innovations.  This project investigates the theory and
application of bilinear model with GARCH errors and present a simulation study and some
applications using real word data.  

4.  Paul Fortuin 

Title: The Bayesian Lasso: Approximation and Standard Errors 

Abstract: The Bayesian Lasso proposed by Park and Casella is a model in Statistics and
Machine learning that has a variety of applications.  Park and Casella’s model utilize
a Laplacian prior to increase sparsity of coefficients in linear regression analysis.
We examine Park and Casella’s methods, the use of Gibbs sampling, a Markov Chain Monte
Carlo Method, to evaluate the model.  Variational Bayesian approximations(Variational
Bayes) are introduced as a method of inference to model the Bayesian Lasso.  The
performance of Variational Bayesian approximations are discussed and their tendency to
under-approximate posterior standard errors is shown.  We propose Bayesian Expectation
Maximisation as an alternative to Variational Bayes and explore its performance in
calculating the standard errors of the Bayesian Lasso.  

5.  Mingxi Huang 

Title: Sampling triangulations of 4-manifolds using the Metropolis-Hastings algorithm 

Abstract: Manifolds form a fundamental class of spaces studied in topology, and the
notion of a manifold generalizes the concept of Euclidean space.  An ndimensional
topological manifold is a space that locally looks like the n-dimensional Euclidean
space.  Consequently, knowing that a space is a manifold does not tell us much about its
global structure.  To study the properties of a manifold, it is helpful to triangulate
it, that is, to construct a homeomorphism to a simplicial complex.  The global structure
of a higher dimensional manifold is hardly as directly observed.  Hence, triangulations
are a good tool to study the global structure and properties of n-dimensional
manifolds.  

4-manifolds are notionally difficult to study in topology, and very little is known
about general 4-manifolds.  Typical representative characteristics of 4- manifolds will
be studied in various fields.  This theoretical difficulty is the reason we choose to
test our method in the 4-dimensional setting.  In order to figure out the structures of
4-manifolds, plenty of triangulations of 4-manifolds are necessary.  Pachner moves
related to the 4-dimensional triangulations are called 4-dimensional Pachner moves.
There are five type 4-dimensional Pachner moves which will be explained in detail.  

Due to the Markov property of the changes between triangulations, in the Pachner graph,
we can sample triangulations of 4-manifolds with the MetropolisHastings algorithm.
Metropolis-Hastings algorithm is a Markov chain Monte Carlo method that simulates models
when the prior probability distribution is unknown.  Many larger triangulations can be
sampled starting with an arbitrary triangulation by constructing the proposal
distribution and acceptance ratio of the Metropolis-Hastings method.  In the thesis, we
describe a MetropolisHastings method to sample large quantities of triangulations of a
4-manifold M using an initial triangulation T of M as a seed.  

6.  Shuyang Mo: 

Title: Semi-parametric estimation of Autoregressive Conditional Duration (ACD) models 

Abstract: With rapid development of computer technology in recent years, more and more
high-frequency data have been collected, especially in finance.  For instance,
transaction duration of stock is an example of high freqency data.  As a result, the
study of a model exploring such data is necessary.  In this thesis, we will discuss a
popular model called Autoregressive Conditional Duration (ACD) model to explore such
data.  We use an estimation method based on estimating functions to estimate unknown
parameters of the ACD model.  We compare this method with the classical approach based
on maximum likelihood to see the performance, by producing some simulations and
applications with real world data.  

7.  Zhoujiajing Wang 

Title: Bayesian Estimation of Long Memory and Related Time Series Models in Finance

Abstract: Long-memory processes, also known as long-range-dependent processes, have
evolved into an essential part of time series analysis during the last several decades.
Long memory processes are characterized by slowly decaying autocorrelations and by a
spectral density exhibiting a pole at the origin.  These features dramatically changed
the statistical behavior and affected estimates and predictions.  As a consequence, many
theoretical detections and methodologies that were formerly applicable to the analysis
of short-memory time series are no longer appropriate for long-memory processes.  The
purpose of this thesis is to provide a study of the theory and methods developed to
estimate with long-memory time series data using classical methods and Bayesian
procedure.  Also, this presents these methodologies with specific applications to
real-life financial time series.  This thesis begins with a review of the work on
long-memory processes and discusses the theory on fractionally integrated autoregressive
moving average (ARFIMA) processes, where we deal with a non-integer differencing
parameter.  We use the popular S&P 500 index and its returns in parameter estimation
with MLE and Bayesian procedure.  

8.  Muzhi Yu: 

Title: Variational Bayes and Horseshoe prior for support vector machine.  

Abstract: Support vector machine is a popular classifier in machine learning nowadays
with many advantages.  It can help us due with many linear problems and even non-linear
ones with some kernel function.  However, the traditional SVM has the assumption that
the samples are from the independent identical distribution.  So Bayesian inference is
introduced to it.  Gibbs sampling is a special case of Markov chain Monte Carlo to
simulate the random values as samples from posterior distribution to achieve the
Bayesian inference, which usually costs too much time.  Therefore, variational Bayes,
which minimizes the gap between the marginal likelihood and the lower bound and uses
lower bound to approximate the posterior distribution, is used to modify it and save
time.  But it can’t do the feature selection when fitting model, which is important when
do the analysis.  So we induce the Horseshoe prior to it, and then it can cost less time
that the Gibbs sampling by variational Bayes and also do the feature selection by
inducing Horseshoe prior.  And we find when facing the problem that when the dimension
of the features is equal or larger than the number of observed samples, our model
behaves better than other models and can do a good feature selection.  It’s a good way
to solve the disadvantages of linear support vector machine, and in future can also
introduce it to non-linear support vector machine as many practical problems are
non-linear ones.