19658 – 978 (240) PhD (Statistics)
See the Faculty Calendar here for a detail description of this program.
Module Content - Honours
Introduction to R (13074-723)
Objectives and content: This module is
an introduction to programming and data analysis within the R open source environment. It is presented as a block course in the first two weeks of the first semester and commences
the week preceding general commencement of classes. The viewpoint of this module as well as of all modules where R plays a role is in agreement with the aim of the R computer
language: "R has a simple goal: To turn ideas into software, quickly and
faithfully".
Biostatistics (10408-712)
Objectives and content: Biostatistics
may be regarded as the study of the application of statistics to medicine. It covers medical terminology, the design of clinical trials, the collection and numerical
analysis of data, the interpretation of the analyses and the drawing of conclusions. Particular emphasis is given to skills relevant to medical literature (the writing, as
well as the understanding of writing by others) and statistical techniques and software that are widely used when doing medical research. It is not a mathematically strenuous
course. It deals primarily with the philosophy and terminology of medical research, as well as the statistical techniques problems encountered in the medical field in particular.
Topics that will be covered are: SAS, Clinical trials, Power and
sample size analysis, Longitudinal data analysis, Handling missing data and Statistical genetics.
Multivariate Methods in Statistics A & B (10600-721 & 10601-751)
Objectives and content: The objective of the course is to teach students the
practical application of multivariate analysis. Various multivariate methods are dealt with. Students learn when and where to apply these techniques. The consequences of the
assumptions made on some of these techniques are also studied. The following topics are studied: Matrix algebra, Characterising and displaying multivariate data, The multivariate
normal distribution, Inferences on one or two mean vectors, Multivariate analysis of variance, Inferences on the covariance matrix, Discriminant analysis, Classification
analysis, Multivariate regression, Canonical correlation, Principal component analysis, Factor analysis and Cluster analysis. The R and SAS software are used in all the
applications to datasets. The MMS A module is a prerequisite for the MMS B module.
Experimental Design (10440-713)
Objectives and content:
This module does not require advanced mathematics and is an option for both statistics and mathematical statistics
students. Focus is mainly on the practical implementation of techniques together with computer packages from consultancy perspective. Attention is given to modeling, design
matrices, least squares and diagnostics.
Data mining (58777-741)
Objectives and content: Statistical learning is a relatively new area in statistics. It is concerned with modeling
and understanding patterns in complex datasets. With to the explosion of "Big Data", there is currently a high demand for individuals with expertise in statistical
learning. The methods studied in this module include regularised regression by means of ridge regression and the lasso; classification using linear discriminant analysis, logistic
regression, quadratic discriminant analysis and k-nearest neighbors; resampling methods such as k-fold cross-validation, leave-one-out cross-validation and the bootstrap; linear
model selection and dimension reduction methods; handling non-linearity via regression splines, smoothing splines, local regression, generalised additive models, bagging, random
forests and boosting; and non-linear classification and regression by means of support vector machines. The objectives of the module are to equip students with the following
knowledge and skills:
- the theory underlying the above statistical learning techniques;
- application of statistical learning methods in a programming environment;
- assessment and comparison of various models;
- interpretation and effective (written and verbal) communication of results.
We extensively make use of the R programming language, therefore note that the R course is a prerequisite.
Sampling Techniques (10705-742)
Objectives and content: The design of a
sample is one of the most important aspects of any survey: no amount of statistical analysis can compensate for a badly-designed sample. Therefore, the emphasis of this
course is the scientific design of samples, determination of sample sizes and is related to methods for analysing the data from a survey. Contents include: Questionnaire design,
sampling techniques (simple random, stratified, systematic, cluster, complex), proportional vs disproportional allocation for stratified sampling, ratio and regression estimation,
estimation of means, totals proportions and their variances, weighting of survey data, dealing with non-response.
Financial Mathematical Statistics (11164-732)
Objectives and Content: In this module an introduction is given to Extreme Value Theory (EVT) and its role in
Financial Risk Management. EVT entails the study of extreme events and for this theory has been developed to describe the behaviour in the tails of distributions. The module will
disduss the theory in a conceptual fashion without proving the results. It will be shown how this theory can be used to carry out inferences on the relevant parameters of the
underlying distribution. Both the classical approach of block maxima based on the Fisher-Tippett Theorem and the more modern threshold approach based on the Pickands-Balkema-de
Haan Theorem will be discussed and applied. Results for both independent and dependent data will be covered.
Applied Time series Analysis (10748-722)
Objectives and content: This module is a continuation of undergraduate time
series analyses and concentrates on more advanced forecasting techniques. Topics that are covered include:
- The Box & Jenkins methodology of tentative model identification, conditional and unconditional parameter estimation and diagnostic methods
for checking the fit of the series.
- ARIMA and Seasonal ARIMA-processes.
- Introduction to Fourier Analysis, spectrum of a periodic time series, estimation of the spectrum, periodogram analysis, smoothing of the
spectrum.
- Case studies using STATISTICA, R and SAS.
- Forecasting with ARMA models and prediction intervals for forecasts.
- Transfer function models and intervention analysis.
- Multiple regression with ARMA errors, cointegration of non-stationary time series.
- Conditional heteroscedastic time series models, ARCH and GARCH.
Applied Stochastic simulation (65269-746)
Objectives and content: In this module
the student learns to understand and apply the underlying mathematical principles behind stochastic simulation. Statistical theory is applied to practical problems which are
modelled on computer. To achieve this, the student revises probability theory, including conditional probability and independence, discrete and continuous distributions, probability integral transformation, and Bayes' theorem. Pseudorandom number
generation is studied to simulate stochastic problems. These pseudorandom numbers are used to evaluate integrals, and to generate discrete and continuous random variables. It is
also important to do a statistical analysis of simulated data using point and interval estimates of the mean and variance, as well as bootstrap techniques. Input data for
simulation models are analysed with the chi-squared and Kolmogorov-Smirnoff goodness-of-fit tests. The student finally develops practical simulation models of discrete-event,
dynamic stochastic processes using a dedicated simulation software package.
Capita Selecta in Statistics A & B (11920-725
& 11921-755)
Objectives and content: Selected and specialised topics to be followed in Applied Statistics. Content varies from year to year when
offered.
Module Content -
Masters
Bootstrap and other Resampling techniques A & B (10694-811 & 10695-841)
Objectives and content: Traditional
procedures of statistical inference in many cases are true only asymptotically or under strict assumptions for small samples. For many problems it is impossible to find
solutions analytically. Re-sampling techniques are computer intensive methods using repeated re-sampling from the original sample in order to obtain solutions for
inferential statistical problems. The aim of this module is to introduce the student to the bootstrap and related computer intensive methods enabling him/her to use
correctly these methods with confidence in practice. The A module is a prerequisite for the B module.
Applied Learning Theory
Objectives and content: The outcomes of
these modules can be summarized as follows:
- The students should develop a holistic view of the subject of statistics, gaining an appreciation of the general principles underlying many
(seemingly unrelated) statistical methods.
- The students should develop an awareness of the size, complexity and diversity of data sets which one encounters in practice.
- The students should develop an appreciation of the challenges and problems posed to a statistician wishing to analyse a data set, especially the problem referred to as the "curse of dimensionality" in high-dimensional problems.
- The students must gain knowledge of the various approaches to the "curse of dimensionality" problem and the manner in which these
approaches compromise between underlying assumptions and sample size requirements.
- The students must understand the important role played by more traditional, established statistical procedures such as multiple regression
analysis, logistic regression analysis and linear discriminant analysis in modern data mining.
- The students should become sensitive to and appreciative of the valuable contributions made to the development of data mining procedures in
areas such as computer science and machine learning.
- The students should be granted the opportunity to enhance their programming skills through writing appropriate programs to solve various data
analysis problems.
Regarding the content of the modules, statistical learning theory is a collective noun for a variety of techniques
that can be used to identify, describe and model important patterns and trends in data sets. The topics which are studied in these modules include techniques that are well
established in traditional statistics, namely regression analysis, discriminant analysis, spline models and smoothing splines, as well as more recently developed approaches such
as regression and classification trees, additive models, bagging, boosting (also from a functional gradient descent point of view), neural networks, random forests and support
vector machines.
Advanced Sampling Techniques (10523-818)
Objectives and content: In practice,
complex sampling techniques are usually applied to design sample surveys. Furthermore, nonresponse and skewness generally manifest in sampling surveys that need to be
addressed scientifically. This course covers both theoretical and practical aspects regarding sampling and include the following: two-stage cluster sampling; design and
estimation of complex sample surveys; design effects; dealing with nonresponse and missing data; weighting of surveys; inferential statistics for complex survey data.
Multi-dimensional Scaling A & B (10597-822 & 11910-852)
Objectives and content: Multi-dimensional scaling (MDS) consists of various techniques from the field of multivariate statistical analysis. MDS focuses on dimension reduction and
graphical displays of multi-dimensional data. This module introduces the theory and practical implementation of classical metrical scaling, non-metrical scaling, various forms of
Procrustes analysis, unfolding techniques, individual differences models, as well as m-mode n-way models. Biplot methodology is emphasised. MDS techniques for both quantitative
and qualitative data are considered. Correspondence analysis, multiple correspondence analysis, homogeneity analysis, analysis of distance as well as non-linear principal
component analysis and canonical variate analysis are also discussed. The A module is a prerequisite for the B module.
Advanced Statistics A & B (10521-821 &
10522-851)
Objectives and content: Selected and specialised topics to be followed in Applied Statistics. Content varies from year to year when offered.