APPLYING FOR ADMISSION TO COURSES OR DEGREES HONOURS IN MATHEMATICAL STATISTICS

Transcription

1 APPLYING FOR ADMISSION TO COURSES OR DEGREES Application forms may be obtained from or from the Student Enrolment centre, ground floor, Senate House. The website also contains information about general registration rules, fees and financial aid, as well as rules applying to international students. The application form needs to include details of what is being applied for: honours, PDS, MCRR or MSc, whether this is as a full-time (FT), part-time (PT) or as an occasional student (taking the course for non-degree purposes), and the area you are applying for, namely either Mathematical Statistics. Students from Universities other than Wits must supply full academic transcripts with their applications no decision as to the offer of a place in the postgraduate program can be made without these. HONOURS IN MATHEMATICAL STATISTICS The short courses offered within the School in 2017 include: Course NQF credits Semester offered Course code Compulsory Research Project: Mathematical Statistics 42 1 STAT4113 Elective courses Advanced Distribution Theory 14 1 STAT4101 Applied Sampling 14 2 STAT4102 Extreme Value Theory 14 2 STAT4104 Operations Research Techniques 14 1 STAT4110 Reliability and Maintenance Theory 14 2 STAT4111 Spatial Statistics 14 2 STAT4107 Statistical Aspects of Data Mining 28 1 STAT4108 Stochastic Processes with Applications in 14 1 STAT4109 Finance Point Processes* 14 2 STAT4106 * May be offered Other elective courses from Computational and Applied Mathematics (CAM) and Mathematics (Maths) may also be included, by special permission of the postgraduate coordinators of the schools involved:

2 MASTERS BY COURSEWORK AND RESEARCH REPORT AND POSTGRADUATE DIPLOMA IN SCIENCE The short courses offered within the School in 2017 include: Short Courses Points Semester offered MCRR code Advanced Sampling 20 2 STAT7030 Data Mining Theory & Application * 40 1 STAT7038 Extreme Value Theory 20 2 STAT7033 Operations Research 20 1 STAT7035 [Point Processes] # 20 2 STAT7036 Reliability & Maintenance Theory 20 2 STAT7004 Spatial Statistics * 20 2 STAT7006 Stochastic Processes with Applications in Finance * 20 1 STAT7037 At least one of the courses marked with an * must be included in the curriculum. Note that not all courses will be offered in a particular year, depending on the demand. # may not be offered. By special permission, appropriate short courses may also be taken from other Mathematical Sciences Schools or from the School of Economic and Business Science, in the Faculty of Commerce, Law and Management. COURSE OUTLINES FOR HONOURS COURSES ADVANCED DISTRIBUTION THEORY Lecturer: Prof GV Kass This course covers the development and underlying theory of: special distributions, inequalities and either quadratic forms or systems of distributions. Transformations and special distributions: General and Orthogonal transformations (including Helmert). Noncentral t, χ2, F. Dirichlet distributions. Asymptotic distributions of Order Statistics ( /\ s, Weibull, Fréchet, Gumbel; Extreme order statistics, Mills Ratio). Moment inequalities: Markov, Chebychev, Kolmogorov, Jensen, Cauchy-Schwartz, Holder, Minkowski, rth root of the rth absolute moment, Bonferroni.Convex Ordering. Quadratic forms: Idempotent Matrices (Properties and more on non-central χ2). Mgf and cumulants of Q.F. conditions for QF~ χ2 (λ). Independence of QFs (and linear functions and QFs). Simultaneous Orthogonal Diagonalization. Cochran s Theorem (and simple applications to ANOVA). Systems of Distributions: Pearson, Gram-Charlier, Johnson systems of distributions. APPLIED SAMPLING Lecturer: Dr A Turasie This course covers both theoretical and practical aspects of Survey Sampling and includes: questionnaire design and piloting; definition of types of sampling (simple random, stratified, systematic, cluster, double, snowball, convenience, complex) and their advantages and disadvantages in theory and practice; proportional vs disproportional allocation for stratified sampling and reasons for their choice; sample size calculation; in general and for different methods, including optimal allocation; estimation of means, totals and proportions, and the

3 variances of the estimators; margin of error tables and nomograms; weighting of surveys; cell weighting vs raking; household vs personal weights; design effects: calculation and implementation of: cost vs efficiency; dealing with missing values; using complete data, imputation via means and regression; an introduction to data fusion: criteria for fusing of data sets, methods of performing data fusion (donor to recipient, one to one, many to one, many to many, transportation algorithm, once-off fusion, customized fusion) and of assessing the quality of the fusion. EXTREME DATA VALUE THEORY Lecturer: Mr M Dowdeswell This course provides the student with an understanding of the modelling and analysis of data concerning the extremes of a distribution, and includes the following topics. Introduction to and examples of extreme value data. Review of the asymptotic likelihood theory required for the analysis of extreme values and of the relevant model diagnostic plots. Distributions of extreme values: Gumbel, Fréchet and Weibull. The Generalized extreme value (GEV) distribution. Inference for the GEV distribution. Threshold data and the Generalized Pareto (GP) distribution for modelling threshold excesses. Inference for the GP distribution. Modelling and analysis of extremes of stationary (dependent) series. Extremes of non-stationary series. MATHEMATICAL STATISTICS PROJECT (COMPULSORY FOR STUDENTS REGISTERED FOR MATHEMATICAL STATISTICS HONOURS) Coordinator: Dr C Chimedza This course provides grounding in the collaborative and independent research skills required for statistical practice, including the skills required to explain statistics intelligibly both to peers and to other students. It comprises the presentation of a seminar, participation in seminars, 2 hours of tutoring a week, and a research report. Part time students may be exempted from the seminars and tutoring, but will have to provide an expanded research report. OPERATIONS RESEARCH TECHNIQUES Lecturer: Mr H Chipoyera Operations Research has been defined as the science of better. In Operations Research, we employ techniques from the mathematical sciences, such as mathematical modelling, statistical analysis, and mathematical optimization to find optimal or near-optimal solutions to complex decision-making problems in manufacturing, finance and supply chain management. This short course provides an introduction to the algorithms and techniques or Operations Research. In this course, we will cover the standard areas of model building, linear programming, the simplex algorithm, optimisation of network problems and integer programming. Branch and bound and other optimisation techniques will be introduced. We will also spend some time on game theory, and decision making as a separate topic. Throughout the course there will be practical and short projects.

4 POINT PROCESSES In operations research, point processes are probabilistic models for describing stochastic phenomena as the arrival of customers at a service station (e.g. supermarkets, banks, workshops), the arrival of claims (time and amount) at an insurance company, the failure time points of machines, fluctuation of staff (manpower planning), the occurrence of nature catastrophes, and so on. Hence, they are basic analytical tools in operations research, risk and safety analysis, management sciences, reliability, maintenance, and environmental sciences. This course deals with those point processes that have proved to be most adequate for applications in these areas: Poisson processes, mixed Poisson processes, Polya-Lundberg processes, and renewal processes. The course mainly focuses on applications to risk analysis and queuing models. s Beichelt, F. (2006): Stochastic Processes in Science, Engineering and Finance. Chapman and Hall/CRC, Boca Raton, New York, London. Beichelt, F. (2015): Applied Probability and Stochastic Processes. Chapman and Hall/CRC, New York, London. RELIABILITY AND MAINTENANCE THEORY Reliability and safety analysis as well as maintenance planning play an important role in engineering, but increasingly also in banking, communication, and manpower planning. The first part of this course deals with the key problem of reliability theory, namely with the investigation of the mutual relationship between reliability criteria of systems and the corresponding reliability criteria of their components (subsystems). Special emphasis is on network reliability analysis (e.g. computer networks, satellite networks). In the second part, classes of lifetime distributions are introduced, which model the aging behavior of technical products (as well that of human beings). Based on the underlying aging behavior, costoptimal maintenance policies are discussed. Beichelt, F. and Tittmann P. (2012): Reliability and Maintenance Systems and Networks. Chapman and Hall/CRC, New York, London. SPATIAL STATISTICS Lecturer: Mr S Salau This course provides the student with an understanding of the modelling and analysis of data which are spatially distributed, and in which the correlation between two observations is a function of the distance between them. Topics covered include: Introduction to spatial random variables and spatial data; definition of the variogram and its properties; models for the variogram and its estimation, either nonparametrically or via maximum likelihood; spatial prediction and kriging. Simple and ordinary kriging; change of support and block kriging; co-kriging and universal kriging, Bayesian methods, as well as more advanced topics to be decided.

5 STATISTICAL ASPECTS OF DATA MINING Lecturers: Dr C Chimedza and Dr D Rose Data mining is the process that attempts to discover patterns in large data sets. It utilizes methods at the intersection of artificial intelligence, machine learning, statistics, and database systems. The overall goal of the data mining process is to extract information from a data set and transform it into an understandable structure for further use. This short course provides the theoretical background for students to understand the techniques and approaches of data mining. These include classical multivariate techniques (including principal component analysis, factor analysis, cluster and discriminant analysis, canonical correlation analysis) as well as tree induction and rule learning, support vector machines and neural nets. It also provides the practical background required to apply these techniques to practical problems using training and validation data subsets, to evaluate the models, and to interpret and present the results. STOCHASTIC PROCESSES WITH APPLICATIONS IN FINANCE This course covers stochastic processes used to model the development in time of share prices, interest rates, exchange rates, prices of precious metals, and other stochastically fluctuating financial parameters. The course includes: basic definitions and concepts from the theory of stochastic processes, discrete and continuous time martingales, the optional stopping theorem, the Brownian motion process and its transformations. The advantages of applying Brownian motion in finance as well as its shortfalls are discussed. An exact derivation of the Black-Scholes-Merton-formula is given, and some other option pricing problems are solved. Beichelt, F. (2006): Stochastic Processes in Science, Engineering and Finance. Chapman and Hall/CRC, Boca Raton, New York, London COURSE OUTLINES FOR MASTERS LEVEL COURSES ADVANCED SAMPLING Lecturer: Dr A Turasie This course provides the theoretical background of and investigates issues in the application of: Calibration weighting methods, and the comparison to standard methods of cell and rim weighting; Methods of estimation: design-based, model-based and model-assisted. Data fusion: methods of combining data sets, ranging from multiple imputation, to single fusion, to customized fusion; comparison of the methods as to advantages and disadvantages; comparison of these types of fusions, and of assessing the quality of the fusion. Students will be required to apply one of more of these classes of techniques to data sets, and to provide a project report on their analyses, including additional literature studied.

6 DATA MINING THEORY & APPLICATION Lecturers: Dr C Chimedza and Dr D Rose Data mining is the process that attempts to discover patterns in large data sets. It utilizes methods at the intersection of artificial intelligence, machine learning, statistics, and database systems. The overall goal of the data mining process is to extract information from a data set and transform it into an understandable structure for further use. This short course provides the theoretical background for students to understand the techniques and approaches of data mining. These include classical multivariate techniques (including principal component analysis, factor analysis, cluster and discriminant analysis, canonical correlation analysis) as well as tree induction and rule learning, support vector machines and neural nets. It also provides the practical background required to apply these techniques to practical problems using training and validation data subsets, to evaluate the models, and to interpret and present the results. Students will be required to submit a project report in which they apply these techniques in an in-depth study of a modelling technique or techniques to a substantial set of data, and provide a literature review of these techniques. DYNAMIC PROGRAMMING Lecturer: Prof GV Kass This theoretical short unit with practical overtones covers the following topics with particular emphasis on stochastic application to: dynamic programming (DP) solutions to path problems, including those with stochastic elements, feedback control and adaptive control (learning); solving standard problems by DP including: equipment replacement with stochastic costs; Bayesian approach to quality control; simple resource allocation; theory and solution of problems with linear dynamics and quadratic criteria including stochastic errors; different approaches to inventory models; Markov decision processes; sensitivity analysis to DP solutions. EXTREME VALUE THEORY Lecturer: Mr M Dowdeswell This short course provides successful candidates with an understanding of the modelling and analysis of extreme values and the ability to apply this theory to the analysis of extreme value data. It covers the Generalized Extreme Value distribution for modelling extremes of independent series and the Generalized Pareto distribution for threshold excesses of such series, as well as their extension to stationary and non-stationary series. Furthermore candidates cover, by self-study, two advanced aspects of extreme value theory, namely the Point Processes, Characterisation of extremes, which provides a unifying theoretical framework for modelling extreme values, and the analysis of Multivariate Extreme value data. Candidates also have to complete a major project in which they either research some advanced aspect or perform an extensive analysis of an extreme value dataset.

7 OPERATIONS RESEARCH Lecturer: Mr H W Chipoyera This short course provides an introduction to the algorithms and techniques behind supply chain optimization. This includes the mathematical background as well the practical application of these techniques. Topics covered include Forecasting, Transportation systems, Transportation models and algorithms; Genetic algorithms and simulated annealing, Inventory management systems and algorithms, Continuous and discrete point location algorithms, Supply chain models, Neural networks for the optimization of supply chains; Manufacturing systems, Manufacturing scheduling models, Material handling models and algorithms, Warehousing systems. An in-depth study of the theoretical grounding of methods involved in, and application of, one of the following topics: transportation systems/models, inventory management systems, supply chain models, manufacturing systems, warehousing systems. POINT PROCESSES Point processes are probabilistic models for describing stochastic phenomena as the arrival of customers at a service station (e.g. supermarkets, banks, workshops), the arrival of claims (time and amount) at an insurance company, the failure time points of machines, fluctuation of staff (manpower planning), the occurrence of nature catastrophes, and so on. Hence, they are basic analytical tools in operations research, risk and safety analysis, management sciences, reliability, maintenance, and environmental sciences. This short course deals with those point processes that have proved to be most adequate for applications in these areas: Poisson processes, mixed Poisson processes, Polya-Lundberg processes, and renewal processes. The course mainly focuses on applications to risk analysis and queuing models. Beichelt, F. (2006): Stochastic Processes in Science, Engineering and Finance. Chapman and Hall/CRC, Boca Raton, New York, London. RELIABILITY AND MAINTENANCE THEORY Reliability and safety analysis as well as maintenance planning play an important role in engineering, but increasingly also in banking, communication, and manpower planning. The first part of this short course deals with the key problem of reliability theory, namely with the investigation of the mutual relationship between reliability criteria of systems and the corresponding reliability criteria of their components (subsystems). Special emphasis is on network reliability analysis (e.g. computer networks, satellite networks). In the second part, classes of lifetime distributions are introduced, which model the aging behavior of technical products (as well that of human beings). Based on the underlying aging behavior, costoptimal maintenance policies are discussed. Beichelt, F. and P. Tittmann (2012): Reliability and Maintenance: Networks and Systems. Chapman and Hall/CRC, Boca Raton, New York, London.

8 SPATIAL STATISTICS Lecturer: Mr S Salau This short course provides successful candidates with an understanding of the modelling and analysis of data which are spatially distributed, and the ability to apply this theory to the analysis of such data. A key concept is that of the variogram, which models the variance of the difference between two observations as a function of the distance between them. Models for the variogram are discussed as are methods of estimating these models. The variogram is used for spatial prediction via the techniques of kriging. The course covers simple and ordinary kriging, change of support and block kriging, co-kriging and universal kriging, as well as indicator and disjunctive kriging. The course also discusses contextual classification for classifying spatially dependent data occurring on a regular grid, such as remotely sensed data, via Markov random fields and Gibbs sampling. Finally, candidates have to complete a major project in which they perform an extensive analysis of a spatial data set. STOCHASTIC PROCESSES WITH APPLICATIONS IN FINANCE This short course covers stochastic processes used to model the development in time of share prices, interest rates, exchange rates, prices of precious metals, and other stochastically fluctuating financial parameters. The course includes: basic definitions and concepts from the theory of stochastic processes, discrete and continuous time martingales, the optional stopping theorem, the Brownian motion process and its transformations. The advantages of applying Brownian motion in finance as well as its shortfalls are discussed. An exact derivation of the Black-Scholes-Merton-formula is given, and some other option pricing problems are solved. Beichelt, F. (2006): Stochastic Processes in Science, Engineering and Finance. Chapman and Hall/CRC, Boca Raton, New York, London