Semester I

Descrpition of aims and contents:

This module aims at (1) familiarizing students with the ideas and methodologies of some multivariate methods together with their applications in data analysis using the SPSS or other computing softwares like R or Excel (2) exploring some of the real-life situations occurring in the fields of agriculture, biology, environment, engineering, industrial experimentation, medicine, social sciences, etc, that can be investigated using multivariate techniques. 

Contents and literature:

             Contents:

1. Introduction to Multivariate Analysis
2. Multivariate distributions
3. Characterizing and displaying multivariate data
4. Multivariate normal distribution and statistical inference based on the multivariate normal distribution
5. Multivariate analysis of variance (MANOVA)
6. Discriminant analysis and Classification Analysis
7. Multivariate Regression and Canonical Correlation Analysis (MR, CCA)
8. Principal Component Analysis (PCA)
9. Factor Analysis and Cluster Analysis

          Literature: 

[1]  ALVIN C. RENCHER and WILLIAM F. CHRISTENSEN. Methods of Multivariate Analysis, Third Edition, John Wiley & Sons, Inc., New York, 2012.

[2]  ALVIN C. RENCHER. Methods of Multivariate Analysis, Second Edition, John Wiley & Sons, Inc., New York, 2002. https://www.ipen.br/biblioteca/slr/cel/0241

[3]  RICHARD A. JOHNSON and DEAN W. WICHERN. Applied Multivariate Statistical Analysis, Sixth Edition, Pearson Prentice Hall, New Jersey, 2007.

The aim of this module is to study the basic theory introduce the stochastic processes in discrete and continuous time.  We use mathematical techniques to explore the behaviour of these processes.  We introduce to the students the Markov chain both discrete and, continuous and their application to Queueing models, Martingale and Gaussian Processes, and finally simulation of some stochastic processes using R, or MATLAB.

The course will introduce modern methods of statistical inference which use computational methods of analysis. These methods such as permutation tests and bootstrapping are nowadays used in business, finance, and scientific research. The focus is on statistical methods that use computation to replace certain assumptions. Students will learn how to manipulate data, design and perform simple Monte Carlo experiments, and be able to use resampling methods such as the Boot-strap. Although the main focus will be put on understanding the methods, programming language R will be used to implement them.

Expected outcomes

1)      Should be able to write R code, and be able to modify and understand existing R code.

2)      Understand basic data structures and algorithms in statistical applications.

3)      Understand basic numerical methods such as optimization, sampling, etc.

4)      Learn some statistics topics such as bootstrap, linear/logistic regression.

Indicative Content

Unit 1: R Programming Introduction

Unit 2: Introduction: Distributions of random variables; Classical parametric hypothesis testing; p-values.

Unit 3: Nonparametric tests: Permutation tests; Rank tests; Matched pairs.

Unit 4: The bootstrap: The jackknife; The empirical distribution; The nonparametric bootstrap; The parametric bootstrap; Bootstrap confidence intervals; Bootstrapping linear models.

Unit 5: Cross-validation: Leave-one-out cross-validation; Cross-validation for smoothing splines; k-fold cross-validation; Cross-validation for likelihood-based models.

Reference

James E. Gentle, Elements of Computational Statistics.

Efron and Tibshirani, An Introduction to the Bootstrap.

Computational Statistics Handbook with MATLAB®.

G.H. Givens and J.A. Hoeting, “Computational Statistics”, 2nd edition, Wiley, 2012.

Functional analysis is the study of infinite-dimensional vector spaces equipped with extra structure.  Such spaces arise naturally as spaces of functions. As well as being a beautiful subject in its own right,
functional-analysis has numerous applications in other areas of both pure and applied mathematics, including Fourier analysis, the study of the solutions of certain differential equations, stochastic processes,   quantum physics,  ordinary and partial differential equations,
numerical analysis, calculus of variations, approximation theory, integral equations, optimization and approximation theory and much more.  Apart from an introductory chapter, where we review basic concepts used in functional analysis, the module develops the theory of metric spaces, normed spaces, Hilbert spaces, linear operators, and linear functionals.  The module will deal with these topics at a basic level appropriate for undergraduates students in Applied Mathematics.

This module aims at studying the abstract theory of measures and Lebesgue Integration. A measure is a generalization of the concepts of length, area, and volume in finite-dimensional Euclidean spaces. The Lebesgue integral is a generalization of the Riemann integral to a large class of functions.

Topics covered in this module are the abstract theory of measures on sigma-algebras, measurable functions, essential properties of the Lebesgue measure, fundamental theorems of the Lebesgue integral, the connection between the Riemann and Lebesgue integrals, product spaces, and Fubini and Tonelli Theorems. The module will deal with these topics at a basic level appropriate for undergraduate students in Applied Mathematics.

The knowledge of the theory of measure and integration is essential for the study of several advanced topics in Functional Analysis, Partial Differential Equations, and many other areas of Mathematics. In particular, the theory of measure and integration is vital to the study of Probability and Stochastic Processes.

The study of Dynamical Systems explores how systems evolve over time, focusing on the long-term behavior of these systems under different conditions. It provides the mathematical framework to analyze how a system's state changes as a function of time, which can model a wide range of real-world phenomena, from physical and biological processes to economics and engineering applications. The course is delivered mainly through lectures backed up by tutorial sessions. The lecture includes interactive elements whereby students in groups apply principles to simple problems to ensure their involvement and so gain understanding.  Handouts are used so that students can concentrate on the material of the lecture, but with gaps where students either have to fill in or make separate notes. Problem sheets will be given out to students and after time, the problems will be discussed in class.  The assignment will require the students to undertake some investigation on their own and to develop ideas and apply them.  They will also produce a report for each.

Differential geometry is a Mathematical discipline that uses tools, techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry

This module covers topics in time series analysis and some statistical techniques on forecasting, such as time series regression, decomposition methods, exponential smoothing and Box-Jenkins forecasting methodology.

 The main aim of the module is to equip students with various forecasting techniques and knowledge on modern statistical methods for analyzing time series data.

The idea is to give students broad knowledge on statistics produced by National Institute of Statistics of Rwanda (NISR) which are not covered in Demographics statistics course.

The study of Dynamical Systems explores how systems evolve over time, focusing on the long-term behavior of these systems under different conditions. It provides the mathematical framework to analyze how a system's state changes as a function of time, which can model a wide range of real-world phenomena, from physical and biological processes to economics and engineering applications. Dynamical Systems  is delivered mainly through lectures backed up by tutorial sessions. The lecture includes interactive elements whereby students in groups apply principles to simple problems to ensure their involvement and so gain understanding.  Handouts are used so that students can concentrate on the material of the lecture, but with gaps where students either have to fill in or make separate notes. Problem sheets will be given out to students and after time, the problems will be discussed in class.  The assignment will require the students to undertake some investigation on their own and to develop ideas and apply them.  They will also produce a report for each.

This module provides a basic understanding of the theory and practice of  research. It describes  both the qualitative and quantitative methods in scientific research. It differentiates the types of research (strategic, applied and fundamental)  and their respective associated designs.  The module starts with the philosophical meaning of research , then objectives of research, motivation of research, research approaches and significance of research. Finally all the process of research starting from Problem identification and formulation to interpretation of results and report are described.

    Differential geometry is a Mathematical discipline that uses tools, techniques of differential   calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry

The Mathematics Project is a Final Year module in the BSc  in Mathematics (Both Applied mathematics and statistical Mathematics).

 By taking this module, students are given the opportunity to undertake independent studies to master a particular topic and use this knowledge to solve a specific problem in applied or pure mathematics.

The Research methodology  required for the Mathematics Project is completed as a separate module in the first trimester, the Mathematics Report, which is evaluated independently. This first module gives students the chance to realise whether undertaking the Mathematics Project in the topic would meet their preference, interest or career prospects. At the the end of Final year project  students should produce a report with the project's outcomes.

All projects are closely supervised by Members of the Academic Staff with expertise in the project's area. While students are encouraged to present their own project proposals, they are also offered a list of proposals from potential supervisors.