MMath Mathematics and Statistics

An introduction to machine learning models and algorithms, both for unsupervised and supervised learning, with a focus on the underlying mathematical and statistical principles, while including some aspects of implementation and practice. Studying the relation of machine learning approaches with traditional statistical methods. 

MATH48292 Pre-Requisites: MATH27720 and MATH27711, and one of MATH38161 or MATH48061

One of either MATH38161 Multivariate Statistics and Machine Learning or MATH48061 Multivariate Statistics required as a pre-requisite.

The unit aims to:

Provide an introduction to machine learning models and algorithms, both for unsupervised and supervised learning, with a focus on the underlying mathematical and statistical principles, while including aspects of implementation and practice.  A further aim is to study the relationship of machine learning approaches to traditional statistical methods. 

On the successful completion of the course, students will be able to:

1. Explain the principles underlying both probabilistic and algorithmic machine learning techniques
2. Select and identify appropriate methodology for data analysis at the interface of statistics and machine learning
3. Apply modern machine learning methods to supervised and unsupervised learning problems
4. Use R and/or Python to analyse data with the methods discussed in the course

Part A - Nonlinear (non-neural network) machine learning approach

• Supervised Machine Learning:

      -    Classification, the Bayes classifier, and K-nearest neighbours

      -   Gaussian processes

      -   Decision trees and random forests 

• Unsupervised Machine Learning:

       -   Nonparametric clustering and   Topic models

      -   Dimensionality reduction including manifold learning 

 

Part B - Nonlinear (neural network) machine learning approaches

• Single-layer neural networks and deep neural networks

   

• Optimising neural networks (gradient descent, backpropagation)

• Statistical properties (overparametrisation, regularisation, universal function approximation)

   

• Standard models (e.g. auto-encoders) and diverse data types 

Teaching is composed of two hours of lectures per week and one tutorial class per week. Teaching materials will be made available online for reference and review. One week is reserved for coursework.

Other = coursework data analysis project, with a 30% weighting

Generic feedback will be provided after marks are released. 

C. M. Bishop and H. Bishop. 2024.  Deep learning: Foundations and concepts. Springer.

https://link.springer.com/book/10.1007/978-3-031-45468-4  

 

Trevor Hastie, Robert Tibshirani, Jerome Friedman The Elements of Statistical Learning

Data Mining, Inference, and Prediction  https://link-springer-com.manchester.idm.oclc.org/book/10.1007/978-0-387-84858-7

 

Carl Edward Rasmussen and Christopher K. I. Williams Gaussian Processes for Machine Learning https://ebookcentral-proquest-com.manchester.idm.oclc.org/lib/manchester/detail.action?docID=3338604

 

Wolfgang Karl Härdle, Léopold Simar, Applied Multivariate Statistical Analysis https://link-springer-com.manchester.idm.oclc.org/book/10.1007/978-3-030-26006-4

 

Ruye Wang, Introduction to Machine Learning From Math to Code https://www.cambridge.org/highereducation/books/introduction-to-machine-learning/D52B7B2F91C909DE461D86B498802155#overview