ES98A15 Fundamentals of Predictive Modelling
Introductory description
This module provides students with fundamental knowledge for predictive modelling and uncertainty quantification. It gives an overview of the essential elements of the mathematical, statistical, and computational techniques needed to provide wellcalibrated predictions for the behaviour of physical systems.
Module aims
Understand how to use statistical modelling to quantify uncertainty arising from computer simulation and epistemic and aleatoric uncertainty in scientific problems. Formulate and solve Bayesian inverse problems of highdimensional quantities of interest. Make wellcalibrated predictions for the behaviour of physical systems described by mathematical models and data.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Univariate probability prereading – discrete and continuous distributions, the Law of Large Numbers, expectations and variance
 Multivariate probability
 Bayesian probability
 Multivariate Gaussians and their conditioning
 State estimation via Kalman filtering
 Forward and inverse UQ
 Quadrature and sampling (MC, MCMC, QMC, LHS, …)
 Finitedimensional linear algebra
 Basic functional analysis (infinitedimensional linear algebra)
 Perspectives on random functions
 Interpolation, regression, orthogonal projection
 Motivation for regularisation
 Tikhonov regularisation / ridge regression
 Sparse approximation (\ell^1 regularisation)
 Optimisation approach to inverse problems, linear and nonlinear case, connect to Tikhonov regularisation
 Bayesian approach to inverse problems, wellposedness of BIPs
 High or infinitedimensional aspects
 Approximate Bayesian inference e.g. Laplace approximation,
Learning outcomes
By the end of the module, students should be able to:
 Understand and apply univariate and multivariate probability to complex scientific modelling problems
 Understand and interpret forward and inverse uncertainty quantification
 Apply and evaluate quadrature and sampling schemes to solve predictive modelling tasks
 Appreciate, synthesise, and apply modelling strategies for uncertain quantities of interest in high dimension
 Contextualise the relations among regression, approximation, and orthogonal projection
 Recognise, formulate, analyse and solve Bayesian inverse problems
Indicative reading list
McClarren, Uncertainty Quantification and Predictive Computational Science, available from SpringerLink on campus at Warwick  aimed at a general Physical Sciences / Engineering audience
Sullivan, Introduction to Uncertainty Quantification, available from SpringerLink on campus at Warwick
Boyd and Vandenberghe, Introduction to Applied Linear Algebra  Vectors, Matrices and Least Squares, freely available online. Companion exercises that implement the material in Python and Julia are available from the same webpage.
View reading list on Talis Aspire
Interdisciplinary
The MSc programme will recruit students with backgrounds across the physical and mathematical sciences, including engineering, and will provide an interdisciplinary perspective on predictive modelling.
The fundamentals of uncertainty quantification and predictive modelling are drawn from the mathematical, statistical and computational sciences, and rely on disciplinary fusion with application domains such as physical/chemical science and engineering.
Subject specific skills
 Computational statistics
 Mathematical modelling
 Predictive modelling
 Reliability assessment of descriptive and predictive scientific models
Transferable skills
 Data analysis and modelling
 Oral presentation skills
 Scientific computing
Study time
Type  Required 

Lectures  30 sessions of 1 hour (20%) 
Supervised practical classes  5 sessions of 3 hours (10%) 
Private study  105 hours (70%) 
Total  150 hours 
Private study description
NB:
Inperson instruction takes place in weeks 15 of module
Students attend WCPM seminar in weeks 110 of module, i.e. throughout the teaching term
Viva takes place in week 10, referring to all taught components, preliminary work on MSc research project, and WCPM seminars attended
Costs
No further costs have been identified for this module.
You must pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Computer Laboratory Assignments  60%  
6 x 1 page assignments based upon lecture topic and computer laboratory work. (One assignment will correspond to "week 0" induction material, and the others to the five taught weeks of the module.) 

Oral Examination  40%  

Feedback on assessment
Annotation of computer workbooks with feedback on individual questions
Written feedback from examiners of viva voce exam
Courses
This module is Core for:
 Year 1 of TESAH1B1 Postgraduate Taught Predictive Modelling and Scientific Computing
This module is Optional for:
 Year 1 of TMAAG1P9 Postgraduate Taught Interdisciplinary Mathematics
 Year 1 of TMAAG1P0 Postgraduate Taught Mathematics
 Year 1 of TMAAG1PF Postgraduate Taught Mathematics of Systems

TPXAF344 Postgraduate Taught Modelling of Heterogeneous Systems
 Year 1 of F344 Modelling of Heterogeneous Systems (MSc)
 Year 2 of F344 Modelling of Heterogeneous Systems (MSc)

TPXAF345 Postgraduate Taught Modelling of Heterogeneous Systems (PGDip)
 Year 1 of F345 Modelling of Heterogeneous Systems (PGDip)
 Year 2 of F345 Modelling of Heterogeneous Systems (PGDip)