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The following are some suggestions, but you are free to choose your own projects and directions. If you choose a project that is not listed here, I would like to have a chat with you and your group first to make sure it is suitable. It is possible that more than one group chooses the same or similar project. It is important that each group works on their own; in particular, two groups working on the same or similar project should not be helping each other on the project. Please feel free to discuss with me for more details and directions regarding the following suggestions.
There is a text (from a prison) that was coded using a Ceasar cipher; this was decoded using just pairwise probabilities and Markov chains. One project I had in mind was to decode this text or encode/decode other suitably available texts, and also see if using a longer memory makes a difference. background
Various projects on card shuffling. One would explore various shuffles and discuss using theory and simulation. For example, see these papers and the references within: overhand shuffle Shuffling cards and stopping times (use UCL VPN for link) How many times do I have to shuffle this deck
Queuing: central versus individual queues: does it make sense that banks use centralized queues and supermarkets often use individual queues? In fact, some supermarkets have a combination of individual and centralized queues; is there some optimal arrangement?
How should planes be boarded and de-boarded? Develop models and test them. Compare with real world data.
How to manage line-ups? Hospitals and other institutions put priority based on need, but you can’t make someone with a broken ankle wait forever?
Various models in statistical physics: eg: Discuss the Ising model, sample it using MCMC. For example, this paper considers exact sampling.
Applications of MCMC to Bayesian statistics: discuss the basic Bayesian framework, and how MCMC is applied, supporting the examples with code.
Random number generation: discuss the theory and practice of random number generation.
Discuss Quasi-Monte Carlo methods and illustrate theory with examples and simulations. There is an introduction here
Hidden Markov chains: Often the observations we see, do not correspond to a Markov chain, but in the background a Markov chain is pulling the strings.
How can the theory we learned be applied and extended to model epidemics? If we understand a model and the model is good, does it allow us to make interventions that help us improve the situation.
Write a report on a research paper, working through some of the theory and illustrating relevant examples via coding/simulation.
Write a report on a more advanced topic that was not covered in the module, and illustrate the relevant example via coding/simulation: eg solutions of PDE via probability theory, random walk and electrical networks.
Version: 05 December 2021