Statistical Data Science - MSc

This applied statistics module focusses on problems that occur in the fields of ecology, biology, genetics and psychology. Motivated by real examples, you will learn how to define and fit stochastic models to the data. In more complex situations this will mean using optimisation routines in MATLAB to obtain maximum likelihood estimates for the parameters. You will also learn how construct, fit and evaluate such stochastic models. Outline Syllabus includes: Function optimisation. Basic likelihood tools. Fundamental features of modelling.  Model selection. The EM algorithm. Simulation techniques. Generalised linear models.

The module enables students to undertake an independent piece of work in a particular area of statistics, or statistical finance/financial econometrics and to write a coherent account of the material.

There is no specific syllabus for this module.

A list of possible topics, together with names of Staff willing to supervise these projects, will be circulated to students in the autumn term. A broad range of projectsis available, encompassing both practical data analysis and more methodological work, although projects that are primarily theoretical will typically have obvious practical applications. Students then choose a topic after consultation and agreement with the relevant member of staff. This is done early in the spring term and some preliminary work is done during the spring term, leading to a short presentation at the end of that term. The main part of the project is then undertaken after the examinations in May.

This course introduces (and revises for some students) the essentials of probability and classical (frequentist) statistical inference, which provide the backbone for later modules.

Syllabus: Probability: axioms, marginal, joint and conditional distributions, Bayes theorem, important distributions, generating functions and various modes of convergence. Classical Inference: Sampling

distributions. Point estimation: consistency, Cramer-Rao inequality, efficiency, sufficiency, minimum variance unbiased estimators. Likelihood. Methods of estimation. Hypothesis tests: maximum likelihood-ratio test, Wald and score tests, profile and test-based confidence intervals.

Linear model. Least squares. General linear model; simple and multiple regression, polynomial regression. Model selection, residuals, outliers, diagnostics. Analysis of variance. Generalised linear model.

Discrete data analysis. Review of Binomial, Poisson, negative binomial and multinomial distributions. Properties, estimation, hypothesis tests.

Contingency tables. Tests for independence. Measures of association. Logistic models.

Multidimensional tables. Log–linear models; fitting and model selection.

Bayes Theorem for density functions; Conjugate models; Predictive distribution; Bayes estimates; Sampling density functions; Gibbs and Metropolis-Hastings samplers; Winbugs/OpenBUGS; Bayesian hierarchical models; Bayesian model choice; Objective priors; Exchangeability; Choice of priors; Applications of hierarchical models.

Sampling: Simple random sampling. Sampling for proportions and percentages. Estimation of sample size. Stratified sampling. Systematic sampling. Ratio and regression estimates. Cluster sampling. Multi-stage sampling and design effect. Questionnaire design. Response bias and non-response.

General principles of experimental design: blocking, randomization, replication. One-way ANOVA. Two-way ANOVA. Orthogonal and non-orthogonal designs. Factorial designs: confounding, fractional replication. Analysis of covariance.

Design of clinical trials: blinding, placebos, eligibility, ethics, data monitoring and interim analysis. Good clinical practice, the statistical analysis plan, the protocol. Equivalence and noninferiority. Sample size. Phase I, II, III and IV trials. Parallel group trials. Multicentre trials.

In addition, level 7 students will study hierarchical designs: fixed and random effects models; split-plot designs; crossover trials; variance components.

Introduction: Machine learning and data visualisation with R.

Classification and prediction: Generalised linear model (GLM), linear discrimination analysis (LDA), k-nearest neighbors (KNN). R-based worked examples.

Resampling methods: Cross-validation (CV) and bootstrap. R-based worked examples.

Regression tree-based methods: Classification and regression trees (CART), bagging, random forests and boosting. R-based worked examples.

Support vector machines (SVM): Support vector classifier, regression SVM. R-based worked examples.

Machine Learning in Action:

(a) Biomedical and health data analysis;

(b) Bond default data analysis;

(c) Insurance data analysis;

(d) Financial data analysis;

(e) Other big data analysis.

Each year three topics will be offered and will reflect recent advances in statistical modelling and statistical methodology. Example topics are:

a) Statistical Ecology: Understanding demographic parameters and how they are used to model population dynamics. Estimating abundance and the effect of heterogeneity. Models for estimating survival probabilities. Multi-site and multi-state models. Classical model-selection. Complex models. Case studies.

b) Survival analysis: Survival data, types of censoring. Failure times and hazard functions; Accelerated failure time model. Parametric models, exponential, piecewise exponential, Weibull. Nonparametric estimates: the Kaplan-Meier estimator, and asymptotic confidence regions. Parametric inference. Survival data with covariates. Proportional hazards. Cox's model and inference. Computer software: R and WinBUGS.

c) Regression models with many variables: Examples of high-dimensional problems; Penalized maximum likelihood; Ridge regression; non-negative garrote; Lasso and adaptive Lasso estimation; LARS algorithm; Oracle property; Elastic Net; Group lasso.

d) Modern nonparametric statistics: Bias-variance trade-off, Kernel density estimation, Kernel smoothing, Locally linear and locally quadratic estimation, basis function methods.

In addition, level 7 students will study advanced applications of these techniques (often using R) in all topics.

Multivariate normal distribution, Inference from multivariate normal samples, principal component analysis, mixture models, factor analysis, clustering methods, discrimination and classification, graphical models, the use of appropriate software.

This module will focus on basic features of stochastic processes and time series analysis. It includes: Markov chains on discrete state spaces, communication classes, transience and recurrence, positive recurrence, stationary distributions. Markov processes on discrete state spaces, exponential distribution, embedded Markov chain, transition graphs, infinitesimal generator, transition probabilities, stationary distributions, skip-free Markov processes. Stationary time series: Stationarity, autocovariance and autocorrelation functions, partial autocorrelation functions, ARMA processes. ARIMA Model Building and Testing: Estimation, Box Jenkins, criteria for choosing between models, diagnostic tests for the residuals of a time series after estimation. Forecasting: Holt-Winters, Box-Jenkins, prediction bounds.