Statistics and Data Analysis

Module aims

For those who have not met probability statistics before: to enable an understanding of the very basics of statistical analyses in common use in the biomedical engineering sciences, whilst at the same time, presenting a fresh view of statistical concepts. For those who have met statistics before: to give some familiarity with the resources available for performing data and statistical analyses which typically enable more flexibility than is normally found in off-the-shelf packages, such as SPSS.

Learning outcomes

Appreciation of the underlying concepts of statistics: probability, random variables Appreciation of joint probability and conditional probability Understanding of the relationship between histograms and probability density and probability mass functions Understanding why mathematical models are required to extract information from experimental data Appreciation of the Fourier transformation as a representation of ordered, regularly sampled data in an alternative basis Ability to tackle simple probabilistic analyses of discrete experiments Reporting experimental data with statistical meaning Ability to seek out the appropriate models for analysing problems related to experimental datasets How to create estimates of probability density and probability mass functions from experimental data Use of MATLAB to perform basic statistical tests Use of MATLAB and other computing packages to perform appropriate statistical analysis on experimental data Application of the Fourier transformations for the spectral analysis of signals Drawing of random numbers standard distributions (uniform and Normal); appreciation of what techniques exist for sampling from outside these distributions; understanding of how to create a bootstrap-world, and simulate simple stochastic processes Learn to use MATLAB on-line documentation Use of statistics in reporting experimental data. Transfer of concepts to other courses, such as neuroscience

Module syllabus

Probability Theory: Experiments and outcomes; random variables, probability, probability density & mass functions; conditional probability, joint probability. Bayes’ theorem. Data distribution and modeling: Standard distributions. Estimating properties of distributions: mean and variance, skewness and kurtosis. Estimating confidence intervals of estimated parameters. Goodness of fit of experimental data to distributions: chi-squared test, tests for normal distributions. Significance tests and inference: mixture models, t-tests, paired and unpaired. Analysis of variance (ANOVA). Parametric vs non-parametric statistics. Sampling methods: Sampling from distributions, Monte-Carlo methods, bootstrapping. Regression: Univariate and multivariate linear regression; significance testing on linear regression; Pearson correlation coefficient; nonlinear regression. Model selection. Cross-Validation. Fourier analysis: Fourier series, Fourier transform, Discrete Fourier Transform, Amplitude and Power Spectra, Sampling theorem, Anti-aliasing, Short-Time Fourier Transform. Applications of Fourier Analysis. Practicals: C/Matlab for implementation of basic data analysis methods, apprimately one third of the course. Applied to medical, epidemiological and biological data sets.

Pre-requisites

Calculus, integrals, linear algebra, an understanding of functions of more than one variable. Some programming experience (any language) is preferrable.

Teaching methods

Lectures: 24 hours
Labs: 6 hours

Assessments

Examinations:
●  Written exam: 100% weighting
    Rubrics: One paper, four questions, answer all four.
     Outline answers to past papers will be available

Feedback : Feedback will be provided during assignment correction and laboratory sessions, which also provide an opportunity for students to ask questions on the lectures. Additionally, office hours will be made available.