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Mathematics for Real-World Systems Centre for Doctoral Training

MA930 - Data Analysis (12 Cats)

Lecturer: David Wild (Systems Biology)

Module Aims

This is one of 4 core modules for the new MSc in Mathematics of Systems. The main aims are to provide the students with a broad knowledge of modern techniques of exploratory data analysis, time series modelling and forecasting, spectral analysis and data assimilation algorithms. By the end of this module, the students will be able to quantitatively summarise and critically assess data from real-world systems, use modern methods of parameter estimation to model and forecast time-series data, compute and interpret spectral representations of time-series data and incorporate observations into mathematical models to reduce the uncertainty in predictions made using these models.

Syllabus

Statistical analysis of data

  • Preprocessing
  • Exploratory data analysis: summary statistics, histograms, scatterplots, correlations, common data transformations
  • Fitting models to data and parameter estimation: regression, maximum likelihood estimator, sensitivity analysis
  • Principal component analysis and dimensionality reduction

Time-series modelling and forecasting

  • Stationary timeseries: Linear correlation, nonlinear correlation and quantification of uncertainty
  • Recurrence relations
  • Linear timeseries models: autoregressive, integrated and moving average models and their combinations
  • Parameter estimation for linear timeseries models
  • Nonstationary time-series: detrending, change-point detection
  • Autoregressive forecasting
  • Nonlinear timeseries models

Spectral analysis and data transforms

  • Fourier transform, the Wiener-Kinchin theorem and uses in applied mathematics
  • Discrete Fourier transform, Nyquist frequency, quantification of uncertainty in power spectra
  • Implementation of the DFT via the FFT algorithm
  • The wavelet transform and time-frequency analysis of timeseries
  • Applications of data transforms in signal processing: flitering, smoothing, denoising, feature extraction and compression.

Data assimilation

  • Some concepts from Bayesian inference
  • Statistical interpolation and the Kalman filter
  • Ensemble Kalman filter
  • 3DVAR and 4DVAR
  • Some applications to simple dynamical systems

Illustrative Bibliography

C.M. Bishop, Pattern Recognition and Machine Learning, Springer 2006
J.D. Hamilton, Time Series Analysis, Princeton University Press 1994
G.E.P. Box, G.M. Jenkins and G.C. Reisel, Time Series Analysis: Forecasting and Control, Prentice Hall 1994

Teaching

See main calendar for timetable

  • Per week: 2 x 2 hours of lectures, 2 x 2 hours of classwork
  • Duration: 5 weeks (first half of term 1)

Assessment

For deadlines see Module Resources page

  • Written homework assignments (25%)
  • Class test (25%) and
  • Oral examination (50%)

 

Module Resources