Mathematical Methods Course Notes

Lecture notes from my MSc course on mathematical methods.

The aim was to give an overview of various mathematical techniques and algorithms, likely to be of use to people doing research, particularly in computer vision and bioinformatics. The intention is not to go into tremendous detail on each method, but to give enough information so that students get the general idea of the approaches, and an understanding of the situations in which each algorithm would be useful.

The recommended course textbook is "Numerical Recipes in C" (or more recent variants) by Press et al. which gives an excellent overview of many useful techniques.

It was recommended that the course assignments were done in MatLab.

• Linear Algebra (1) (pdf)
  • Vectors, Matricies and Geometric Interpretation
• Linear Algebra (2) (pdf)
  • Linear Equations, Determinants
  • Cholesky, LU, Eigen-Decomposition, SVD
• Linear Algebra (3) (pdf)
  • Over/Under-determined Systems, Linear Subspaces
• Probability Theory (1) (pdf)
  • Basics, 1-D Distributions, Central Limit Theorem
• Probability Theory (2) (pdf)
  • Mixture Models, n-D Distributions
• Optimisation in 1D (pdf)
  • Golden Section, Brent, Newton Raphson
• Optimisation in n-D (1) (pdf)
  • Downhill Simplex, Powell's, Steepest Descent
  • Fletcher-Reeves, Polak-Ribiere, Levenberg-Marquardt
• Optimisation in n-D (2) (pdf)
  • Linear Programming, Quadratic Programming, Dynamic Programming
  • Simulated Annealing, Population Based Methods (GAs, EDAs)
• Classification Theory (1) (pdf)
  • Basics, Likelihood Ratio Test, ROC Curves
  • LDA, Nearest Neighbour Classifiers
• Classification Theory (2) (pdf)
  • Support Vector Machines, the "Kernel Trick"
  • Relevance Vector Machines
• Modelling Data(pdf)
• Parameter Estimation, Error Propogation, Quality of Fit

Any comments on content, exposition or important topics missing would be gratefully recieved.

Tim Cootes