Project 1: Graphics and Matlab: Due Feb. 14th

The first series of projects are due to be handed in on Friday, February 14th (so that I can take the long weekend to play with the programs).

You may make up a subject of your own, but that requires my prior agreement, NO projects can be accepted on subjects I have not sent an email agreeing with.

Here are the subjects I would like you to choose from: (I have quite a precise idea about these projects, so any queries should be sent to me by email).

Correlation images/movies

1. Write a program that displays the correlations of a $p$ variable data set,( input as a $n\times p$ matrix) as grey scales or color instead of numbers.
2. Make the graphic interactif, so that you can change the color scaling, click on a cell to find out the value.
3. Optional: Make it so that we can follow as a movie the evolution of several correlation matrices for a cube of data taken as `time' evolves.

Movie of Singular Value Decomposition

1. Write a matlab function, a little like rrefmovie that shows a step by step singular value decomposition.
2. Replace the actual values in the svd movie by colors to watch how the decomposition builds up.
3. Option: Set a tolerance value so that you can ignore small residuals.

Principal Components with Matlab

Write a Matlab function that implements Principal Component Analysis (PCA), with the following steps:

1. Display the eigenvalues as a scree plot, let the user look at the graph and then choose what number of components to keep.
2. Compute the coordinates of the observations in the new basis, display them on a plot with the axes normalized to the sqrt of the eigenvalues.
3. If the PCA was done on a correlation matrix, compute the coordinates of the old variables in the new basis, draw the unit circle so that the user can compare to the ideal representation.
4. Optional: Choose to weight the variables, and the observations.

Following iterations of Markov Chains Matrices

1. For a sparse Markov Chain transition matrix, make a movie that shows how and when the sparse structure disappears.
2. Compare several methods of taking high powers of matrices.
3. Find empirically the 'worst' starting point of a Markov Chain, the ones that takes the longest to coverge to stationarity.

Matlab Correspondence analysis program

Write a complete matlab script with all the functions needed to implement Correspondence analysis.

Alternatives to the QQ plot

Program some of Tukey and Parzen's alternatives to QQplots.

Random orthogonal matrices and their eigenvalues

......more to come here................