report01101 Overview Helper Maple Code Library Calls and Global Parameters Set Random Number Seed General Purpose Helper Procedures Reduce the precision of a Matrix to Aid Printing Generate and Manipulate a Data Matrix A Data Matrix of Gaussian Random Variable Samples Generate a 4x4 Rotation Matrix Generate a 4x4 Scale Matrix Generate a 4x4 Translation Matrix Generate a 4x4 Scale, Rotation, Translation Matrix Generate a 4x4 homogenous Matrix from 3x3 specifying upper left Generate a Gaussiant Distributed data matrix with mean zero, standard deviation one Generate a scaled, rotated and then translated Gaussian data matrix Three classes from means, standard deviations and angles. Append Class Data Matrices to form one Common Data Matrix Sample Mean and Covariance Matrices Compute the Mean Vector from a Data Matrix Compute the x, y and z ranges for a Data Matrix Sample Covariance from a Data Matrix Scatter Matrix within and Between Classes Diagonilize a Covariance Matrix Return R and S Matrices given Sample Covariance/Scatter Routines for Plotting Classes and Discriminants View Specification Basis vector endpoints Plot the points and basis vectors Compute R and S using the Generalized Eigenvector Method Compute The Fisher Criterion Function Project onto 2D Fisher Basis Vectors Introduction Gaussian Random Variables and Principal Components Axis Aligned Gaussians Example of an Axis Aligned Gaussian 3D Plot of Points 3D Animated Plot of Points The Scattter Matrix, Sample Mean and Sample Covariance Example of a Gaussian Rotated with respect to Principal Axes 3D plot of points. 3D annimated plot of points. Scatter and Covariance Matrices for Rotated Distribution Recovering the Rotation and Original Principal Axes 3D Plot of Points after Rotation to Principal Components 3D Animated Plot of Points after Rotation to Principal Components Interpreting Variance as Scale 3D Plot of Points after Rotation and Scaling 3D Animated Plot of Points after Rotation to Principal Components Principal Components Subpace Maximizes Variance Fisher Discriminants Transformation to a Standard Eigenvector Problem How this Transformation Acts Upon Fisher's Criterion Example of a Three Class Problem Different Covariance Structures 3D Plot of Points from Three Classes 3D Animated Plot of Points from Three Classes Illustrating the Fisher Linear Discriminants The Within and Between Class Scatter Matrices Computer R and S for the within class and between class matrices Test that diagonalization generates proper matrices Change Coordinates by the G Transformation Look at within and between scatter matrices after transformation Confirm which are eigenvectors, rows or columns? Test if W and Lambda are solution to Generarl Eigenvector Problem The Fisher Basis Vectors in Transformed and Original Space Plot Data Matrix with Fisher Basis Vectors Fisher Basis Vectors in Transformed Space Animation of Fisher Basis Vectors in Transformed Space Fisher Basis Vectors in Original Space Animation of Fisher Basis Vectors in Original Space What is the fischerCriterion for the resulting W Compare to Generalized Eigenvector Method 3D Plot of Fisher Basis Vectors Found using Generalized Eigenvector Method 3D Animation of Fisher Basis Vectors Found using Generalized Eigenvector Method View 2D projection using Alternative Bases 2D Plot of classes projected onto Fisher Discriminants References