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