1. Linear Discriminant Analysis

2. Linear Discriminant Analysis
Why
To identify variables into one of two or more mutually exclusive and exhaustive categories.
To examine whether significant differences exist among the groups in terms of the predictor variables.
What
The analysis helps determine what predictor variables contribute most to intergroup differences.
It then classifies cases to one of the groups based on the values of the predictor variables.
How
Using a combination of MANOVA, PCA and MLP.

3. LDA Assumptions Absence of outliers Equal samples size Many data
Homogeneity of variance-covariance
Linear relationship
No multicolinearity

4. LDA
Toy example
IVs
DVs
= X

5. LDA
First step: Significance testing of the overall classifier in order to know if a set of discriminant functions can significantly predict group membership or not
Second step: Significance testing for each discriminant function.
Third step: Computation of the (standardized, unstandardized) discriminant functions

6. LDA - Overall Testing
Sum of Square and Cross Product
SSCP=

7. LDA - Overall Testing Canonical Correlation Matrix
Error and hypothesis matrices

8. LDA - Overall Testing
Computing W (WLR)
where s = min(df, q), lk is kth eigenvalue extracted from HiE-1 and |E| (as well as |E+Hi|) is the determinant.
The overall test is significant

9. LDA - Individual Testing
Eigenvalues and eigenvectors decomposition of the matrix: E-1H
E-1H=
PCA
E-1H

10. LDA - Individual Testing
Canonical Discriminant Analysis
Squared canonical correlation (Can also obtained from the eigenvalues of the correlation matrix R)
Canonical correlation

11. LDA - Individual Testing
Significance test for the canonical correlations
A significant output indicates that there is a variance share between IV and DV sets
Procedure:
We test for all the variables (m=1,…,min(p,q))
If significant, we removed the first variable (canonical correlate) and test for the remaining ones (m=2,…, min(p,q)
Repeat

12. LDA - Individual Testing
Significance test for the canonical correlations
Since all canonical variables are significant, we will keep them all.

13. LDA – Projection of the solution
Second group
First group
P=VY
Third group
Second discriminant function
First discriminant function

14. LDA – Discriminant Functionsb0 b1 b2 b3 b4
Class membership is given by: Max(D1, D2, D3)
Example x=(86, 6, 35, 6.5);
D1= (MAX)
D2=
D3=