1. Principal Components Analysis

2. Principal Components - Description
Data: p Variables observed on n Experimental/Sampling units
Goal: Describe the Variance-Covariance structure of the variables through uncorrelated linear combinations of the variables
Data Reduction: Fewer linear combinations (k) that “explain” large portion of total variation than the number of variables (p)
Interpretation: New variables (principal components) can simplify the summary of the original variables.
Used as input to other procedures such as Factor Analysis

3. Population Principal Components - I

4. Population Principal Components - II

5. Population Principal Components - III

6. Population Principal Components - IV

7. Population Principal Components - V

8. Principal Components for Standardized Variables

9. Special Covariance Matrices

10. Principal Components with Sample Data

11. Choosing the Number of Principal Components to Retain
Goal: Data Reduction  Keeping k < p Components
PCA Based on Correlation Matrix  Keep Components with Eigenvalues exceeding 1 (the average of the Eigenvalues)
Scree Plot – Plot the Eigenvalues versus their component number and visually determine where curve flattens out
Percentage of Variation Explained – If a given percentage is to be explained (say 75%), keep components that cover the threshold

12. Standardizing Sample Principal Components - I

13. Standardizing Sample Principal Components - II

14. Graphical Methods
Scatterplots of Units by their first few principal components allows to see which units are similar and different with respect to their primary component scores . Also can be used to assess normality
Q-Q Plots to assess normality
Biplot – Plot representing where experimental units and variables lie with respect to 2 principal components

15. Large-Sample Inferences

16. Lawley’s Test for Equal Correlation Structure