1. Principal Component Analysis

2. Consider a collection of points

3. Suppose you want to fit a line

4. Consider variance of distribution on the line Project onto the Line

5. different variance Different line...

6. Maximum Variance

7. Minimum Variance

8. Given by eigenvectors of covariance matrix of coordinates of original points

9. PCA notes… Input data set Subtract the mean to get data set with 0- mean Compute the covariance matrix Compute the eigenvalues and eigenvectors of the covariance matrix Choose components and form a feature vector. Order by eigenvalues – highest to lowest

10. PCA To compress, ignore components of lesser significance The feature vector F is a matrix is the matrix of ordered eigenvectors Derive the data set in the new coordinates: new_data = F T old_data

11. Covariance C, of 2 random variables X and Y where

12. Example

16. Choose bounding box oriented this way OOBB

17. OOBB: Fitting Covariance matrix of point coordinates describes statistical spread of cloud. OBB is aligned with directions of greatest and least spread (which are guaranteed to be orthogonal).

18. Good Box OOBB

19. Add points: worse Box OOBB

20. More points: terrible box OOBB