1. Principal Curve Lin Binbin Oct. 16th

2. Prom PCA to Principal curve 1904 Spearman PCA 1983 Trevor Hastie Principal curve “Principal curve and surface”

3. What is principal curve? Continuous Middle …

4. What is principal curve? (Hastie and Stuetzle) HS principal curve 1. f does not intersect itself, 2. f has finite length inside any bounded subset of and 3. f is self-consistent, i.e.,

5. Projecting a points to a curve

6. HS principal algorithm

7. HS Result

9. Summary Measure middle-self consistent One dimensional structure of the manifold Bias Existence Uniqueness Convergence High dimensional generalization

10. Problem of principal curve No exact object function Theoretical foundation: differential manifold and Riemannian geometry

11. Kegl principal curve Length constrain Existence

12. Kegl principal curve Data fittingSmoothness Average curvatureAverage distance

13. Projection distribution

14. Result

19. Failure case

20. Local and global Top-down strategies – Start with straight line Bottom-up strategies – Start with local neighborhood

21. Principal oriented points(POPS) POCP: principal curve of oriented points

22. POPS(Delicado) Each point xOrthogonal vectorDefine a function And a vector b, we have a hyperplane Minimizing the total variance Fixed points of is called principal oriented points. x: center b: direction

23. LPC algorithm Local principal curve

31. Summary Theory High dimension