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Principal Curve

Principal Curve. Lin Binbin Oct. 16th. Prom PCA to Principal curve. 1904 Spearman PCA 1983 Trevor Hastie Principal curve “Principal curve and surface” . What is principal curve?. Continuous Middle …. What is principal curve?. ( H astie and S tuetzle) HS principal curve

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Principal Curve

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  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

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

  9. Problem of principal curve • No exact object function • Theoretical foundation: differential manifoldand Riemannian geometry

  10. Kegl principal curve • Length constrain • Existence

  11. Kegl principal curve Data fitting Smoothness Average distance Average curvature

  12. Projection distribution

  13. Result

  14. Failure case

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

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

  17. POPS(Delicado) Each point x Orthogonal vector Define 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

  18. LPC algorithm • Local principal curve

  19. Summary • Theory • High dimension

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