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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 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? • (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.,
Summary • Measure middle-self consistent • One dimensional structure of the manifold • Bias • Existence • Uniqueness • Convergence • High dimensional generalization
Problem of principal curve • No exact object function • Theoretical foundation: differential manifoldand Riemannian geometry
Kegl principal curve • Length constrain • Existence
Kegl principal curve Data fitting Smoothness Average distance Average curvature
Local and global • Top-down strategies • Start with straight line • Bottom-up strategies • Start with local neighborhood
Principal oriented points(POPS) POCP: principal curve of oriented points
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
LPC algorithm • Local principal curve
Summary • Theory • High dimension