Principal Curve Lin Binbin Oct. 16th. Prom PCA to Principal curve 1904 Spearman PCA 1983 Trevor Hastie Principal curve “Principal curve and surface”

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Presentation transcript:

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.,

Projecting a points to a curve

HS principal algorithm

HS Result

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 manifold and Riemannian geometry

Kegl principal curve Length constrain Existence

Kegl principal curve Data fittingSmoothness Average curvatureAverage distance

Projection distribution

Result

Failure case

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 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

LPC algorithm Local principal curve

Summary Theory High dimension