An Interesting Question

1. An Interesting Question How generally applicable is Backwards approach to PCA? An Attractive Answer: James Damon, UNC Mathematics Key Idea: Express Backwards PCA as Nested Series of Constraints

2. General View of Backwards PCA Define Nested Spaces via Constraints E.g. SVD Now Define: ConstraintGives Nested Reduction of Dim’n

3. Vectors of Angles Vectors of Angles as Data Objects Slice space with hyperplanes???? (ala Principal Nested Spheres)

4. Vectors of Angles E.g. , Data w/ “Single Mode of Var’n” Best Fitting Planar Slice gives Bimodal Dist’n Special Thanks to Eduardo García-Portugués

5. Torus Space Try To Fit A Geodesic Challenge: Can Get Arbitrarily Close

6. Torus Space Fit Nested Sub-Manifold

7. PNS Main Idea Data Objects: Where is a dimensional manifold Consider a nested series of sub-manifolds: where for and Goal: Fit all of simultaneously to

8. General Background Call each a stratum, so is a manifold stratification To be fit to New Approach: Simultaneously fit Nested Submanifold (NS)

9. Projection Notation For let denote the telescoping projection onto I.e. for Note: This projection is fundamental to Backwards PCA methods

10. PNS Components For a given , represent a point by its Nested Submanifold components: where for In the sense that “” means the shortest geodesic arc between &

11. Nested Submanifold Fits Simultaneous Fit Criteria? Based on Stratum-Wise Sums of Squares For define Uses “lengths” of NS Components:

12. NS Components in NS Candidate 2 (Shifted to Sample Mean) Note: Both & Decrease

13. NS Components in NS based On PC1 Note: Yet is Constant (Pythagorean Thm)

14. NS Components in NS based On PC2 Note: is Constant (Pythagorean Thm)

15. NS Components in NS Candidate 1

16. NS Components in NS Candidate 2

17. NS Components in NS based On PC1

18. NS Components in NS based On PC2

19. Nested Submanifold Fits Simultaneously fit Simultaneous Fit Criterion? Above Suggests Want: Works for Euclidean PCA (?)

20. Nested Submanifold Fits Simultaneous Fit Criterion? Above Suggests Want: Important Predecessor Pennec(2016) AUC Criterion:

21. Pennec’s Area Under the Curve Based on Scree Plot 2 Component Index

22. Pennec’s Area Under the Curve Based on Scree Plot Cumulative 2 Component Index

23. Pennec’s Area Under the Curve Based on Scree Plot Cumulative Area = Component Index

24. Torus Space Fit Nested Sub-Manifold Choice of & in: ???

25. Torus Space Tiled embedding is complicated (maybe OK for low rank approx.) Instead Consider Nested Sub-Torii Work in Progress with Garcia, Wood, Le Key Factor: Important Modes of Variation

26. OODA Big Picture New Topic: Curve Registration Main Reference: Srivastava et al (2011)

27. Collaborators • AnujSrivastava(Florida State U.) • Wei Wu (Florida State U.) • Derek Tucker (Florida State U.) • Xiaosun Lu (U. N. C.) • Inge Koch (U. Adelaide) • Peter Hoffmann (U. Adelaide) • J. O. Ramsay (McGill U.) • Laura Sangalli (Milano Polytech.)

28. Context Functional Data Analysis Curves as Data Objects Toy Example:

29. Context Functional Data Analysis Curves as Data Objects Toy Example: How Can We Understand Variation?

30. Context Functional Data Analysis Curves as Data Objects Toy Example: How Can We Understand Variation?

31. Context Functional Data Analysis Curves as Data Objects Toy Example: How Can We Understand Variation?

32. Functional Data Analysis Insightful Decomposition

33. Functional Data Analysis Insightful Decomposition • Horiz’l • Var’n

34. Functional Data Analysis Insightful Decomposition Vertical Variation • Horiz’l • Var’n

35. Challenge • Fairly Large Literature • Many (Diverse) Past Attempts • Limited Success (in General) • Surprisingly Slippery (even mathematical formulation)

36. Challenge (Illustrated) Thanks to Wei Wu

37. Challenge (Illustrated) Thanks to Wei Wu

38. Functional Data Analysis Appropriate Mathematical Framework? Vertical Variation • Horiz’l • Var’n

39. Landmark Based Shape Analysis Approach: Identify objects that are: • Translations • Rotations • Scalings of each other Mathematics: Equivalence Relation Results in: Equivalence Classes Which become the Data Objects

40. Landmark Based Shape Analysis Equivalence Classes become Data Objects a.k.a. “Orbits” Mathematics: Called “Quotient Space” , , , , , ,

41. Curve Registration What are the Data Objects? Vertical Variation • Horiz’l • Var’n

42. Curve Registration What are the Data Objects? Consider “Time Warpings” (smooth) More Precisely: Diffeomorphisms

43. Curve Registration Diffeomorphisms • is 1 to 1 • is onto (thus is invertible) • Differentiable • is Differentiable

44. Time Warping Intuition Elastically Stretch & Compress Axis

45. Time Warping Intuition Elastically Stretch & Compress Axis (identity)

46. Time Warping Intuition Elastically Stretch & Compress Axis

47. Time Warping Intuition Elastically Stretch & Compress Axis

48. Time Warping Intuition Elastically Stretch & Compress Axis

49. Curve Registration Say curves and are equivalent, When so that

50. Curve Registration Toy Example: Starting Curve,