1. 1 UNC, Stat & OR Nonnegative Matrix Factorization

2. 2 UNC, Stat & OR Standard NMF But Note Not Nested No “Multi-scale” Analysis Possible (Scores Plot?!?) Nonnegative Matrix Factorization

3. 3 UNC, Stat & OR Same Toy Data Set Rank 1 Approx. Properly Nested Nonnegative Nested Cone Analysis

4. 4 UNC, Stat & OR 5-d Toy Example Rank 3 NNCA Approx. Current Research: How Many Nonneg. Basis El’ts Needed? Nonnegative Nested Cone Analysis

5. 5 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Potential Application: Principal Curves Hastie & Stuetzle, (1989) (Foundation of Manifold Learning) An Interesting Question

6. 6 UNC, Stat & OR 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 An Interesting Question

7. 7 UNC, Stat & OR New Topic Curve Registration

8. 8 UNC, Stat & OR Context Functional Data Analysis Curves as Data Objects Toy Example: How Can We Understand Variation?

9. 9 UNC, Stat & OR Context Functional Data Analysis Curves as Data Objects Toy Example: How Can We Understand Variation?

10. 10 UNC, Stat & OR Functional Data Analysis Insightful Decomposition Vertical Variation Horiz’l Var’n

11. 11 UNC, Stat & OR Challenge  Fairly Large Literature  Many (Diverse) Past Attempts  Limited Success (in General)  Surprisingly Slippery (even mathematical formulation)

12. 12 UNC, Stat & OR Challenge (Illustrated) Thanks to Wei Wu

13. 13 UNC, Stat & OR Challenge (Illustrated) Thanks to Wei Wu

14. 14 UNC, Stat & OR Functional Data Analysis Appropriate Mathematical Framework? Vertical Variation Horiz’l Var’n

15. 15 UNC, Stat & OR 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

16. 16 UNC, Stat & OR Curve Registration

17. 17 UNC, Stat & OR Time Warping Intuition

18. 18 UNC, Stat & OR Curve Registration

19. 19 UNC, Stat & OR Data Objects I

20. 20 UNC, Stat & OR Data Objects I Equivalence Classes of Curves (Set of All Warps of Given Curve) Next Task: Find Metric on Equivalence Classes

21. 21 UNC, Stat & OR Metrics in Curve Space Find Metric on Equivalence Classes Start with Warp Invariance on Curves & Extend

22. 22 UNC, Stat & OR Metrics in Curve Space

23. 23 UNC, Stat & OR Metrics in Curve Space

24. 24 UNC, Stat & OR Metrics in Curve Space

25. 25 UNC, Stat & OR Metrics in Curve Space

26. 26 UNC, Stat & OR Metrics in Curve Space

27. 27 UNC, Stat & OR Metrics in Curve Space

28. 28 UNC, Stat & OR Metrics in Curve Space

29. 29 UNC, Stat & OR Metrics in Curve Space

30. 30 UNC, Stat & OR Metrics in Curve Space

31. 31 UNC, Stat & OR Metrics in Curve Space

32. 32 UNC, Stat & OR Metrics in Curve Space Why square roots?

33. 33 UNC, Stat & OR Metrics in Curve Space Why square roots? Thanks to Xiaosun Lu

34. 34 UNC, Stat & OR Metrics in Curve Space Why square roots?

35. 35 UNC, Stat & OR Metrics in Curve Space Why square roots?

36. 36 UNC, Stat & OR Metrics in Curve Space Why square roots?

37. 37 UNC, Stat & OR Metrics in Curve Space Why square roots?

38. 38 UNC, Stat & OR Metrics in Curve Space Why square roots?

39. 39 UNC, Stat & OR Metrics in Curve Space Why square roots?

40. 40 UNC, Stat & OR Metrics in Curve Space Why square roots?

41. 41 UNC, Stat & OR Metrics in Curve Space Why square roots?

42. 42 UNC, Stat & OR Metrics in Curve Space Why square roots? Dislikes Pinching Focusses Well On Peaks of Unequal Height

43. 43 UNC, Stat & OR Metrics in Curve Space

44. 44 UNC, Stat & OR Metrics in Curve Quotient Space Above was Invariance for Individual Curves Now extend to:  Equivalence Classes of Curves  I.e. Orbits as Data Objects  I.e. Quotient Space

45. 45 UNC, Stat & OR Metrics in Curve Quotient Space

46. 46 UNC, Stat & OR Mean in Curve Quotient Space Benefit of a Metric: Allows Definition of a “Mean” Fréchet Mean Geodesic Mean Barycenter Karcher Mean

47. 47 UNC, Stat & OR Mean in Curve Quotient Space

48. 48 UNC, Stat & OR Mean in Curve Quotient Space

49. 49 UNC, Stat & OR Mean in Curve Quotient Space

50. 50 UNC, Stat & OR Mean in Curve Quotient Space

51. 51 UNC, Stat & OR Mean in Curve Quotient Space Thanks to Anuj Srivastava

52. 52 UNC, Stat & OR Toy Example – (Details Later) Estimated Warps (Note: Represented With Karcher Mean At Identity)

53. 53 UNC, Stat & OR Mean in Curve Quotient Space

54. 54 UNC, Stat & OR More Data Objects

55. 55 UNC, Stat & OR More Data Objects Data Objects II

56. 56 UNC, Stat & OR More Data Objects Data Objects II ~ Kendall’s Shapes

57. 57 UNC, Stat & OR More Data Objects Data Objects III

58. 58 UNC, Stat & OR More Data Objects Data Objects III ~ Chang’s Transfo’s

59. 59 UNC, Stat & OR Computation Several Variations of Dynamic Programming Done by Eric Klassen, Wei Wu

60. 60 UNC, Stat & OR Toy Example Raw Data

61. 61 UNC, Stat & OR Toy Example Raw Data Both Horizontal And Vertical Variation

62. 62 UNC, Stat & OR Toy Example Conventional PCA Projections

63. 63 UNC, Stat & OR Toy Example Conventional PCA Projections Power Spread Across Spectrum

64. 64 UNC, Stat & OR Toy Example Conventional PCA Projections Power Spread Across Spectrum

65. 65 UNC, Stat & OR Toy Example Conventional PCA Scores

66. 66 UNC, Stat & OR Toy Example Conventional PCA Scores Views of 1-d Curve Bending Through 4 Dim’ns’

67. 67 UNC, Stat & OR Toy Example Conventional PCA Scores Patterns Are “Harmonics” In Scores

68. 68 UNC, Stat & OR Toy Example Scores Plot Shows Data Are “1” Dimensional So Need Improved PCA Decomp.

69. 69 UNC, Stat & OR Visualization

70. 70 Toy Example Aligned Curves (Clear 1-d Vertical Var’n)

71. 71 UNC, Stat & OR Toy Example Aligned Curve PCA Projections All Var’n In 1 st Component

72. 72 UNC, Stat & OR Visualization

73. 73 Toy Example Estimated Warps

74. 74 UNC, Stat & OR Toy Example Warps, PC Projections

75. 75 UNC, Stat & OR Toy Example Warps, PC Projections Mostly 1 st PC

76. 76 UNC, Stat & OR Toy Example Warps, PC Projections Mostly 1 st PC, But 2 nd Helps Some

77. 77 UNC, Stat & OR Toy Example Warps, PC Projections Rest is Not Important

78. 78 UNC, Stat & OR Toy Example Horizontal Var’n Visualization Challenge: (Complicated) Warps Hard to Interpret Approach: Apply Warps to Template Mean (PCA components)

79. 79 UNC, Stat & OR Toy Example Warp Compon’ts (+ Mean) Applied to Template Mean

80. 80 UNC, Stat & OR Refined Calculations

81. 81 UNC, Stat & OR PNS on SRVF Sphere Toy Example Tangent Space PCA (on Horiz. Var’n) Thanks to Xiaosun Lu

82. 82 UNC, Stat & OR PNS on SRVF Sphere Toy Example PNS Projections (Fewer Modes)

83. 83 UNC, Stat & OR PNS on SRVF Sphere Toy Example Tangent Space PCA Note: 3 Comp’s Needed for This

84. 84 UNC, Stat & OR PNS on SRVF Sphere Toy Example PNS Projections Only 2 for This

85. 85 UNC, Stat & OR TIC testbed Serious Data Challenge: TIC (Total Ion Count) Chromatograms Modern type of “chemical spectra” Thanks to Peter Hoffmann

86. 86 UNC, Stat & OR TIC testbed Raw Data: 15 TIC Curves (5 Colors)

87. 87 UNC, Stat & OR TIC testbed Special Feature: Answer Key of Known Peaks Found by Major Time & Labor Investment

88. 88 UNC, Stat & OR TIC testbed Special Feature: Answer Key of Known Peaks Goal: Find Warps To Align These

89. 89 UNC, Stat & OR TIC testbed Fisher – Rao Alignment

90. 90 UNC, Stat & OR TIC testbed Fisher – Rao Alignment Spike-In Peaks

91. 91 UNC, Stat & OR TIC testbed Next Zoom in on This Region

92. 92 UNC, Stat & OR TIC testbed Zoomed Fisher – Rao Alignment

93. 93 UNC, Stat & OR TIC testbed Before Alignment

94. 94 UNC, Stat & OR TIC testbed Next Zoom in on This Region

95. 95 UNC, Stat & OR TIC testbed Zoomed Fisher – Rao Alignment

96. 96 UNC, Stat & OR TIC testbed Before Alignment

97. 97 UNC, Stat & OR TIC testbed Next Zoom in on This Region

98. 98 UNC, Stat & OR TIC testbed Zoomed Fisher – Rao Alignment

99. 99 UNC, Stat & OR TIC testbed Before Fisher-Rao Alignment

100. 100 UNC, Stat & OR TIC testbed Next Zoom in on This Region

101. 101 UNC, Stat & OR TIC testbed Zoomed Fisher – Rao Alignment

102. 102 UNC, Stat & OR TIC testbed Zoomed Fisher – Rao Alignment Note: Very Challenging

103. 103 UNC, Stat & OR TIC testbed Before Alignment

104. 104 UNC, Stat & OR TIC testbed Next Zoom in on This Region

105. 105 UNC, Stat & OR TIC testbed Zoomed Fisher – Rao Alignment

106. 106 UNC, Stat & OR TIC testbed Before Alignment

107. 107 UNC, Stat & OR TIC testbed Warping Functions

108. 108 UNC, Stat & OR References for Much More Big Picture Survey: Marron, Ramsay, Sangalli & Srivastava (2014) TIC Proteomics Example: Koch, Hoffman & Marron (2014)

109. 109 UNC, Stat & OR Take Away Message Curve Registration is Slippery Thus, Careful Mathematics is Useful Fisher-Rao Approach: Gets the Math Right Intuitively Sensible Computable Generalizable Worth the Complication