1. Statistics – O. R. 891 Object Oriented Data Analysis J. S. Marron Dept. of Statistics and Operations Research University of North Carolina

2. Manifold Feature Spaces x x x x

3. Fréchet Mean in General: Big Advantage: Works in Any Metric Space Manifold Feature Spaces

5. Directional Data Examples of Fréchet Mean: Also of interest is Fréchet Variance: Works like Euclidean sample variance Note values in movie, reflecting spread in data Note theoretical version: Useful for Laws of Large Numbers, etc. Manifold Feature Spaces

6. OODA in Image Analysis First Generation Problems: Denoising Segmentation Registration (all about single images, still interesting challenges)

7. OODA in Image Analysis Second Generation Problems: Populations of Images –Understanding Population Variation –Discrimination (a.k.a. Classification) Complex Data Structures (& Spaces) HDLSS Statistics

8. Image Object Representation Major Approaches for Image Data Objects: Landmark Representations Boundary Representations Medial Representations

9. Boundary Representations Main Drawback: Correspondence For OODA (on vectors of parameters): Need to “match up points” Easy to find triangular mesh –Lots of research on this driven by gamers Challenge to match mesh across objects –There are some interesting ideas…

10. Medial Representations Main Idea: Represent Objects as: Discretized skeletons (medial atoms) Plus spokes from center to edge Which imply a boundary Very accessible early reference: Yushkevich, et al (2001)

11. A Challenging Example Male Pelvis –Bladder – Prostate – Rectum –How do they move over time (days)? –Critical to Radiation Treatment (cancer) Work with 3-d CT –Very Challenging to Segment –Find boundary of each object? –Represent each Object?

12. Male Pelvis – Raw Data One CT Slice (in 3d image) Tail Bone Rectum Bladder Prostate

13. 13 UNC, Stat & OR PGA for m-reps, Bladder-Prostate-Rectum Bladder – Prostate – Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong)

14. 14 UNC, Stat & OR PGA for m-reps, Bladder-Prostate-Rectum Bladder – Prostate – Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong)

15. 15 UNC, Stat & OR PGA for m-reps, Bladder-Prostate-Rectum Bladder – Prostate – Rectum, 1 person, 17 days PG 1 PG 2 PG 3 (analysis by Ja Yeon Jeong)

16. 16 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean

17. 17 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Counterexample: Data on sphere, along equator

18. 18 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data

19. 19 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data Counterexample: Data follows Tropic of Capricorn

20. 20 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data Jung, Foskey & Marron (Princ. Arc Anal.)

21. 21 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic mean Huckemann, Hotz & Munk (Geod. PCA) Best fit of any geodesic to data Jung, Foskey & Marron (Princ. Arc Anal.) Best fit of any circle to data (motivated by conformal maps)

22. 22 UNC, Stat & OR PCA Extensions for Data on Manifolds

23. 23 UNC, Stat & OR Principal Arc Analysis Jung, Foskey & Marron (2011) Best fit of any circle to data Can give better fit than geodesics

24. 24 UNC, Stat & OR Principal Arc Analysis Jung, Foskey & Marron (2011) Best fit of any circle to data Can give better fit than geodesics Observed for simulated m-rep example

25. 25 UNC, Stat & OR Challenge being addressed

26. 26 UNC, Stat & OR Composite Nested Spheres

27. 27 UNC, Stat & OR Landmark Based Shape Analysis Kendall Bookstein Dryden & Mardia (recall major monographs)

28. 28 UNC, Stat & OR Landmark Based Shape Analysis Kendall Bookstein Dryden & Mardia Digit 3 Data

29. 29 UNC, Stat & OR Landmark Based Shape Analysis Kendall Bookstein Dryden & Mardia Digit 3 Data

30. 30 UNC, Stat & OR Landmark Based Shape Analysis Kendall Bookstein Dryden & Mardia Digit 3 Data (digitized to 13 landmarks)

31. 31 UNC, Stat & OR Landmark Based Shape Analysis Key Step: mod out Translation Scaling Rotation

32. 32 UNC, Stat & OR Landmark Based Shape Analysis Key Step: mod out Translation Scaling Rotation Result: Data Objects   points on Manifold ( ~ S 2k-4 )

33. 33 UNC, Stat & OR Landmark Based Shape Analysis Currently popular approaches to PCA on S k :  Early: PCA on projections (Tangent Plane Analysis)

34. 34 UNC, Stat & OR Landmark Based Shape Analysis Currently popular approaches to PCA on S k :  Early: PCA on projections  Fletcher: Geodesics through mean

35. 35 UNC, Stat & OR Landmark Based Shape Analysis Currently popular approaches to PCA on S k :  Early: PCA on projections  Fletcher: Geodesics through mean  Huckemann, et al: Any Geodesic

36. 36 UNC, Stat & OR Landmark Based Shape Analysis Currently popular approaches to PCA on S k :  Early: PCA on projections  Fletcher: Geodesics through mean  Huckemann, et al: Any Geodesic New Approach: Principal Nested Sphere Analysis Jung, Dryden & Marron (2012)

37. 37 UNC, Stat & OR Principal Nested Spheres Analysis Main Goal: Extend Principal Arc Analysis (S 2 to S k )

38. 38 UNC, Stat & OR Principal Nested Spheres Analysis Key Idea: Replace usual forwards view of PCA With a backwards approach to PCA

39. 39 UNC, Stat & OR Terminology Multiple linear regression:

40. 40 UNC, Stat & OR Terminology Multiple linear regression: Stepwise approaches:  Forwards: Start small, iteratively add variables to model

41. 41 UNC, Stat & OR Terminology Multiple linear regression: Stepwise approaches:  Forwards: Start small, iteratively add variables to model  Backwards: Start with all, iteratively remove variables from model

42. 42 UNC, Stat & OR Illust’n of PCA View: Recall Raw Data

43. 43 UNC, Stat & OR Illust’n of PCA View: PC1 Projections

44. 44 UNC, Stat & OR Illust’n of PCA View: PC2 Projections

45. 45 UNC, Stat & OR Illust’n of PCA View: Projections on PC1,2 plane

46. 46 UNC, Stat & OR Principal Nested Spheres Analysis Replace usual forwards view of PCA Data  PC1 (1-d approx)

47. 47 UNC, Stat & OR Principal Nested Spheres Analysis Replace usual forwards view of PCA Data  PC1 (1-d approx)  PC2 (1-d approx of Data-PC1)  PC1 U PC2 (2-d approx)

48. 48 UNC, Stat & OR Principal Nested Spheres Analysis Replace usual forwards view of PCA Data  PC1 (1-d approx)  PC2 (1-d approx of Data-PC1)  PC1 U PC2 (2-d approx)  PC1 U … U PCr (r-d approx)

49. 49 UNC, Stat & OR Principal Nested Spheres Analysis With a backwards approach to PCA Data  PC1 U … U PCr (r-d approx)

50. 50 UNC, Stat & OR Principal Nested Spheres Analysis With a backwards approach to PCA Data  PC1 U … U PCr (r-d approx)  PC1 U … U PC(r-1)

51. 51 UNC, Stat & OR Principal Nested Spheres Analysis With a backwards approach to PCA Data  PC1 U … U PCr (r-d approx)  PC1 U … U PC(r-1)  PC1 U PC2 (2-d approx)

52. 52 UNC, Stat & OR Principal Nested Spheres Analysis With a backwards approach to PCA Data  PC1 U … U PCr (r-d approx)  PC1 U … U PC(r-1)  PC1 U PC2 (2-d approx)  PC1 (1-d approx)

53. 53 UNC, Stat & OR Principal Nested Spheres Analysis Top Down Nested (small) spheres

54. 54 UNC, Stat & OR Digit 3 data: Principal variations of shape Princ. geodesics by PNSPrincipal arcs by PNS

55. 55 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Discussion: Jung et al (2010) Pizer et al (2012) An Interesting Question

56. 56 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Anywhere this is already being done??? An Interesting Question

57. 57 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

58. 58 UNC, Stat & OR Goal: Find lower dimensional manifold that well approximates data  ISOmap Tennenbaum (2000)  Local Linear Embedding Roweis & Saul (2000) Manifold Learning

59. 59 UNC, Stat & OR 1 st Principal Curve Linear Reg’n Usual Smooth

60. 60 UNC, Stat & OR 1 st Principal Curve Linear Reg’n Proj’s Reg’n Usual Smooth

61. 61 UNC, Stat & OR 1 st Principal Curve Linear Reg’n Proj’s Reg’n Usual Smooth Princ’l Curve

62. 62 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Potential Application: Principal Curves Perceived Major Challenge: How to find 2 nd Principal Curve? Backwards approach??? An Interesting Question

63. 63 UNC, Stat & OR Key Component: Principal Surfaces LeBlanc & Tibshirani (1994) An Interesting Question

64. 64 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Another Potential Application: Nonnegative Matrix Factorization = PCA in Positive Orthant (early days) An Interesting Question

65. 65 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Nonnegative Matrix Factorization Discussed in Guest Lecture: Tuesday, November 13 Lingsong Zhang Thanks to stat.purdue.edu An Interesting Question

66. 66 UNC, Stat & OR How generally applicable is Backwards approach to PCA? Another Potential Application: Trees as Data (early days) An Interesting Question

67. 67 UNC, Stat & OR How generally applicable is Backwards approach to PCA? An Attractive Answer An Interesting Question

68. 68 UNC, Stat & OR How generally applicable is Backwards approach to PCA? An Attractive Answer: James Damon, UNC Mathematics Geometry Singularity Theory An Interesting Question

69. 69 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

70. 70 UNC, Stat & OR General View of Backwards PCA

71. 71 UNC, Stat & OR Define Nested Spaces via Constraints E.g. SVD (Singular Value Decomposition = = Not Mean Centered PCA) (notationally very clean) General View of Backwards PCA

72. 72 UNC, Stat & OR General View of Backwards PCA

73. 73 UNC, Stat & OR General View of Backwards PCA

74. 74 UNC, Stat & OR General View of Backwards PCA

75. 75 UNC, Stat & OR General View of Backwards PCA

76. 76 UNC, Stat & OR Define Nested Spaces via Constraints Backwards PCA Reduce Using Affine Constraints General View of Backwards PCA

77. 77 UNC, Stat & OR Define Nested Spaces via Constraints Backwards PCA Principal Nested Spheres Use Affine Constraints (Planar Slices) In Ambient Space General View of Backwards PCA

78. 78 UNC, Stat & OR Define Nested Spaces via Constraints Backwards PCA Principal Nested Spheres Principal Surfaces Spline Constraint Within Previous? General View of Backwards PCA

79. 79 UNC, Stat & OR Define Nested Spaces via Constraints Backwards PCA Principal Nested Spheres Principal Surfaces Spline Constraint Within Previous? {Been Done Already???} General View of Backwards PCA

80. 80 UNC, Stat & OR Define Nested Spaces via Constraints Backwards PCA Principal Nested Spheres Principal Surfaces Other Manifold Data Spaces Sub-Manifold Constraints?? (Algebraic Geometry) General View of Backwards PCA

81. 81 UNC, Stat & OR Define Nested Spaces via Constraints Backwards PCA Principal Nested Spheres Principal Surfaces Other Manifold Data Spaces Tree Spaces Suitable Constraints??? General View of Backwards PCA

82. 82 UNC, Stat & OR Why does Backwards Work Better? General View of Backwards PCA

83. 83 UNC, Stat & OR Why does Backwards Work Better?  Natural to Sequentially Add Constraints (I.e. Add Constraints, Using Information in Data) General View of Backwards PCA

84. 84 UNC, Stat & OR Why does Backwards Work Better?  Natural to Sequentially Add Constraints  Hard to Start With Complete Set, And Sequentially Remove General View of Backwards PCA

85. 85 UNC, Stat & OR Recall Main Idea:  Represent Shapes as Coordinates  “Mod Out” Transl’n, Rotat’n, Scale Variation on Landmark Based Shape

86. 86 UNC, Stat & OR Typical Viewpoint:  Variation in Shape is Goal  Other Variation+ is Nuisance Recall Main Idea:  Represent Shapes as Coordinates  “Mod Out” Transl’n, Rotat’n, Scale Variation on Landmark Based Shape

87. 87 UNC, Stat & OR Typical Viewpoint:  Variation in Shape is Goal  Other Variation+ is Nuisance Interesting Alternative:  Study Variation in Transformation  Treat Shape as Nuisance Variation on Landmark Based Shape

88. 88 UNC, Stat & OR Context: Study of Tectonic Plates Movement of Earth’s Crust (over time) Take Motions as Data Objects Interesting Alternative:  Study Variation in Transformation  Treat Shape as Nuisance Variation on Landmark Based Shape

89. 89 UNC, Stat & OR Context: Study of Tectonic Plates Movement of Earth’s Crust (over time) Take Motions as Data Objects Royer & Chang (1991) Thanks to Wikipedia Variation on Landmark Based Shape

90. 90 UNC, Stat & OR Non - Euclidean Data Spaces What is “Strongly Non-Euclidean” Case? Trees as Data

91. 91 UNC, Stat & OR Non - Euclidean Data Spaces What is “Strongly Non-Euclidean” Case? Trees as Data Special Challenge: No Tangent Plane Must Re-Invent Data Analysis

92. 92 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects Thanks to Burcu Aydin

93. 93 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects From Graph Theory: Graph is set of nodes and edges Thanks to Burcu Aydin

94. 94 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects From Graph Theory: Graph is set of nodes and edges Tree has root and direction Thanks to Burcu Aydin

95. 95 UNC, Stat & OR Strongly Non-Euclidean Spaces Trees as Data Objects From Graph Theory: Graph is set of nodes and edges Tree has root and direction Data Objects: set of trees Thanks to Burcu Aydin

96. 96 UNC, Stat & OR Strongly Non-Euclidean Spaces General Graph: Thanks to Sean Skwerer

97. 97 UNC, Stat & OR Strongly Non-Euclidean Spaces Special Case Called “Tree” Directed Acyclic 5 4 3 21 0 Graphical note: Sometimes “grow up” Others “grow down”

98. 98 UNC, Stat & OR Strongly Non-Euclidean Spaces Motivating Example: From Dr. Elizabeth BullittDr. Elizabeth Bullitt Dept. of Neurosurgery, UNC Blood Vessel Trees in Brains Segmented from MRAs Study population of trees Forest of Trees

99. 99 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRI view  Single Slice  From 3-d Image

100. 100 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

101. 101 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

102. 102 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

103. 103 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

104. 104 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

105. 105 UNC, Stat & OR Blood vessel tree data Marron’s brain:  MRA view  “A” for “Angiography”  Finds blood vessels (show up as white)  Track through 3d

106. 106 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Segment tree  of vessel segments  Using tube tracking  Bullitt and Aylward (2002)

107. 107 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

108. 108 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

109. 109 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

110. 110 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

111. 111 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

112. 112 UNC, Stat & OR Blood vessel tree data Marron’s brain:  From MRA  Reconstruct trees  in 3d  Rotate to view

113. 113 UNC, Stat & OR Blood vessel tree data Now look over many people (data objects) Structure of population (understand variation?) PCA in strongly non-Euclidean Space???,...,,

114. 114 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals,...,,

115. 115 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals (not accessible by traditional methods) Predict Stroke Tendency (Collateral Circulation),...,,

116. 116 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals (not accessible by traditional methods) Predict Stroke Tendency (Collateral Circulation) Screen for Loci of Pathology,...,,

117. 117 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals (not accessible by traditional methods) Predict Stroke Tendency (Collateral Circulation) Screen for Loci of Pathology Explore how age affects connectivity,...,,

118. 118 UNC, Stat & OR Blood vessel tree data Examples of Potential Specific Goals (not accessible by traditional methods) Predict Stroke Tendency (Collateral Circulation) Screen for Loci of Pathology Explore how age affects connectivity,...,,

119. 119 UNC, Stat & OR Blood vessel tree data Big Picture: 3 Approaches 1.Purely Combinatorial 2.Euclidean Orthant 3.Dyck Path