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SAMSI AOOD Opening Workshop

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  1. SAMSI AOOD Opening Workshop Tutorial OODA of Tree Structured Objects J. S. Marron Dept. of Statistics and O. R., UNC September 17, 2014

  2. Workshop Big Picture An investment by: • Provided Funding to Bring Us Together • Has Specific Goal: Generating Collaborative Research

  3. Workshop Big Picture An investment by: Workshop Aim: Kickoff Ongoing Research (through whole program year)

  4. Workshop Big Picture Thus different format: • Fewer Main Talks • Main Talks Aimed at Collaborations • “2-Minute Madness” Talks – Introductory • Wed. Afternoon: Form “Working Groups”

  5. Working Groups Usual Structure • Conceived of at Opening Workshop • Agreed upon on Wednesday Afternoon • First Meeting: Thursday or Friday • Followed by weekly meetings • Can Skype or WebEx in remotely

  6. Working Groups Goals: • Collaborative Research • Among unexpected partners Our hope: • This group unusually well suited for this

  7. Working Groups Program “Areas of Emphasis”: • Functional Data Analysis • Time Dynamics • Image Analysis • Trees as Data • Shape and Manifold Data Where are potential (new) connections?

  8. Working Groups Program “Areas of Emphasis”: • Functional Data Analysis • Time Dynamics • Image Analysis • Trees as Data • Shape and Manifold Data fMRI Where are potential (new) connections?

  9. Working Groups Program “Areas of Emphasis”: • Functional Data Analysis • Time Dynamics • Image Analysis • Trees as Data DTI • Shape and Manifold Data Where are potential (new) connections?

  10. Working Groups Program “Areas of Emphasis”: • Functional Data Analysis • Time Dynamics • Image Analysis Brain Development • Trees as Data • Shape and Manifold Data Where are potential (new) connections?

  11. Working Groups Program “Areas of Emphasis”: • Functional Data Analysis • Time Dynamics “Atlas” of Human Body • Image Analysis • Trees as Data • Shape and Manifold Data Where are potential (new) connections?

  12. Working Groups Where are potential (new) connections? Requests of you: • Look for more of these • Discuss with others • Bring up on Wednesday Afternoon • Join in on Thursday +

  13. Object Oriented Data Analysis What is the “atom” of a statistical analysis? • First Course: Numbers • Multivariate Analysis: Vectors • Functional Data Analysis: Curves • OODA: More Complicated Objects • Images • Movies • Shapes • Tree Structured Objects

  14. An Aside on Acronyms What is it? OODA or AOOD ???

  15. SAMSI AOOD Opening Workshop Tutorial OODA of Tree Structured Objects J. S. Marron Dept. of Statistics and O. R., UNC September 17, 2014

  16. Acronym History Original SAMSI Proposal: Object Oriented Data Analysis (OODA)

  17. Acronym History Original SAMSI Proposal: Object Oriented Data Analysis (OODA) SAMSI Directors’ Suggestion: Analysis of Object Oriented Data (AOOD)

  18. Acronym History Original SAMSI Proposal: Object Oriented Data Analysis (OODA) SAMSI Directors’ Suggestion: Analysis of Object Oriented Data (AOOD) NISS Board Suggestion: Analysis Of Object Data (AOOD)

  19. An Aside on Acronyms What is it? OODA or AOOD Suggestion: Treat these as synonyms

  20. Object Oriented Data Analysis What is the “atom” of a statistical analysis? • First Course: Numbers • Multivariate Analysis: Vectors • Functional Data Analysis: Curves • OODA: More Complicated Objects • Images • Movies • Shapes • Tree Structured Objects

  21. Euclidean Data Spaces Data are vectors, in Effective (and Traditional) Analysis: • Linear Methods • Mean • Covariance • Principal Component Analysis • Gaussian Distribution

  22. Euclidean Data Spaces Data are vectors, in Challenges: • High Dimension, Low Sample Size (Classical Methods Fail) • Visualization: • Find Structure (Expected & Unknown) • Understand range of “normal cases” • Find anomalies

  23. Non - Euclidean Data Spaces “Simple” Example: m-reps for shapes • Data involve angles • Thus lie in “manifold” • i.e. “curved feature space” • Typical Approach: Tangent Plane Approx. • e.g. PGA • Personal Terminology: “Mildly non-Euclidean”

  24. 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)

  25. 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)

  26. 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)

  27. Non - Euclidean Data Spaces What is “Strongly Non-Euclidean” Case? Trees as Data Special Challenge: • No Tangent Plane • Must Re-Invent Data Analysis

  28. 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

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

  30. Blood vessel tree data • Marron’s brain: • MRI view • Single Slice • From 3-d Image

  31. Blood vessel tree data • Marron’s brain: • MRA view • “A” for “Angiography” • Finds blood vessels • (show up as white) • Track through 3d

  32. Blood vessel tree data • Marron’s brain: • MRA view • “A” for “Angiography” • Finds blood vessels • (show up as white) • Track through 3d

  33. Blood vessel tree data • Marron’s brain: • MRA view • “A” for “Angiography” • Finds blood vessels • (show up as white) • Track through 3d

  34. Blood vessel tree data • Marron’s brain: • MRA view • “A” for “Angiography” • Finds blood vessels • (show up as white) • Track through 3d

  35. Blood vessel tree data • Marron’s brain: • MRA view • “A” for “Angiography” • Finds blood vessels • (show up as white) • Track through 3d

  36. Blood vessel tree data • Marron’s brain: • MRA view • “A” for “Angiography” • Finds blood vessels • (show up as white) • Track through 3d

  37. Blood vessel tree data • Marron’s brain: • From MRA • Segment tree • of vessel segments • Using tube tracking • Bullitt and Aylward (2002)

  38. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

  39. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

  40. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

  41. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

  42. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

  43. Blood vessel tree data • Marron’s brain: • From MRA • Reconstruct trees • in 3d • Rotate to view

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

  45. 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

  46. Blood vessel tree data Big Picture: 3 Approaches Purely Combinatorial Folded Euclidean Dyck Path

  47. Blood vessel tree data Big Picture: 3 Approaches Purely Combinatorial Folded Euclidean Dyck Path

  48. Blood vessel tree data , , ... , • Possible focus of analysis: • Connectivity structure only (topology) • Location, size, orientation of segments • Structure within each vessel segment

  49. Blood vessel tree data • Present Focus: • Topology only • Already challenging • Later address additional challenges • By adding attributes • (locations, thicknesses, curvature, …) • To tree nodes • And extend analysis

  50. Blood vessel tree data • Topological Representation: • Each Vessel Segment (up to 1st Split) • is a node • Split Segments are child nodes • Connecting lines show relationship