1 UNC, Stat & OR Hailuoto Workshop Object Oriented Data Analysis, I J. S. Marron Dept. of Statistics and Operations Research, University of North Carolina.

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1 UNC, Stat & OR Hailuoto Workshop Object Oriented Data Analysis, I J. S. Marron Dept. of Statistics and Operations Research, University of North Carolina March 6, 2016

2 UNC, Stat & OR Object Oriented Data Analysis, I What is the “atom” of a statistical analysis? 1 st Course: Numbers Multivariate Analysis Course : Vectors Functional Data Analysis: Curves More generally: Data Objects

3 UNC, Stat & OR Functional Data Analysis, I Curves as Data Objects Important Duality: Curve Space  Point Cloud Space Illustrate with Travis Gaydos Graphics 2 dim’al curves (easy to visualize)

4 UNC, Stat & OR Functional Data Analysis, Toy EG I

5 UNC, Stat & OR Functional Data Analysis, Toy EG II

6 UNC, Stat & OR Functional Data Analysis, Toy EG III

7 UNC, Stat & OR Functional Data Analysis, Toy EG IV

8 UNC, Stat & OR Functional Data Analysis, Toy EG V

9 UNC, Stat & OR Functional Data Analysis, Toy EG VI

10 UNC, Stat & OR Functional Data Analysis, Toy EG VII

11 UNC, Stat & OR Functional Data Analysis, Toy EG VIII

12 UNC, Stat & OR Functional Data Analysis, Toy EG VIII

13 UNC, Stat & OR Functional Data Analysis, Toy EG IX

14 UNC, Stat & OR Functional Data Analysis, Toy EG X

15 UNC, Stat & OR Functional Data Analysis, 10-d Toy EG 1

16 UNC, Stat & OR Functional Data Analysis, 10-d Toy EG 1

17 UNC, Stat & OR Functional Data Analysis, 10-d Toy EG 2

18 UNC, Stat & OR Functional Data Analysis, 10-d Toy EG 2

19 UNC, Stat & OR Object Oriented Data Analysis, I What is the “atom” of a statistical analysis? 1 st Course: Numbers Multivariate Analysis Course : Vectors Functional Data Analysis: Curves More generally: Data Objects

20 UNC, Stat & OR Object Oriented Data Analysis, II Examples: Medical Image Analysis Images as Data Objects? Shape Representations as Objects Micro-arrays Just multivariate analysis?

21 UNC, Stat & OR Object Oriented Data Analysis, III Typical Goals: Understanding population variation Principal Component Analysis + Discrimination (a.k.a. Classification) Time Series of Data Objects

22 UNC, Stat & OR Object Oriented Data Analysis, IV Major Statistical Challenge, I: High Dimension Low Sample Size (HDLSS) Dimension d >> sample size n “Multivariate Analysis” nearly useless Can’t “normalize the data” Land of Opportunity for Statisticians Need for “creative statisticians”

23 UNC, Stat & OR Object Oriented Data Analysis, V Major Statistical Challenge, II: Data may live in non-Euclidean space Lie Group / Symmetric Spaces Trees/Graphs as data objects Interesting Issues: What is “the mean” (pop’n center)? How do we quantify “pop’n variation”?

24 UNC, Stat & OR Statistics in Image Analysis, I First Generation Problems: Denoising Segmentation Registration (all about single images)

25 UNC, Stat & OR Statistics in Image Analysis, II Second Generation Problems: Populations of Images Understanding Population Variation Discrimination (a.k.a. Classification) Complex Data Structures (& Spaces) HDLSS Statistics

26 UNC, Stat & OR HDLSS Statistics in Imaging Why HDLSS (High Dim, Low Sample Size)? Complex 3-d Objects Hard to Represent Often need d = 100’s of parameters Complex 3-d Objects Costly to Segment Often have n = 10’s cases

27 UNC, Stat & OR Object Representation Landmarks (hard to find) Boundary Rep’ns (no correspondence) Medial representations Find “skeleton” Discretize as “atoms” called M-reps

28 UNC, Stat & OR 3-d m-reps Bladder – Prostate – Rectum (multiple objects, J. Y. Jeong) Medial Atoms provide “skeleton” Implied Boundary from “spokes”  “surface”

29 UNC, Stat & OR Personal HDLSS Viewpoint: Data Images (cases) are “Points” In Feature Space Features are Axes Data set is “Point Clouds” Use Proj’ns to visualize

30 UNC, Stat & OR Personal HDLSS Viewpoint: PCA Rotated Axes Often Insightful One set of Dir’ns Others Useful, too

31 UNC, Stat & OR Cornea Data, I Images as data ~42 Cornea Images Outer surface of eye Heat map of curvature (in radial direction) Hard to understand “population structure”

32 UNC, Stat & OR Cornea Data, II PC 1 Starts at Pop’n Mean Overall Curvature Vertical Astigmatism Correlated! Gaussian Projections Visualization: Can’t Overlay (so use movie)

33 UNC, Stat & OR Cornea Data, III PC 2 Horrible Outlier! (present in data) But look only in center: Steep at top -- bottom Want Robust PCA For HDLSS data ???

34 UNC, Stat & OR Cornea Data, IV Robust PC 2 No outlier impact See top – bottom variation Projections now Gaussian

35 UNC, Stat & OR PCA for m-reps, I Major issue: m-reps live in (locations, radius and angles) E.g. “average” of: = ??? Natural Data Structure is: Lie Groups ~ Symmetric spaces (smooth, curved manifolds)

36 UNC, Stat & OR PCA for m-reps, II PCA on non-Euclidean spaces? (i.e. on Lie Groups / Symmetric Spaces) T. Fletcher: Principal Geodesic Analysis Idea: replace “linear summary of data” With “geodesic summary of data”…

37 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)

38 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)

39 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)