![Page 1: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/1.jpg)
1
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)
![Page 2: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/2.jpg)
2
UNC, Stat & OR
PCA Extensions for Data on Manifolds
![Page 3: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/3.jpg)
3
UNC, Stat & OR
Landmark Based Shape Analysis
Key Step: mod out
• Translation
• Scaling
• Rotation
Result:
Data Objects
points on Manifold ( ~ S2k-
4)
![Page 4: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/4.jpg)
4
UNC, Stat & OR
Principal Nested Spheres Analysis
Main Goal:
Extend Principal Arc Analysis (S2 to Sk)
Jung, Dryden & Marron (2012)
![Page 5: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/5.jpg)
5
UNC, Stat & OR
Principal Nested Spheres Analysis
Top Down Nested (small) spheres
![Page 6: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/6.jpg)
6
UNC, Stat & OR
Principal Nested Spheres Analysis
Main Goal:
Extend Principal Arc Analysis (S2 to Sk)
Jung, Dryden & Marron (2012)
Important Landmark: This Motivated
Backwards PCA
![Page 7: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/7.jpg)
7
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)
![Page 8: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/8.jpg)
8
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)
![Page 9: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/9.jpg)
9
UNC, Stat & OR
Principal Component Analysis
Euclidean Settings:
Forwards PCA = Backwards PCA
(Pythagorean Theorem,
ANOVA Decomposition)
So Not Interesting
But Very Different in Non-Euclidean Settings
(Backwards is Better !?!)
![Page 10: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/10.jpg)
10
UNC, Stat & OR
Principal Component Analysis
Important Property of PCA:
Nested Series of Approximations
(Often taken for granted)
(Desirable in Non-Euclidean Settings)
![Page 11: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/11.jpg)
11
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
Where is this already being done???
An Interesting Question
![Page 12: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/12.jpg)
12
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
An Application:
Nonnegative Matrix Factorization
= PCA in Positive Orthant
Think
With ≥ 0 Constraints (on both & )
An Interesting Question
![Page 13: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/13.jpg)
13
UNC, Stat & OR
Standard Approach:
Lee et al (1999):
Formulate & Solve Optimization
Major Challenge:
Not Nested, ()
Nonnegative Matrix Factorization
![Page 14: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/14.jpg)
14
UNC, Stat & OR
Standard NMF
(Projections
All Inside
Orthant)
Nonnegative Matrix Factorization
![Page 15: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/15.jpg)
15
UNC, Stat & OR
Standard NMF
But Note
Not Nested
No “Multi-scale”
Analysis
Possible (Scores Plot?!?)
Nonnegative Matrix Factorization
![Page 16: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/16.jpg)
16
UNC, Stat & OR
Improved Version:
Use Backwards PCA Idea
“Nonnegative Nested Cone
Analysis”
Collaborator:
Lingsong Zhang (Purdue)
Zhang, Marron, Lu (2013)
Nonnegative Matrix Factorization
![Page 17: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/17.jpg)
17
UNC, Stat & OR
Same Toy
Data Set
All
Projections
In Orthant
Nonnegative Nested Cone Analysis
![Page 18: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/18.jpg)
18
UNC, Stat & OR
Same Toy
Data Set
Rank 1
Approx.
Properly
Nested
Nonnegative Nested Cone Analysis
![Page 19: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/19.jpg)
19
UNC, Stat & OR
Chemical
Spectral
Data
Gives
Clearer
View
Nonnegative Nested Cone Analysis
![Page 20: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/20.jpg)
20
UNC, Stat & OR
Chemical
Spectral
Data
Rank 3
Approximation Highlights Lab Early Error
Nonnegative Nested Cone Analysis
![Page 21: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/21.jpg)
21
UNC, Stat & OR
5-d Toy
Example
(Rainbow
Colored
by Peak
Order)
Nonnegative Nested Cone Analysis
![Page 22: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/22.jpg)
22
UNC, Stat & OR
5-d Toy Example Rank 1 NNCA Approx.
Nonnegative Nested Cone Analysis
![Page 23: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/23.jpg)
23
UNC, Stat & OR
5-d Toy Example Rank 2 NNCA Approx.
Nonnegative Nested Cone Analysis
![Page 24: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/24.jpg)
24
UNC, Stat & OR
5-d Toy Example Rank 2 NNCA Approx.
Nonneg.
Basis
Elements
(Not Trivial)
Nonnegative Nested Cone Analysis
![Page 25: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/25.jpg)
25
UNC, Stat & OR
5-d Toy Example Rank 3 NNCA Approx.
Current
Research:
How Many
Nonneg.
Basis El’ts
Needed?
Nonnegative Nested Cone Analysis
![Page 26: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/26.jpg)
26
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
![Page 27: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/27.jpg)
27
UNC, Stat & OR
Goal: Find lower dimensional manifold that well approximates data
ISOmap
Tennenbaum (2000)
Local Linear Embedding
Roweis & Saul (2000)
Manifold Learning
![Page 28: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/28.jpg)
28
UNC, Stat & OR
1st Principal Curve
Linear Reg’n
Usual Smooth
![Page 29: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/29.jpg)
29
UNC, Stat & OR
1st Principal Curve
Linear Reg’n
Proj’s Reg’n
Usual Smooth
![Page 30: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/30.jpg)
30
UNC, Stat & OR
1st Principal Curve
Linear Reg’n
Proj’s Reg’n
Usual Smooth
Princ’l Curve
![Page 31: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/31.jpg)
31
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
Potential Application: Principal Curves
Perceived Major Challenge:
How to find 2nd Principal Curve?
Backwards approach???
An Interesting Question
![Page 32: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/32.jpg)
32
UNC, Stat & OR
Key Component:
Principal Surfaces
LeBlanc & Tibshirani (1996)
An Interesting Question
![Page 33: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/33.jpg)
33
UNC, Stat & OR
Key Component:
Principal Surfaces
LeBlanc & Tibshirani (1996)
Challenge:
Can have any dimensional surface,
But how to nest???
An Interesting Question
![Page 34: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/34.jpg)
34
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
Another Potential Application:
Trees as Data
(early days)
An Interesting Question
![Page 35: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/35.jpg)
35
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
An Attractive Answer
An Interesting Question
![Page 36: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/36.jpg)
36
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
An Attractive Answer:
James Damon, UNC Mathematics
Geometry
Singularity
Theory
An Interesting Question
![Page 37: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/37.jpg)
37
UNC, Stat & OR
How generally applicable is
Backwards approach to PCA?
An Attractive Answer:
James Damon, UNC Mathematics
Damon and Marron (2013)
An Interesting Question
![Page 38: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/38.jpg)
38
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
![Page 39: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/39.jpg)
39
UNC, Stat & OR
Define Nested Spaces via Constraints
Satisfying More Constraints
Smaller Subspaces
General View of Backwards PCA
![Page 40: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/40.jpg)
40
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
![Page 41: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/41.jpg)
41
UNC, Stat & OR
Define Nested Spaces via Constraints
E.g. SVD
Have Nested Subspaces:
General View of Backwards PCA
![Page 42: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/42.jpg)
42
UNC, Stat & OR
Define Nested Spaces via Constraints
E.g. SVD
-th SVD Subspace
Scores
Loading Vectors
General View of Backwards PCA
![Page 43: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/43.jpg)
43
UNC, Stat & OR
Define Nested Spaces via Constraints
E.g. SVD
Now Define:
General View of Backwards PCA
![Page 44: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/44.jpg)
44
UNC, Stat & OR
Define Nested Spaces via Constraints
E.g. SVD
Now Define:
Constraint Gives Nested Reduction of Dim’n
General View of Backwards PCA
![Page 45: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/45.jpg)
45
UNC, Stat & OR
Define Nested Spaces via Constraints
• Backwards PCA
Reduce Using Affine Constraints
General View of Backwards PCA
![Page 46: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/46.jpg)
46
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
![Page 47: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/47.jpg)
47
UNC, Stat & OR
Define Nested Spaces via Constraints
• Backwards PCA
• Principal Nested Spheres
• Principal Surfaces
Spline Constraint Within Previous?
General View of Backwards PCA
![Page 48: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/48.jpg)
48
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
![Page 49: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/49.jpg)
49
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
![Page 50: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/50.jpg)
50
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
![Page 51: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/51.jpg)
51
UNC, Stat & OR
New Topic
Curve Registration
![Page 52: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/52.jpg)
52
UNC, Stat & OR
Collaborators
• Anuj Srivastava (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)
![Page 53: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/53.jpg)
53
UNC, Stat & OR
Context
Functional Data AnalysisCurves as Data Objects
Toy Example:
![Page 54: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/54.jpg)
54
UNC, Stat & OR
Context
Functional Data AnalysisCurves as Data Objects
Toy Example:
How Can WeUnderstandVariation?
![Page 55: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/55.jpg)
55
UNC, Stat & OR
Context
Functional Data AnalysisCurves as Data Objects
Toy Example:
How Can WeUnderstandVariation?
![Page 56: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/56.jpg)
56
UNC, Stat & OR
Context
Functional Data AnalysisCurves as Data Objects
Toy Example:
How Can WeUnderstandVariation?
![Page 57: 1 UNC, Stat & OR PCA Extensions for Data on Manifolds Fletcher (Principal Geodesic Anal.) Best fit of geodesic to data Constrained to go through geodesic](https://reader035.vdocuments.us/reader035/viewer/2022062315/5697bfdc1a28abf838cb11e2/html5/thumbnails/57.jpg)
57
UNC, Stat & OR
Functional Data Analysis
InsightfulDecomposition
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58
UNC, Stat & OR
Functional Data Analysis
InsightfulDecomposition
Horiz’l Var’n
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59
UNC, Stat & OR
Functional Data Analysis
InsightfulDecomposition
Vertical Variation
Horiz’l Var’n
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60
UNC, Stat & OR
Challenge
Fairly Large Literature
Many (Diverse) Past Attempts
Limited Success (in General)
Surprisingly Slippery
(even mathematical formulation)
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UNC, Stat & OR
Challenge (Illustrated)
Thanks to Wei Wu
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62
UNC, Stat & OR
Challenge (Illustrated)
Thanks to Wei Wu
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63
UNC, Stat & OR
Functional Data Analysis
AppropriateMathematicalFramework? Vertical Variation
Horiz’l Var’n
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64
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
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65
UNC, Stat & OR
Landmark Based Shape Analysis
Equivalence Classes become Data Objects
a.k.a. “Orbits”
Mathematics: Called “Quotient Space”
, , , , , ,
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66
UNC, Stat & OR
Curve Registration
What are theData Objects?
Vertical Variation
Horiz’l Var’n
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67
UNC, Stat & OR
Curve Registration
What are the Data Objects?
Consider “Time Warpings”
(smooth)
More Precisely: Diffeomorphisms
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UNC, Stat & OR
Curve Registration
Diffeomorphisms
is 1 to 1 is onto
(thus is invertible) Differentiable is Differentiable
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UNC, Stat & OR
Time Warping Intuition
Elastically Stretch & Compress Axis
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70
UNC, Stat & OR
Time Warping Intuition
Elastically Stretch & Compress Axis
(identity)
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71
UNC, Stat & OR
Time Warping Intuition
Elastically Stretch & Compress Axis
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72
UNC, Stat & OR
Time Warping Intuition
Elastically Stretch & Compress Axis
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73
UNC, Stat & OR
Time Warping Intuition
Elastically Stretch & Compress Axis
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UNC, Stat & OR
Curve Registration
Say curves and are equivalent,
When so that
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UNC, Stat & OR
Curve Registration
Toy Example: Starting Curve,
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76
UNC, Stat & OR
Curve Registration
Toy Example: Equivalent Curves,
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UNC, Stat & OR
Curve Registration
Toy Example: Warping Functions
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UNC, Stat & OR
Curve Registration
Toy Example: Non-Equivalent Curves
CannotWarpIntoEach Other
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79
UNC, Stat & OR
Data Objects I
Equivalence Classes of Curves
(parallel to Kendall shape analysis)
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UNC, Stat & OR
Data Objects I
Equivalence Classes of Curves
(Set of AllWarps ofGiven Curve)
Notation: for a “representor”
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81
UNC, Stat & OR
Data Objects I
Equivalence Classes of Curves
(Set of AllWarps ofGiven Curve)
Next Task: Find Metric on Equivalence Classes
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UNC, Stat & OR
Metrics in Curve Space
Find Metric on Equivalence Classes
Start with Warp Invariance on Curves& Extend
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83
UNC, Stat & OR
Metrics in Curve Space
Traditional Approach to Curve
Registration:
• Align curves, say and
• By finding optimal time warp, , so:
• Vertical var’n: PCA after alignment
• Horizontal var’n: PCA on s
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84
UNC, Stat & OR
Metrics in Curve Space
Problem:
Don’t have proper metric
Since:
Because:
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UNC, Stat & OR
Metrics in Curve Space
Thanks toXiaosun Lu
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86
UNC, Stat & OR
Metrics in Curve Space
Note:VeryDifferentL2 norms
Thanks toXiaosun Lu