1. Multiple Change Point Detection for Symmetric Positive Definite Matrices
Dehan Kong
University of Toronto
JSM 2018
July 30, 2018

2. Joint work with
Zhenhua Lin
Qiang Sun

3. Roadmap Motivation and Background Modeling Covariance Matrices
Change Point Detection
Numerical Examples

4. Functional Magnetic Resonance Imaging

5. Functional Magnetic Resonance Imaging
brain activity changes of cerebral blood blow
BOLD fMRI
blood-oxygen-level dependent (BOLD) contrast
color coding
yellow area shows increased activity compared with a control condition

6. Functional Magnetic Resonance Imaging
Each image consists of around 100,000 brain voxels
T = several hundred

7. Regions of Interest

8. Estimated Brain Network: Intro Music Scene
visual
auditory
Occipital and auditory: process basic visual information and sound

9. Estimated Brain Network: Mystery And Poison
precuneus
Precuneus: high level region, not well understood. Represents high-level
concepts, integrating information from other parts of the brain.
References: Lernet et al. (2011) and Ames et al. (2015)

10. Estimated Brain Network: Scientific Deduction
prefrontal
cortex
Prefrontal cortex: executive function such as expectation based on actions,
differentiate among conflicting thoughts, determine good and bad, etc

11. Task-introduced Dynamic Functional Connectivities
Intro Music Scene
mystery and poison
scientific deduction
Task fMRI: different tasks (scenes) introduce changes of the brain networks

12. Modeling Connectivity: Graphical Model
conditional
dependence
variable
conditional dependence is difficult to model and compute !!!
other variables
Goal:
Estimate the conditional dependence relationships among the variables

13. Modeling Connectivity: Covariance Matrix
pairwise dependence relationships among the variables
Covariance matrix: covariances between all pairs of regions
SPD
SPD: nonlinear Riemannian manifold !!!
Note:
Estimate the conditional dependence relationships among the variables
pairwise dependence

14. Dynamic Functional Connectivity: A Time Series of Covariance Matrices

15. Dynamic Functional Connectivity: A Time Series of Covariance Matrices

16. How to Model Covariance Matrices?

17. How to Model Covariance Matrices?
，the straightforward and simple way:
lose positive definiteness
Better ways?
to preserve positive definiteness

18. Preserving Positive Definiteness
consider the eigen-decomposition of a matrix
positive definiteness
define matrix-Log:
matrix-Log
symmetric matrices
symmetric positive definite matrices
References: Lin, Kong and Sun (2017+), preprint.

19. The Heterogeneous Matrix-Log Mean Model
d-by-d symmetric matrices is isomorphic to
References: Lin, Kong and Sun (2017+), preprint.

20. The Heterogeneous Matrix-Log Mean Model
we embed the space of covariance matrices into
a Euclidean space
References: Lin, Kong and Sun (2017+), preprint.

21. The Heterogeneous Matrix-Log Mean Model
The Heterogeneous Matrix-Log Covariance Model
is essentially a linear model in Euclidean space!
So that we can do all those Euclidean operations!
References: Lin, Kong and Sun (2017+), preprint.

22. We have now finished the modeling part…
We move to the problem of change point detection

23. Model for Change Points

24. Change Point Problem
a local scan statistic k

25. Change Point Problem
a local scan statistic h k

26. if k is not a change point:
Change Point Problem
if k is not a change point:
||G(k,h)|| is close to 0 k j
if j is a change point:
||G(j,h)|| is large

27. Theoretical Property Theorem (Lin, Kong and Sun, 2017+)
Under proper scaling conditions, with probability going to 1，
our procedure can recover all change points.

28. Different combinations of (n, d, J)
Toy Example
dimension
Different combinations of
(n, d, J)
sample size
# of change points
(100, 6, 2), (200, 6, 2)
J=2
J=4: (200, 6, 4), (400, 6, 4), ……

29. Toy Example probability of recovering CP 1
probability of estimated #= true #
our method
comparison method

30. Human Connectome Project (HPC) Data
Data: ~850 subjects, 274 time points
triangles
circles
rectangles
Video: 5 video blocks, social cognition task
Question: Estimate change points
Reference: Wheatley et al. (2007), Psychological Science

31. Cortical Surface fMRI Data
Volumetric fMRI mapped onto the cortical surface
Stronger signals

32. Cortical Surface fMRI Data Regions of Interest
Desikan-Killiany parcellation, 68 ROIs
Reference: Desikan et al. (2006), NeuroImage

33. Cortical Surface fMRI Data Regions of Interest
8 Selected ROIs related to social cognition
Left and Right superior temporal, inferior parietal, temporal pole and precuneus
Reference: Green et al. (2006), Nature Reviews. Neuroscience

34. Result 5 video blocks in the task design => 4 change points
at time 35, 70, 105, 140
sliding window =100 to calculate the cross covariances
the mean number of change points detected is 3.66 (0.03)
estimated change points: 39.6, 72.0, 106.7, 138.5

35. Manifold Learning ?
manifold learning -> Euclidean learning
heterogeneous errors can account for the curvature structure of SPD manifold
mean model -> regression model?

36. Thank You!