Nonparametric regression for topology. applied to brain imaging data

Transcription

1 , applied to brain imaging data Cleveland State University October 15, 2010

2 Motivation from Brain Imaging

3 MRI Data Topology Statistics Application

4 MRI Data Topology Statistics Application

5 Cortical surface

6 Cortex thickness

7 Our model The cortex is modeled by a surface M, and the cortical thickness by a function f : M R. Our parameter of interest is the persistence diagram of f, which captures the topology of f.

8 What is topology? Overview Homology Persistence Diagrams Distance Definition Topology is the branch of mathematics that studies spatial features which do not change under continuous deformations (can stretch, twist and compress, but not tear or glue together).

9 0-dimensional features Overview Homology Persistence Diagrams Distance Example Connected components are 0-dimensional topological features. Source: R.G. Miller, Jr. Annals of Statistics. 13 (2) (1985) p.511.

10 1-dimensional features Overview Homology Persistence Diagrams Distance Example Holes and tunnels are 1-dimensional topological features.

11 1-dimensional features Overview Homology Persistence Diagrams Distance Example Periodic motion can be revealed by 1-dimensional topological features.

12 2-dimensional features Overview Homology Persistence Diagrams Distance Example Voids are 2-dimensional topological features.

13 Betti numbers and homology Overview Homology Persistence Diagrams Distance For each dimension, homology gives a vector space. The Betti numbers give the dimension of this vector space. Example the torus

14 Betti numbers and homology Overview Homology Persistence Diagrams Distance For each dimension, homology gives a vector space. The Betti numbers give the dimension of this vector space. Example the torus β 0 = 1,

15 Betti numbers and homology Overview Homology Persistence Diagrams Distance For each dimension, homology gives a vector space. The Betti numbers give the dimension of this vector space. Example the torus β 0 = 1, β 1 = 2,

16 Betti numbers and homology Overview Homology Persistence Diagrams Distance For each dimension, homology gives a vector space. The Betti numbers give the dimension of this vector space. Example the torus β 0 = 1, β 1 = 2, β 2 = 1, β k = 0 for k 3

17 From topology to statistical topology Overview Homology Persistence Diagrams Distance Instead of considering the topology of geometric objects, we will consider the topology of a function on a geometric object. Our calculational tool will be persistent homology. Persistent homology describes the homological features which persist as a single parameter changes. Here, we take this parameter to be a threshold on a function on the space from which we are sampling.

18 Persistent homology of functions on R Overview Homology Persistence Diagrams Distance Let f : R R. Assume: if f (x) = 0, then f (x) 0, Each critical point is a local minimum or a local maximum.

19 Persistent homology of functions on R Overview Homology Persistence Diagrams Distance Let f : R R. Assume: if f (x) = 0, then f (x) 0, Each critical point is a local minimum or a local maximum. Define the lower excursion sets R t = {x R f(x) t}, t R. Fact As t increases, the topology of R t does not change as long as we do not pass a critical value of f.

20 Persistent homology of functions on R Overview Homology Persistence Diagrams Distance At the critical values we have the following effects: At a local minimum, a new component appears. At a local maximum, two components are merged. Pair critical values: Pair a local maximum, with the higher of the local minimums associated with the two components which it joins. We graph all such pairs to obtain the Persistence Diagram.

21 Persistence Diagram Overview Homology Persistence Diagrams Distance For example: y y = f(x) x

22 Data Topology Statistics Application Persistence Diagram Overview Homology Persistence Diagrams Distance For example: y y = f(x) R 1 x

23 Data Topology Statistics Application Persistence Diagram Overview Homology Persistence Diagrams Distance For example: y y = f(x) R 2 x

24 Persistence Diagram Overview Homology Persistence Diagrams Distance For example: y y = f(x) x R 3

25 Persistence Diagram Overview Homology Persistence Diagrams Distance For example: y y = f(x) x R 4

26 Data Topology Statistics Application Persistence Diagram Overview Homology Persistence Diagrams Distance For example: y y = f(x) death x birth

27 Overview Homology Persistence Diagrams Distance Persistent homology for functions on surfaces For a function f : M R on a surface (or manifold), we can also consider lower excursion sets (sublevel sets) M r = {x M f(x) r}. The persistence diagram keeps track of how the topology of these sublevel sets changes as the parameter r increases.

28 A Function Data Topology Statistics Application Overview Homology Persistence Diagrams Distance For example x y

29 Data Topology Statistics Application Overview Homology Persistence Diagrams Distance Lower excursion sets and H x y x y x x x y 0.0 y y

30 Data Topology Statistics Application Persistence Diagram of the function Overview Homology Persistence Diagrams Distance death birth

31 The Bottleneck Distance Overview Homology Persistence Diagrams Distance A useful metric on the space of Persistence Diagrams: Let f,g : M R be two functions, with associated Persistence Diagrams D(f) and D(g). Definition The Bottleneck distance is given by d B (D(f),D(g)) = inf sup x η(x), η where the infimum is taken over all bijections η : D(f) D(g) and the supremum is taken over all points x D(f). x

32 Bottleneck Distance Overview Homology Persistence Diagrams Distance For example: y y = f(x) x

33 Data Topology Statistics Application Bottleneck Distance Overview Homology Persistence Diagrams Distance For example: y y = f(x) death x birth

34 Stability Data Topology Statistics Application Overview Homology Persistence Diagrams Distance The following fundamental result bounds the bottleneck distance for persistence diagrams with the supremum norm. Theorem (Cohen-Steiner, Edelsbrunner, Harer) d B (D(f),D(g)) f g Take f to be an unknown function and ˆf to its statistical estimator. Then, d B (D(f),D(ˆf)) f ˆf

35 Nonparametric regression Nonparametric regression Results Let M be a manifold (e.g. surface). Problem (Nonparametric regression) Assume that there exists a function f : M R such that y = f(x)+ǫ, x M where ǫ is a normal random variable with mean zero and variance σ 2 > 0. Given a sample (x 1,y 1 ),...,(x n,y n ), find an estimator ˆf of f.

36 The precise setup Nonparametric regression Results M is a compact Riemannian manifold with metric ρ(, ) (given by geodesic distance).

37 The precise setup Nonparametric regression Results M is a compact Riemannian manifold with metric ρ(, ) (given by geodesic distance). Parameter space: the Hölder class of functions, { } Λ(β,L) = f : M R : f(x) f(y) Lρ(x,y) β,x,y M, where 0 < β 1.

38 Expected loss and minimax risk Nonparametric regression Results Definition The expected loss (risk) of an estimator f is given by The minimax risk is given by inf f E f f. sup E f f. f Λ(β,L)

39 The main result Data Topology Statistics Application Nonparametric regression Results Theorem (BCKL, 2010) For the regression model, there exists an estimator ˆf (constructive) that attains the minimax risk, and sup E ˆf f C f Λ(β,L) ( ) log(n) β/(2β+d), as n, ( ) where C = L d/(2β+d) σ 2 vol(m)(β+d)d 2 β (2β+d). vol(s d 1 )β 2 n

40 The construction of the estimator Nonparametric regression Results A sketch of the construction of the estimator: Partition M by {A i M} i and let ˆf(x) = i â i I Ai (x). By a suitable choice of A i and â i (constructive) one obtains the desired estimator.

41 Estimating persistent homology Nonparametric regression Results Corollary In the regression model, as n. ( log(n) Ed B (D(ˆf),D(f)) C n ) β/(2β+d)

42 Cortex thickness Data Topology Statistics Application Brain Imaging The Estimator Results Summary

43 Constructing our estimator Brain Imaging The Estimator Results Summary Construct an estimator: First, smooth the data on S 2 using the kernel K xi (x) = max(1 κarccos(x ix),0), and the Nadaraya-Watson kernel-weighted average f(x) = i y ik xi (x) i K x i (x). (1)

44 Constructing our estimator Brain Imaging The Estimator Results Summary Construct an estimator: First, smooth the data on S 2 using the kernel K xi (x) = max(1 κarccos(x ix),0), and the Nadaraya-Watson kernel-weighted average f(x) = i y ik xi (x) i K x i (x). (1) Next, choose design points from a triangulation of the sphere: take an iterated subdivision of the icosahedron, which has 1280 faces and 642 vertices. Define ˆf on vertices using (1) and extend by linear interpolation.

45 Triangulated sphere Brain Imaging The Estimator Results Summary

46 Cortex thickness estimator Brain Imaging The Estimator Results Summary

47 Calculating Persistent Homology Brain Imaging The Estimator Results Summary Remark Critical points only occur at vertices. The values of the estimator at the vertices, induce a filtration of the triangulation of the sphere. The persistent homology of this filtered complex is identical to the persistent homology of the estimator. Use the software Plex to calculate the persistent homology of the filtered complex.

48 Persistence diagrams Brain Imaging The Estimator Results Summary 6 Persistence Diagram in degree

49 Space of Persistence Diagrams Brain Imaging The Estimator Results Summary The bottleneck distance endows the space of persistence diagrams with a metric. The same is true for the (L p ) Wasserstein distance. We can measure the pairwise distance between our persistence diagrams to get a distance matrix. We use Classical Multidimensional Scaling to project from this metric space to R 2.

50 Brain Imaging The Estimator Results Summary Eigenvalues for CMDS for Wasserstein distance

51 CMDS for Wasserstein distance Brain Imaging The Estimator Results Summary autistic control

52 Summary Data Topology Statistics Application Brain Imaging The Estimator Results Summary Given a function one can study the persistent homology of its lower excursion sets. The sup norm minimax estimator gives an estimate for persistent homology. This method shows promise for detecting global nonlinear features in nonlinear settings.

53 Future work Data Topology Statistics Application Brain Imaging The Estimator Results Summary We applied our techniques to the thickness of the cortex. There are other interesting biological possibilities, such as the size and number of minicolumns (a basic organizational unit of brain cells). There are also interesting geometric possibilities, such as the curvature of the cortical surface. Please let me know if you have any data for which you think these ideas might be appropriate.

54 References Data Topology Statistics Application Brain Imaging The Estimator Results Summary P. Bubenik, G. Carlsson, P.T. Kim, Z. Luo. Statistical Topology via Morse Theory Persistence and Nonparametric Estimation. Contemporary Mathematics, 516 (2010), pp M.K. Chung, P. Bubenik, P.T. Kim. Persistence Diagrams of Cortical Surface Data. Image Processing in Medical Imaging Lecture Notes in Computer Science, 2009.