Multivariate General Linear Models (MGLM) on Riemannian Manifolds

Transcription

1 Multivariate General Linear Models (MGLM) on Riemannian Manifolds Hyunwoo J. Kim, Nagesh Adluru, Maxwell D. Collins, Moo K. Chung,! Barbara B. Bendlin, Sterling C. Johnson, Richard J. Davidson, Vikas Singh Poster session ID: O-3C-384

2 General Linear Model for Group Analysis Disease Normal Template Registration Registration

3 General Linear Model for Group Analysis Disease Normal M D M D AGE All subjects AGE

4 General Linear Model for Group Analysis Disease Normal P-value map M D AGE All subjects

5 General Linear Model (GLM) f : R! R : Group (patient or normal)

6 General Linear Model (GLM) f : R 2! R : Group (patient or normal) : Age

7 General Linear Model (GLM) f : R 3! R : Group (patient or normal) : Age : Gender

8 GLM on scalar valued summaries DTI P-value map D = A

9 GLM on scalar valued summaries DTI FA P-value map MD

10 GLM on scalar valued summaries DTI? P-value map D = A x T Dx > 0,x6= 0

11 GLM on scalar valued summaries DTI? P-value Response Y is manifold-valued

12 Manifold -valued data Unit sphere, quotient spaces of spheres SPD matrix i.e covariance matrix, diffusion tensors Probability density functions (PDFs), orientation density functions (ODFs) Lie groups i.e. O(n), SO(n), GL(n), SL(n) Kendall shape manifolds

13 Can we directly use the ordinary general linear model with manifold data?

14 Euclidean model for manifolds

15 Euclidean model for manifolds Test Training Training Test

16 Euclidean model for manifolds Problem 1: Model in ambient space Predictions in ambient space Predictions need to be projected onto manifolds

17 Euclidean model for manifolds

18 Euclidean model for manifolds Problem 2: Distance metric in ambient space

19 Euclidean model for manifolds Problem 2: Distance metric in ambient space Geodesic distance is needed.

20 Model on manifolds

21 Regression with a Single Covariate f : R!M Fletcher, P. Thomas. "Geodesic regression and the theory of least squares on Riemannian manifolds. IJCV, 2013.

22 Model on manifolds f : R!M Jia Du, Alvina Goh, Sergey Kushnareva, Anqi Qiu, "Geodesic regression on orientation distribution functions with its application to an aging study." NeuroImage, 2014.

23 Model on manifolds f : R!M Jia Du, Alvina Goh, Sergey Kushnareva, Anqi Qiu, "Geodesic regression on orientation distribution functions with its application to an aging study." NeuroImage, f : R n!m Our MGLM

24 Single covariate E(p, v) = 1 2 NX i=1 d(exp(p, x i v),y i ) 2 y 1 y 2 y 4 M y 3

25 Single covariate E(p, v) = 1 2 NX i=1 d(exp(p, x i v),y i ) 2 p y 1 y 2 y 4 M y 3

26 Single covariate E(p, v) = 1 2 NX i=1 d(exp(p, x i v),y i ) 2 T p M p y 1 y 2 y4 v M y 3

27 Single covariate E(p, v) = 1 2 NX i=1 d(exp(p, x i v),y i ) 2 T p M p y 1 y 2 y4 v M y 3

28 Single covariate E(p, v) = 1 2 NX i=1 d(exp(p, x i v),y i ) 2 p y 1 x 1 x 2 x 3 x 4 y 2 y4 v T p M M y 3

29 Single covariate E(p, v) = 1 2 NX i=1 d(exp(p, x i v),y i ) 2 p x 1 x 2 x 4 x 3 y 1 y 2 y4 T p M v M y 3

30 Multiple covariates E(p, V )= 1 2 NX i=1 d(exp(p, X j x j i vj ),y i ) 2 v 2 x 2 1 p x 1 1 T p M v 1 M

31 Multiple covariates E(p, V )= 1 2 NX i=1 d(exp(p, X j x j i vj ),y i ) 2 v 2 x 2 2 x 2 1 p x 1 1 x 1 2 T p M v 1 M

32 Multiple covariates E(p, V )= 1 2 NX i=1 d(exp(p, X j x j i vj ),y i ) 2 v 2 x 2 2 x 2 1 p x 1 1 x 1 2 T p M v 1

33 Multiple covariates E(p, V )= 1 2 NX i=1 d(exp(p, X j x j i vj ),y i ) 2 v 2 x 2 2 x 2 1 p x 1 1 x 1 2 T p M v 1

34 Multiple covariates E(p, V )= 1 2 NX i=1 d(exp(p, X j x j i vj ),y i ) 2 v 2 x 2 2 x 2 1 p x 1 1 x 1 2 T p M v 1

35 Optimization (iterative method) E(p, V )= 1 2 NX i=1 d(exp(p, X j x j i vj ),y i ) 2

36 Optimization (iterative method) Step 1 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j p k+1 Exp (p k, i ŷi!p k Log(ŷ i,y i ))

37 Optimization (iterative method) Step 1 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j p k+1 Exp (p k, i ŷi!p k Log(ŷ i,y i )) Error term p k v 1 ŷ 1 v 2 y 1 Error term

38 Optimization (iterative method) Step 1 : E(p, V )= 1 NX d(exp(p, X x j i 2 ),y i ) 2 i=1 j p k+1 Exp (p k, i ŷi!p k Log(ŷ i,y i )) Parallel transported error p k v 1 Parallel transported error v 2 ŷ 1 y 1

39 Optimization (iterative method) Step 1 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j p k+1 Exp (p k, i ŷi!p k Log(ŷ i,y i )) p k p k+1 v 1 v 2 y 1

40 Optimization (iterative method) Step 1 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j p k+1 Exp (p k, i ŷi!p k Log(ŷ i,y i )) p k pk+1 v 1 v 2 ŷ 1 y 1

41 Optimization (iterative method) Step 2 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 v j k+1/2 = vj k + X j x j i ŷ i!plog(ŷ i,y i ) Error term i p k v 1 k v 2 k y 1 ŷ 1 Error term

42 Optimization (iterative method) Step 2 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 v j k+1/2 = vj k + X j x j i ŷ i!plog(ŷ i,y i ) i Parallel transported Error p k v 1 k v 2 k y 1 ŷ 1

43 Optimization (iterative method) Step 2 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 v j k+1/2 = vj k + X j x j i ŷ i!plog(ŷ i,y i ) i p k v 1 k v 1 k+ 1 2 v 2 k+ 1 2 v 2 k y 1 ŷ 1

44 Optimization (iterative method) Step 3 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j V k+1 = pk!p k+1 (V k+1/2 ) p k p k+1 v 1 k+ 1 2 v 2 k+ 1 2 v 1 k+1 v 2 k+1

45 Optimization (iterative method) Step 3 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j V k+1 = pk!p k+1 (V k+1/2 ) Must we solve for p or can p k we approximate it? p k+1 v 1 k+ 1 2 v 2 k+ 1 2 v 1 k+1 v 2 k+1

46 Optimization (iterative method) Step 3 : E(p, V )= 1 NX d(exp(p, X x j i 2 vj ),y i ) 2 i=1 j V k+1 = pk!p k+1 (V k+1/2 ) Must we solve for p or can p k we approximate it? p k+1 v 1 k+ 1 2 v 2 k+ 1 2 v 1 k+1 Xie, Yuchen, Baba C. Vemuri, and Jeffrey Ho Statistical analysis of tensor fields,miccai 10 v 2 k+1

47 Experiments Synthetic data Neuroimaging data GLM Full : y = Exp(p, v 1 Group + v 2 Gender + v 3 Age)

48 Experiments (synthetic ODF)

49 Experiments (synthetic ODF)

50 Experiments (synthetic ODF)

51 Experiments (synthetic ODF)

52 Experiments (synthetic ODF)

53 Experiments Synthetic data Neuroimaging data GLM Full : y = Exp(p, v 1 Group + v 2 Gender + v 3 Age)

54 Experiments (neuroimaging study) LTM (ODF) AD risk (DTI) Subjects Group LTM WLC APOE4+ APOE4- Gender Female Male Female Male Age GLM Full : y =Exp(p, v 1 Group + v 2 Gender + V 3 Age) GLM Age : y =Exp(p, v 2 Gender + v 3 Age) GLMFull : y = Exp(p, v 1 Group + v 2 Gender + v 3 Age) GLM Group : y =Exp(p, v 1 Group + v 2 Gender

55 Experiments (neuroimaging study) p-values maps and histograms for effect of age and group computed from simulating the Null distribution of the F ratio statistic using 20,000 permutations. F ratio statistic is defined for a pair of nested GLMs as! F = RSS 1 RSS 2 p 2 p 1 RSS 2 N p 2

56 Neuroimaging study 1 Age effect in study 1 (long-term meditators, ODFs)! GLM Full : y = Exp(p, v 1 Group + v 2 Gender + v 3 Age) GLM Group : y = Exp(p, v 1 Group + v 2 Gender)!!!!

57 Neuroimaging study 1 Group effect in study 1 (long-term meditators, ODFs)! GLM Full : y = Exp(p, v 1 Group + v 2 Gender + v 3 Age) GLM Age : y = Exp(p, v 2 Gender + v 3 Age)!!!! Group 2 {LTM, WLC}.

58 !!! GLM Full : y = Exp(p, v 1 Group + v 2 Gender + v 3 Age) GLM Group : y = Exp(p, v 1 Group + v 2 Gender)!!!! Neuroimaging study 2 Age model of study 2 (AD, DTI)

59 Conclusion Generalization of multivariate general linear model (MGLM) to Riemannian manifolds Especially useful when response is manifold valued and we want to control for one or more covariates. Here, the analysis obtains significantly improved statistical power Applicable to other manifold-valued statistical inference problems Code is available. See me at the posters! Poster session ID: O-3C-384

60 Thank you Research supported in part by! NSF CAREER RI ! NIH R01 AG Poster session ID: O-3C-384