Mixed effect model for the spatiotemporal analysis of longitudinal - PowerPoint PPT Presentation

Mixed effect model for the spatiotemporal analysis of longitudinal manifold valued data Stéphanie Allassonnière with J.B. Schiratti, J. Chevallier, I. Koval, V. Debavalaere and S. Durrleman Université Paris Descartes & Ecole Polytechnique

Computational Anatomy Represent and analyse geometrical elements upon which deformations • can act Describe the observed objects as geometrical variations of one or several • representative elements Quantify this variability inside a population • Deformable template model from Grenander How does the deformation act? • What is a representative element? • How to quantify the geometrical variability ? • 2

Computational Anatomy One solution : Quantify the distance between observations using deformations • Provide a statistical model to approximate the generation of the observed • population from the atlas Propose a statistical learning algorithm • Optimise the numerical estimation • 3

Bayesian Mixed Effect model First model : • – One observation per subject – Image or shape (viewed as currents) – Deformations either linearized or diffeomorphic – Homogeneous or heterogeneous populations (mixture models) 4

Bayesian Mixed Effect model 5

Bayesian Mixed Effect model 6

Bayesian Mixed Effect model First model : • – One observation per subject – Image or shape (viewed as currents) – Deformations either linearized or diffeomorphic – Homogeneous or heterogeneous populations (mixture models) Ø Limitations Ø One observation per subject Ø Corresponding acquistion time 7

Longitudinal Data Analysis Longitudinal model : • – Several observation per subject – Image, shape, etc – Atlas = representative trajectory and population variability 8

Longitudinal Data Analysis Time (age) subject #1 subject #2 subject #3 How to learn representative trajectories of data changes from longitudinal data? Linear mixed-effects models Temporal marker of progression Regression vector-valued data [Laird&Ware’82, Diggle et al., (e.g. time since drug injection, seeding, birth, (e.g. compare measurements at Fitzmaurice et al.] etc..) same time-point) Needs to disentangle Learning spatiotemporal differences in: distribution of trajectories No temporal marker of progression manifold-valued data - Measurements (e.g. in aging, neurodegenerative diseases, - Find temporal correspondences (normalized data, positive - Compare data at corresponding etc..) matrices, shapes, etc;;) - Dynamics of measurement stages of progression changes 9

Spatiotemporal Statistical Model • Statistical model inclinding: • a representative trajectory of data changes • spatiotemporal variations in: t 0 v 0 • measurement values p 0 • pace of measurement t v i changes p ij T 0 ( t ) = Exp p 0 ,t 0 ( v 0 )( t ) v ij • Orthogonality condition ensures T i ( t ) = Exp T 0 ( t ) (P T 0 t 0 ,t ( v i )) identifiability (unique space/time y ij = T i ( ψ i ( t )) + ε ij decomposition) ψ i ( t ) = t 0 + α i ( t − t 0 − τ i ) • Time is not a covariate but a random variable Space-shift Time-shift Acceleration factor Random effects: α i ∼ log N (0 , σ 2 τ i ∼ N (0 , σ 2 v i = ( A 1 | . . . | A K ) s i α ) τ ) A k ⊥ v 0 Fixed effects: ( σ 2 α , σ 2 and ( p 0 , t 0 , v 0 ) τ , A 1 , ...A K ) 10 [Schiratti et al. IPMI’15, NIPS’15]

Spatiotemporal Statistical Model Submanifold value observations { y ij = T i ( ψ i ( t )) + ε ij Parallel curve T i ( t ) = Exp T 0 ( t ) (P T 0 t 0 ,t ( v i )) Representative trajectory T 0 ( t ) = Exp p 0 ,t 0 ( v 0 )( t ) Linear time reparametrization ψ i ( t ) = t 0 + α i ( t − t 0 − τ i ) α i ∼ log N (0 , σ 2 { α ) Hidden random variables: Acceleration factor τ i ∼ N (0 , σ 2 τ ) Time shift Space shift v i = ( A 1 | . . . | A K ) s i A k ⊥ v 0 Parameters: { ( p 0 , t 0 , v 0 ) Mean trajectory parametrization and ( σ 2 α , σ 2 τ , A 1 , ...A K ) prior parameter 11

Spatiotemporal Statistical Model Comparison with previous work : v 0 v 0 p 0 p 0 v i v i p ij v ij Interest : Parallel transport keep invariant the structure of the distribution, but updated it in time Σ P 0 tT Σ P 0 t p 0 v i p ij v ij 12

Spatiotemporal Statistical Model • The straight line model M = R y b b t 0 t 0 time time x x y ij = ( a + a i )( t i,j − t 0 − τ i } ) + b + ε i,j y ij = ( a + a i )( t i,j − t 0 ) + b + b i | {z } + ε i,j | {z Time at which Measurement of the i th measurement of the i th subject at time t 0 subject reaches ¯ b 13 Schiratti et al. (2015) Laird & Ware (1982)

Spatiotemporal Statistical Model uv • The logistic curve model M =]0 , 1[ , g ( p )( u, v ) = p 2 (1 − p ) 2 • Geodesic are logistic curves (1 − p 0 ) /p 0 ⇣ ⌘ γ 0 ( t ) = 1 + y ij = γ 0 t 0 + α i ( t − t 0 − τ i ) + ε ij ⇣ ⌘ exp p 0 (1 − p 0 ) ( t − t 0 ) v 0 − • It is not equivalent to a linear model on the logit of the observations (i.e. the Riemannian log at p 0 = 0.5), since p 0 is estimated • If we fix p 0 = 0.5 in our model à end up with our previous linear case (different from Laird&Ware) 14

Spatiotemporal Statistical Model N u k v k • The propagation model X M =]0 , 1[ N , g ( p )( u, v ) = p 2 k (1 − p k ) 2 k =1 • Geodesics are logistic curves in each coordinate • Parametric family of geodesics seen as a model of propagation of an effect ⇣ ⌘ γ δ ( t ) = γ 0 ( t ) , γ 0 ( t − δ 1 ) , . . . , γ 0 ( t − δ N − 1 ) • The parallel curve in the direction of the space-shift v i writes ✓ ✓ ◆ ✓ ◆ ✓ ◆◆ t + v i, 1 t − δ 1 + v i, 2 t − δ N − 1 + v i,N γ 0 , γ 0 , ..., γ 0 v 0 v 0 v 0 à The parallel changes the relative timing of the effect onset across coordinates 15

Parameter Estimation y = ( y 1 , ..., y N ) , z = ( z 1 , ...z N ) , θ = ( σ 2 z , σ 2 ε , A 1 , ..., A K , p 0 , t 0 , v 0 ) • Maximum Likelihood: Z max θ p ( y | θ ) = p ( y, z | θ ) dz N ✓ ◆ Z X • EM: θ k +1 = argmax θ log p ( y i , z i | θ ) p ( z i | y i , θ k ) dz i | {z } i =1 p ( y i | z i , θ ) p ( z i | θ ) • Distribution from the curved exponential family log p ( y i , z i | θ ) = φ ( θ ) T S ( y i , z i ) − log ( C ( θ )) ( N ) Z X φ ( θ ) T θ k +1 = argmax θ S ( y i , z i ) p ( z i | y i , θ k ) dz i − N log( C ( θ )) i =1 16

<latexit sha1_base64="Z9NCyt3yIeWKeHQl9WaurXkrylo=">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</latexit> Parameter Estimation: stochastic algorithm • SA-EM : replaces integration by one simulation of the hidden variable: sample from , z i,k +1 p ( z i | y i , θ k ) and a stochastic approximation of the sufficient statistics N ! 1 X S k +1 = (1 − ∆ k ) S k + ∆ k S ( y i , z i,k +1 ) N i =1 Maximization step (unchanged) φ ( θ ) T S k +1 − log( C ( θ )) � θ k +1 = argmax θ • MCMC -SAEM: replaces sampling by a single Markov Chain step • For each subject, sample the random effect w.r.t a transition kernel of a geometrically ergodic Markov chain targeting the conditional distribution q ( z i | y i , θ k ) [Delyon, Lavielle, Moulines.’99] 17 [Allassonnière et al.’10]