Visualizing Outliers in High Dimensional Functional Data for task - PowerPoint PPT Presentation

Visualizing Outliers in High Dimensional Functional Data for task fMRI data Exploration Yasser Alem´ an-G´ omez, Ana Arribas Gil, Manuel Desco, Antonio El´ ıas and Juan Romo Instituto UC3M-Banco Santander de Big Data, IBiDat Universidad Carlos III de Madrid, Spain III International Workshop on Advances in FDA Castro Urdiales, 23rd of May 2019

Outline 1. Task Functional magnetic resonance imaging (tfMRI) 1.1 The problem 2. Robust tools for high-dimensional functional data 2.1 Outliers in Multivariate and high-dimensional functional data 2.2 Depth and epigraph index matrices 2.3 The Depthgrams 3. Simulation Study 4. tfMRI data analysis

Task Functional magnetic resonance imaging (tfMRI) • Non-invasive technique for the study of brain function. • Relies on detecting blood flow changes related to brain activity while executing different tasks (reading, moving fingers, etc...). • Higher spatial resolution than any other Source: http://wrrp.psy.ohio-state.edu/img/fmri.jpg non-invasive method ( 1 to 6 mm ). • Signal can be recorded along time. Case study: n = 100 subjects measured over N = 284 or N = 316 seconds at a motor task experiment. Voxel resolution: 2 mm In collaboration with Laboratorio de Imagen 2mm M´ edica, Hospital General Universitario Gregorio on, Spain . Mara˜ n´ 2 / 34

Functional Magnetic Resonance Imaging (fMRI): data n p -dimensional functional objects measured over N time points p ≈ 200000 vxls (from 91 × 109 × 91 cube) time 1 2 3 T-1 T=284 1 subjects 2 n=100 3 / 34

tfMRI: data 1 individual, 1 time point, slices over the axial plane 4 / 34

tfMRI: data All individuals, some voxels: Voxel 51,40,29 Voxel 17,58,29 Voxel 48,82,29 20000 15000 14000 12000 Intensity Intensity Intensity 10000 15000 10000 5000 10000 8000 0 0 100 200 0 100 200 0 100 200 Time (sec.) Time (sec.) Time (sec.) Voxel 31,33,30 Voxel 33,51,30 Voxel 32,70,30 14000 15000 15000 12000 Intensity Intensity Intensity 12000 12000 10000 9000 8000 9000 6000 6000 0 100 200 0 100 200 0 100 200 Time (sec.) Time (sec.) Time (sec.) 5 / 34

tfMRI: data 6 / 34

The problem • Getting insight throuth visualization/dimension reduction • Estimating the central (most representative) brain in a robust way (jointly over voxels and time). • Looking for atypical (outlying) brains. • Assessing heterogeneity and sample composition (association patterns...) 7 / 34

High-dimensional functional data Multivariate functional setting: • We observe x 1 , . . . , x n p -dimensional functions • i.i.d. realizations of a p -variate functional random variable { X ( t ) : t ∈ T } in C ( T ) p . • discretely observed in { t 1 , . . . , t N } ⊆ T High-dimensional functional setting: Multivariate functional data with n << p 8 / 34

Outline 1. Task Functional magnetic resonance imaging (tfMRI) 1.1 The problem 2. Robust tools for high-dimensional functional data 2.1 Outliers in Multivariate and high-dimensional functional data 2.2 Depth and epigraph index matrices 2.3 The Depthgrams 3. Simulation Study 4. tfMRI data analysis

Outliers in (low/high) multivariate functional data • Marginal outliers : magnitude, shape,... functional outliers • Joint outliers : typical observations through marginals but outyling when considered jointly First dimension Second dimension 6 6 5 4 4 x 1 (t) x 2 (t) 3 2 2 1 0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t 10 / 34

Outliers in (low/high) multivariate functional data • Depths/Outlyingness for multivariate functional data: � o MF ( x i ; x 1 , . . . , x n ) ≡ w ( t ) o p ( x i ( t ); x 1 ( t ) , . . . , x n ( t )) dt T 1 z, z 1 , . . . , z n ∈ R p o p ( z ; z 1 , . . . , z n ) = 1 + d p ( z ; z 1 , . . . , z n ) , • Graphical representation of o MF ( x i ) vs dispersion ( o p ( x i ( t j )) j ) • Able to cope with marginal and joint outliers • To be used with any multivariate depth function • Not dessigned to distinguish between different types of outliers • Not dessigned to provide insight on sample composition • Computationally heavy for high-dimensional functional data Hubert, Rousseeuw and Segaert, SMA , 2015 Nieto-Reyes and Cuesta-Albertos, SMA , 2015 Dai and Genton, JCGS , 2018 Rousseeuw, Raymaekers and Hubert, JCGS , 2018 11 / 34

Robust tools for high-dim functional data: depth matrices x 1 , . . . , x n n p -dimensional functions observed in { t 1 , . . . , t N } ⊆ T 1 ( t ) , . . . , x p ( x 1 x 1 ( t ) = 1 ( t )) 2 ( t ) , . . . , x p ( x 1 x 2 ( t ) = 2 ( t )) . . . . . . . . . ( x 1 n ( t ) , . . . , x p x n ( t ) = n ( t )) Depth dimensions matrix � � d F ( x j i ; x j 1 , . . . , x j D d ( x ) = n ) i =1 ,...,n, j =1 ,...,p Depth time matrix D t ( x ) = ( d F ( x i ( t j ); x 1 ( t j ) , . . . , x n ( t j ))) i =1 ,...,n, j =1 ,...,N depth of depths: D d ( x ) and D t ( x ) treated as functional data sets where functional depth can be applied 12 / 34

Robust tools for high-dim functional data: depth matrices In particular we use band-depth related measures (L´ opez-Pintado and Romo, 2009, 2011): • modified band depth for functional data , MBD • modified epigraph index for functional data , MEI • quadratic relationship between MBD and MEI and get MBD d ( x ) and MEI d ( x ) • ij element is the depth/index of curve x j i with respect to j -th marginal sample. • columns of MBD d ( x ) are the individual depths across marginals. (d ≡ dimensions) MBD t ( x ) and MBI t ( x ) • ij element is the depth /index of curve x i ( t j ) with respect to t j time point p -variate sample. • columns of MBD t ( x ) are the individual depths across time points. (t ≡ time) 13 / 34

Relation between MBD(MEI) and MEI(MBD) Result: If a) ( x j i ( t k 1 ) − x j h ( t k 1 ))( x j i ( t k 2 ) − x j h ( t k 2 )) > 0 , k 1 , k 2 ∈ 1 , . . . , N , i � = h , for all j = 1 , . . . , p holds then MBD ( MEI d ( x )) ≤ P (1 − MEI ( MBD d ( x ))) , ∀ x ∈ { x 1 , . . . , x n } where P : [0 , 1] → R is the parabola P ( z ) = a 0 + a 1 z + a 2 n 2 z 2 and a 0 = a 2 = − 1 /n ( n − 1) , a 1 = ( n + 1) / ( n − 1) . Moreover if b) ( x j i ( t k ) − x j h ( t k ))( x ℓ i ( t k ) − x ℓ h ( t k )) > 0 , k = 1 , . . . , N , i � = h , j � = ℓ also holds then MBD ( MEI d ( x )) = P (1 − MEI ( MBD d ( x ))) , ∀ x ∈ { x 1 , . . . , x n } . 14 / 34

Relation between MBD(MEI) and MEI(MBD) First dimension Second dimension Third dimension 2.0 8 6 1.5 6 4 x 1 (t) x 2 (t) x 3 (t) 1.0 4 2 0.5 2 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t t Across Dimensions Across Time ●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ●●●●●●●●●●●●●●●●●●●●●●●●●● ● ● ● ● ● ● ● ● 0.5 ● 0.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 0.4 ● ● ● ● ● ● ● ● ● ● MBD( MEI d ) ● ● MBD( MEI t ) ● ● ● ● ● ● 0.3 0.3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 1−MEI( MBD d ) 1−MEI( MBD t ) 15 / 34

Relation between MBD(MEI) and MEI(MBD) First dimension Second dimension Third dimension 2.0 8 6 1.5 6 4 x 1 (t) x 2 (t) x 3 (t) 1.0 4 2 0.5 2 0 0.0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t t Across Dimensions Across Time ● ● ● ● ● ● ● ● ● ● ● ● ● 0.5 0.5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.4 ● ● ● 0.4 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● MBD( MEI d ) MBD( MEI t ) ● ● ● ● ● ● ● ● ● ● ● 0.3 0.3 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.2 0.2 ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.1 ● ● ● ● ● ● ● ● 0.0 0.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 1−MEI( MBD d ) 1−MEI( MBD t ) 16 / 34