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´

• mez, 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)

2 / 34

• 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

non-invasive method (1 to 6mm).

• Signal can be recorded along time.

Source: http://wrrp.psy.ohio-state.edu/img/fmri.jpg

Case study: n = 100 subjects measured over N = 284 or N = 316 seconds at a motor task

• experiment. Voxel resolution: 2mm

In collaboration with Laboratorio de Imagen M´ edica, Hospital General Universitario Gregorio Mara˜ n´

• n, Spain.

2mm

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 2 subjects n=100

3 / 34

tfMRI: data

1 individual, 1 time point, slices over the axial plane

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tfMRI: data

All individuals, some voxels:

10000 15000 20000 100 200

Time (sec.) Intensity

Voxel 51,40,29

8000 10000 12000 14000 100 200

Time (sec.) Intensity

Voxel 17,58,29

5000 10000 15000 100 200

Time (sec.) Intensity

Voxel 48,82,29

6000 8000 10000 12000 14000 100 200

Time (sec.) Intensity

Voxel 31,33,30

6000 9000 12000 15000 100 200

Time (sec.) Intensity

Voxel 33,51,30

9000 12000 15000 100 200

Time (sec.) Intensity

Voxel 32,70,30

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...)

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High-dimensional functional data

Multivariate functional setting:

• We observe x1, . . . , xn 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 {t1, . . . , tN} ⊆ 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

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6

First dimension

t x1(t)

0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6

Second dimension

t x2(t)

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Outliers in (low/high) multivariate functional data

• Depths/Outlyingness for multivariate functional data:
• MF (xi; x1, . . . , xn) ≡
• T

w(t)op(xi(t); x1(t), . . . , xn(t))dt

• p(z; z1, . . . , zn) =

1 1 + dp(z; z1, . . . , zn), z, z1, . . . , zn ∈ Rp

• Graphical representation of oMF (xi) vs dispersion(op(xi(tj))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

x1, . . . , xn n p-dimensional functions observed in {t1, . . . , tN} ⊆ T x1(t) = (x1

1(t), . . . , xp 1(t))

x2(t) = (x1

2(t), . . . , xp 2(t))

. . . . . . . . . xn(t) = (x1

n(t), . . . , xp n(t))

Depth dimensions matrix Dd(x) =

• dF (xj

i; xj 1, . . . , xj n)

• i=1,...,n, j=1,...,p

Depth time matrix Dt(x) = (dF (xi(tj); x1(tj), . . . , xn(tj)))i=1,...,n, j=1,...,N depth of depths: Dd(x) and Dt(x) treated as functional data sets where functional depth can be applied

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Robust tools for high-dim functional data: depth matrices

In particular we use band-depth related measures (L´

• pez-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 MBDd(x) and MEId(x)

• ij element is the depth/index of curve xj

i with respect to j-th

marginal sample.

• columns of MBDd(x) are the individual depths across marginals.

(d≡ dimensions) MBDt(x) and MBIt(x)

• ij element is the depth /index of curve xi(tj) with respect to tj

time point p-variate sample.

• columns of MBDt(x) are the individual depths across time points.

(t≡ time)

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Relation between MBD(MEI) and MEI(MBD)

Result:

If a) (xj

i(tk1) − xj h(tk1))(xj i(tk2) − xj h(tk2)) > 0, k1, k2 ∈ 1, . . . , N,

i = h, for all j = 1, . . . , p holds then MBD (MEId(x)) ≤ P (1 − MEI (MBDd(x))) , ∀x ∈ {x1, . . . , xn} where P : [0, 1] → R is the parabola P(z) = a0 + a1z + a2n2z2 and a0 = a2 = −1/n(n − 1), a1 = (n + 1)/(n − 1). Moreover if b) (xj

i(tk) − xj h(tk))(xℓ i(tk) − xℓ h(tk)) > 0, k = 1, . . . , N, i = h, j = ℓ

also holds then MBD (MEId(x)) = P (1 − MEI (MBDd(x))) , ∀x ∈ {x1, . . . , xn}.

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Relation between MBD(MEI) and MEI(MBD)

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6

First dimension

t x1(t)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

Second dimension

t x2(t)

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8

Third dimension

t x3(t)

• 0.0

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDd ) MBD( MEId )

Across Dimensions

• 0.0

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Across Time

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Relation between MBD(MEI) and MEI(MBD)

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6

First dimension

t x1(t)

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0

Second dimension

t x2(t)

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 8

Third dimension

t x3(t)

• ●
• 0.0

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDd ) MBD( MEId )

Across Dimensions

• 0.0

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Across Time

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Relation between MBD(MEI) and MEI(MBD)

0.0 0.2 0.4 0.6 0.8 1.0 −2 2 4 6 8

First dimension

t x1(t)

0.0 0.2 0.4 0.6 0.8 1.0 −4 −2 2 4

Second dimension

t x2(t)

0.0 0.2 0.4 0.6 0.8 1.0 −2 2 4 6 8 10

Third dimension

t x3(t)

• 0.0

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDd ) MBD( MEId )

Across Dimensions

• ●

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Across Time

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The Depthgrams

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6

First dimension

t x1(t)

0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6

Second dimension

t x2(t)

• ●
• 42

86 87 88

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−epi(mbd vector) mbd(epi vector)

Dimensions DepthGram

• 42

86 87 88

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−epi(mbd vector) mbd(epi vector)

Time DepthGram

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The Depthgrams

• Dimensions Depthgram: 2-dimensional representation of

(MBD (MEId(x)) , 1 − MEI (MBDd(x)))

• Time Depthgram: 2-dimensional representation of

(MBD (MEIt(x)) , 1 − MEI (MBDt(x)))

• Time/Correlation Depthgram: 2-dimensional representation of

(MBD (MEIt(˜ x)) , 1 − MEI (MBDt(˜ x))) where ˜ x is an unfolded version of x: ˜ xj

i(t) = xj i(t) j

• k=2

sign(ρ(MEId(x)· k−1, MEId(x)· k)) ρ(·, ·): Pearson’s correlation function

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The Depthgrams

0.0 0.2 0.4 0.6 0.8 1.0 2 4 6

First dimension

t x1(t)

0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 4 5 6

Second dimension

t x2(t)

1 2 3 4 5 6

x(t20) over dimensions

Dimension x(t20)

1 2 −1 1 2 3 4 5 6

x(t50) over dimensions

Dimension x(t50)

1 2

• ●●
• 42

86 87 88

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−epi(mbd vector) mbd(epi vector)

Dimensions DepthGram

• 42

86 87 88

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−epi(mbd vector) mbd(epi vector)

Time DepthGram

• ●
• ●
• ●
• 42

86 87 88

0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−epi(mbd vector) mbd(epi vector)

Time/Correlation DepthGram 20 / 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

Simulation Study: settings

• n = 100, N = 100, p = 10000 and 50000
• nout = 15: nb. of atypical observations
• Contamination rates: c = 0, 0.25, 0.5, 0.75, 1 (proportion of

components) xj

i(t) =

       x0

i (t)hj(t) + εij(t),

i typical 10 + x0

i (t)hj(t) + εij(t),

i mag. out. in jth component s0s

i (t)hj(t) + εij(t),

i shape out. in jth component s0

ℓij(t)hj(t) + εij(t),

i joint out. in jth component εij(t) indp. realizations of a Gaussian process with zero mean and covariance function γ(s, t) = 0.3 exp{−|s − t|/0.3}.

• x0, x0s, h vary from model to model.

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Simulation Study: settings

• Model 1: t ∈ [0, 1]

x0

i (t)

= sin(4πt) + αi, i = 1, . . . , n x0s

i (t)

= cos(4πt + π/2) + αi, i shape out. hj(t) = 1 + 2t1+j/p(1 − t)2−j/p, j = 1, . . . , p αi ∼ N(0, 1) i.i.d

• Model 2: t ∈ [0, 1]

x0

i (t)

= 4t + αi, i = 1, . . . , n x0s

i (t)

= 4t + 2sin(4(t + 0.5)π) + αi, i shape out. hj(t) = 1 + 2t1+j/p(1 − t)2−j/p if j is odd −1 − 2t1+j/p(1 − t)2−j/p if j is even j = 1, . . . , p αi ∼ N(0, 1) i.i.d

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Simulation Study: data

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15

1st dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 −5 5 10

3000th dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 −5 5 10

7001st dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 −5 5 10

10000th dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15

1st dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 −5 5 10

3000th dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 5 10 15

7001st dimension

t x(t)

0.0 0.2 0.4 0.6 0.8 1.0 −5 5 10

10000th dimension

t x(t)

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Simulation Study: results Model 1

DepthGram on Dimensions DepthGram on Time DepthGram on Time/Correlation c = 0 c = 0.25 c = 0.5 c = 0.75 c = 1

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

1−epi(mbd vector) mbd(epi vector) Observations

• Mag. outliers

Shape outliers

• Corr. outliers

Rest of the sample

Frequency on each group

0.25 0.50 0.75 1.00

Simulation Summary − Model 1 , p= 50000 25 / 34

Simulation Study: results Model 2

DepthGram on Dimensions DepthGram on Time DepthGram on Time/Correlation c = 0 c = 0.25 c = 0.5 c = 0.75 c = 1

0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5

1−epi(mbd vector) mbd(epi vector) Observations

• Mag. outliers

Shape outliers

• Corr. outliers

Rest of the sample

Frequency on each group

0.25 0.50 0.75 1.00

Simulation Summary − Model 2 , p= 50000 26 / 34

Simulation Study: low dimensional settings - comparison

• Models 1 and 2 as before with p = 50 and c = 1
• Comparison with outlier detection rules provided by:
• 1. FOM, Functional Outlier Map applied to fDO (functional directional
• utlyingness)

2-dim representation of functional (mean, integrated over time) directional outlyingness vs variability of directional outlyingness. Rousseeuw, Raymaekers and Hubert, JCGS, 2018.

• 2. MS-plot, Magnitude-Shape Plot applied to Directional outlyingness

(p + 1)-dim representation of functional (mean, integrated over time, p dimensional) directional outlyingness vs variability of directional

• utlyingness. Dai and Genton, JCGS, 2018.
• Both p-dimensional and “1-dimensional” versions of both methods
• Non data-driven empirical rule for outlier detection with the

DepthGram (for comparison purposes only)

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Simulation Study: low dimensional setting Model 1

• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDd ) MBD( MEId )

Dimensions DepthGram

• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Time DepthGram

• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Time/Correlation DepthGram

• ●
• ●
• 86

87 88 89 90 91 92 93 94 9596 97 98 99 100

10 20 30 40 0e+00 1e+06 2e+06 3e+06 4e+06

fDO sd(fDO) / (1+fDO)

FOM p−dim

• ●
• ●
• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.00 0.25 0.50 0.75 2 4 6

fDO sd(fDO) / (1+fDO)

FOM 1−dim

• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

5 10 15 4 8 12

||MO|| VO

MS−Plot p−dim

• ●
• ●
• ● ●
• ●
• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.5 1.0 1.5 −4 4 8

MO VO

MS−Plot 1−dim

Green-blue: shape outliers, Orange-brown: magnitude outliers, Pink-purple: joint outliers

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Simulation Study: low dimensional setting Model 2

• ●
• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDd ) MBD( MEId )

Dimensions DepthGram

• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Time DepthGram

• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.1 0.2 0.3 0.4 0.5 0.00 0.25 0.50 0.75 1.00

1−MEI( MBDt ) MBD( MEIt )

Time/Correlation DepthGram

• ●
• 86

8788 89 90 91 92 93 94 95 96 97 9899 100

2 4 6 0e+00 2e+05 4e+05 6e+05

fDO sd(fDO) / (1+fDO)

FOM p−dim

• ●
• ●
• ●
• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

0.0 0.2 0.4 0.6 2 4 6

fDO sd(fDO) / (1+fDO)

FOM 1−dim

• ●
• ●
• ●
• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

1 2 3 4 5 1 2 3 4

log 1+||MO|| log 1+VO

MS−Plot p−dim (log−scale)

• 86

87 88 89 90 91 92 93 94 95 96 97 98 99 100

−2 2 4 2 4 6

MO VO

MS−Plot 1−dim

Green-blue: shape outliers, Orange-brown: magnitude outliers, Pink-purple: joint outliers

29 / 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

tfMRI experiments: description

• n = 100 healthy adults patients
• Brain mask size: p = 192631 voxels
• 2 types of experiments:
• Motor task: N = 284
• Language task: N = 316
• Clean data set: Human conectome project

31 / 34

tfMRI data analysis

• 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Dimensions DepthGram

• 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Time DepthGram

• 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Time/Correlation DepthGram

• 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 6465 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Dimensions DepthGram

• ●
• 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Time DepthGram

• ●
• 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

Time/Correlation DepthGram

32 / 34

tfMRI experiments: description

• Analysis took around 30 min to run on each experiment (personal

computer)

• Voxel heterogeneity much more important in the language than in

the moto experiment

• Central individuals are different in the two experiments, as expected.
• Identification of:
• 39 motor experiment: motion issues
• 84 language experiment: motion issues
• 81 language experiment: mild form of squizophrenia

33 / 34

Conclusions

34 / 34

• Dimension reduction and visualization tool for high-dimensional

functional settings

• Computationally feasable
• Insight on outlier presence, sample composition, association patterns

across dimensions

Conclusions

34 / 34

• Dimension reduction and visualization tool for high-dimensional

functional settings

• Computationally feasable
• Insight on outlier presence, sample composition, association patterns

across dimensions

ana.arribas@uc3m.es

Thanks!