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Robust and Nonparametric StatisticalTools for Big Data in Neuroscience
Henry Laniado.Seminar
PhD in Mathematical Engineering
August 18, 2017
Motivation: fMRI problem
Figure: Functional Magnetic Resonance
Motivation: fMRI problem
Figure: Stimulus
Motivation: fMRI problem
Figure: 900000 Voxels
Motivation: fMRI problemThe theoretical activity is compared with that observed in e achvoxel.
Figure: Activation and non-activation
Motivation: fMRI problemThe theoretical activity is compared with that observed in e achvoxel.
Figure: Activation and non-activation
Motivation: fMRI problem
Figure: Activation and non-activation
Motivation: fMRI problem
Figure: Activation and non-activation
Motivation: fMRI problem
Figure: Identification
Main GoalFind robust and non parametric estimators to explain lineardependence and apply them in the construction of covariancematrices to improve the significance of linear multivariatestatistical models in the presence of outliers
Main GoalFind robust and non parametric estimators to explain lineardependence and apply them in the construction of covariancematrices to improve the significance of linear multivariatestatistical models in the presence of outliers
Hypothesis 1Significance in linear models is too sensitive in the presenc e ofoutliers.
Main GoalFind robust and non parametric estimators to explain lineardependence and apply them in the construction of covariancematrices to improve the significance of linear multivariatestatistical models in the presence of outliers
Hypothesis 1Significance in linear models is too sensitive in the presenc e ofoutliers.
Hypothesis 2Significance in linear models can be improved with robustestimators or non parametric estimators
Outline
1. Robustness: a fast review
2. General Lineal Model (GLM)
3. Robust GLM, with an explanatory variables
4. Robust GLM, multivariate case
5. Non-parametric estimators
6. Conclusions
7. Research lines
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Population: unknown parameters. θ =∫
Population: unknown parameters. θ =∫
Population: unknown parameters. θ =∫
Sample: estimators. θ̂ =∑
Population: unknown parameters. θ =∫
Sample: estimators. θ̂ =∑
µ =
∫
f(x)dx the estimator X̄ =
∑
xi
n
Example
Table: Simulated data from N(0, 1)
0.354 -0.838 0.1204 -0.353 -0.3853 2.1261 -0.268-0.16 -1.038 -0.273 0.7939 0.45153 0.858 0.6697
-1.838 2.5976 -0.282 0.1757 -0.5986 0.9407 -0.036-0.531 0.6208 1.4303 -1.044 -0.1349 0.2476 -0.397-1.432 1.0849 -1.12 0.6788 -1.1078 -0.245 0.5031-1.136 -0.011 1.5668 0.6654 -0.6194 -1.481 -0.4462.0077 -0.898 -0.886 -2.145 -2.435 -0.396 -0.116
Example
Table: Simulated data from N(0, 1)
0.354 -0.838 0.1204 -0.353 -0.3853 2.1261 -0.268-0.16 -1.038 -0.273 0.7939 0.45153 0.858 0.6697
-1.838 2.5976 -0.282 0.1757 -0.5986 0.9407 -0.036-0.531 0.6208 1.4303 -1.044 -0.1349 0.2476 -0.397-1.432 1.0849 -1.12 0.6788 -1.1078 -0.245 0.5031-1.136 -0.011 1.5668 0.6654 -0.6194 -1.481 -0.4462.0077 -0.898 -0.886 -2.145 -2.435 -0.396 -0.116
X̄ =
∑49i=1 xi
49= −0.09
Example
Table: Data from N(0, 1). Contaminated data
0.354 -0.838 0.1204 -0.353 -0.3853 2.1261 -0.268-0.16 -1.038 -0.273 8.2356 0.45153 0.858 0.6697
-1.838 2.5976 -0.282 0.1757 -0.5986 0.9407 12.266-3.82 0.6208 1.4303 -1.044 -0.1349 0.2476 -0.397
-1.432 1.0849 -1.12 0.6788 -1.1078 -0.245 0.5031-1.136 -0.011 1.5668 0.6654 -0.6194 -1.481 -0.4462.0077 -0.898 -0.886 -2.145 -5.1321 -0.396 9.266
Example
Table: Data from N(0, 1). Contaminated data
0.354 -0.838 0.1204 -0.353 -0.3853 2.1261 -0.268-0.16 -1.038 -0.273 8.2356 0.45153 0.858 0.6697
-1.838 2.5976 -0.282 0.1757 -0.5986 0.9407 12.266-3.82 0.6208 1.4303 -1.044 -0.1349 0.2476 -0.397
-1.432 1.0849 -1.12 0.6788 -1.1078 -0.245 0.5031-1.136 -0.011 1.5668 0.6654 -0.6194 -1.481 -0.4462.0077 -0.898 -0.886 -2.145 -5.1321 -0.396 9.266
X̄ =
∑49i=1 xi
49= 0.4252
Example
Table: Data from N(0, 1). Contaminated data
-5.132 -3.82 -2.1448 -1.4809 -1.4318 -1.14 -1.1199-1.108 -1.044 -1.0384 -0.8979 -0.886 -0.84 -0.6194-0.599 -0.531 -0.4462 -0.3974 -0.3956 -0.39 -0.2819-0.273 -0.268 -0.2455 -0.1604 -0.1349 -0.01 0.12040.176 0.2476 0.354 0.45153 0.5031 0.621 0.66540.67 0.6788 0.7939 0.85803 0.9407 1.085 1.4303
1.567 2.0077 2.1261 2.59761 8.2356 9.266 12.266
X̄ =
∑49i=1 xi
49= 0.4252
Example
Table: Data from N(0, 1). Contaminated data
X X X -1.4809 -1.4318 -1.14 -1.1199-1.108 -1.044 -1.0384 -0.8979 -0.886 -0.84 -0.6194-0.599 -0.531 -0.4462 -0.3974 -0.3956 -0.39 -0.2819-0.273 -0.268 -0.2455 -0.1604 -0.1349 -0.01 0.12040.176 0.2476 0.354 0.45153 0.5031 0.621 0.66540.67 0.6788 0.7939 0.85803 0.9407 1.085 1.4303
1.567 2.0077 2.1261 2.59761 X X X
Trimmed mean
X̄α =
∑k2
i=k1x[i]
k2 − k1 + 1
k1 = [α2 n] k2 = [n(1− α2 )]
X̄0.1 = 0.0503
Depth
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Dependent uniform distribution in dimension 2
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1Central Region
Figure: Random Tukey Depth
Normal distribution in dimension 3
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Central Region
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Projection on X3 and X2
Figure: Random Tukey Depth
Central Region: univariate case
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Figure: Data
Central Region: univariate case
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Figure: Median
Central Region: univariate case
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Figure: Central Region 10%
Central Region: univariate case
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Figure: Central Region 20%
Central Region: univariate case
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Figure: Central Region 30%
Central Region: univariate case
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Figure: Central Region 40%
Central Region: univariate case
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Figure: Central Region 50%
Central Region: univariate case
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Values
Col
umn
Num
ber
Figure: Box plot
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
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Figure: Data
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
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Figure: deepest point
Central Region: bivariate case
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Figure: Central Region 10%
Central Region: bivariate case
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Figure: Central Region 30%
Central Region: bivariate case
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Figure: Central Region 40%
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
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Figure: Central Region 50%
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Central Region: functional case
0 10 20 30 40 50 60 70−2
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Figure: Data
Central Region: functional case
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Figure: deepest curve
Central Region: functional case
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Central Region: functional case
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Central Region: functional case
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Central Region: functional case
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Figure: Central Region 50%
Central Region: functional case
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Figure: Central Region , Extremes
Central Region: functional case
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Figure: Data
Central Region: functional case
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Figure: Contaminated Data
Central Region: functional case
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Figure: Central Region , Extremes
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
General Linear Model
0 0.2 0.4 0.6 0.8 1−0.2
0
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y
General Linear Model
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Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Significant variable
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1onsets
0 20 40 60 80 100 120−2
0
2
4
6Explanatory Signal (HRF)
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
Y = X +N(0, 1)
Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
0 20 40 60 80 100 120−5
0
5
10Points on the time where the signal is contaminated
0 20 40 60 80 100 120−20
−10
0
10Signal contaminated
The red points will be generated by:◮ Normal (0, σ2), σ2 varying from 1 to 100.
Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
0 20 40 60 80 100 120−5
0
5
10Points on the time where the signal is contaminated
0 20 40 60 80 100 120−10
0
10
20
30Signal contaminated
The red points will be generated by:◮ Normal (µ, σ2), both varying from 1 to 100.
Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
0 20 40 60 80 100 120−5
0
5
10Points on the time where the signal is contaminated
0 20 40 60 80 100 120−10
0
10
20Signal contaminated
The red points will be generated by:◮ Normal (µ, 1), µ varying from 1 to 100.
Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−10
0
10
20
30T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−2
0
2
4B1 Variations
GLMRobustGLMReal−Beta1=1.01
0 20 40 60 80 100 120−4
−2
0
2
4B0 Variations
GLMRobustGLMReal−Beta0=0.1
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
Significant variable and Contaminating a 5%
0 20 40 60 80 100 1200
10
20
30T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 1200
2
4
6B1 Variations
GLMRobustGLMReal−Beta1
0 20 40 60 80 100 120−2
0
2
4
6B0 Variations
GLMRobustGLMReal−Beta0
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
Significant variable and Contaminating a 5%
0 20 40 60 80 100 1200
10
20
30T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 1200
1
2
3B1 Variations
GLMRobustGLMReal−Beta1
0 20 40 60 80 100 1200
1
2
3B0 Variations
GLMRobustGLMReal−Beta0
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
No Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−4
−2
0
2
4T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−4
−2
0
2B1 Variations
GLMRobustGLMReal−Beta1
0 20 40 60 80 100 120−4
−2
0
2
4B0 Variations
GLMRobustGLMReal−Beta0
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
No Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−1
0
1
2
3T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−2
0
2
4
6B1 Variations
GLMRobustGLMReal−Beta1
0 20 40 60 80 100 120−2
0
2
4
6B0 Variations
GLMRobustGLMReal−Beta0
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
No Significant variable and Contaminating a 5%
0 20 40 60 80 100 120−1
0
1
2T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−1
0
1
2B1 Variations
GLMRobustGLMReal−Beta1
0 20 40 60 80 100 120−1
0
1
2
3B0 Variations
GLMRobustGLMReal−Beta0
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
0 20 40 60 80 100 120−5
0
5
10Points on the time where the signal is contaminated
0 20 40 60 80 100 120−20
−10
0
10
20Signal contaminated
The red points will be generated by:◮ Normal (0, σ2), σ2 varying from 1 to 100.
Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
0 20 40 60 80 100 120−5
0
5
10Points on the time where the signal is contaminated
0 20 40 60 80 100 120−20
0
20
40Signal contaminated
The red points will be generated by:◮ Normal (µ, σ2), both varying from 1 to 100.
Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−5
0
5
10Explained Signal (YfMRI)
0 20 40 60 80 100 120−5
0
5
10Points on the time where the signal is contaminated
0 20 40 60 80 100 120−5
0
5
10
15Signal contaminated
The red points will be generated by:◮ Normal (µ, 1), µ varying from 1 to 100.
Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−10
0
10
20
30T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−4
−2
0
2
4B1 Variations
GLMRobustGLMReal−Beta1=1.01
0 20 40 60 80 100 120−10
−5
0
5
10B0 Variations
GLMRobustGLMReal−Beta0=0.1
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−10
0
10
20
30T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−4
−2
0
2
4B1 Variations
GLMRobustGLMReal−Beta1=1.01
0 20 40 60 80 100 120−10
0
10
20
30B0 Variations
GLMRobustGLMReal−Beta0=0.1
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−10
0
10
20
30T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−1
0
1
2B1 Variations
GLMRobustGLMReal−Beta1=1.01
0 20 40 60 80 100 1200
5
10
15B0 Variations
GLMRobustGLMReal−Beta0=0.1
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
No Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−4
−2
0
2T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−4
−2
0
2B1 Variations
GLMRobustGLMReal−Beta1=−0.0381
0 20 40 60 80 100 120−10
−5
0
5
10B0 Variations
GLMRobustGLMReal−Beta0=0.01
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
No Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−4
−2
0
2T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−4
−2
0
2B1 Variations
GLMRobustGLMReal−Beta1=−0.0381
0 20 40 60 80 100 120−10
0
10
20B0 Variations
GLMRobustGLMReal−Beta0=0.01
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
No Significant variable and Contaminating a 10%
0 20 40 60 80 100 120−2
−1
0
1
2T−value Variations
GLMRobustGLMT−value=2
0 20 40 60 80 100 120−2
−1
0
1B1 Variations
GLMRobustGLMReal−Beta1=−0.0381
0 20 40 60 80 100 1200
5
10
15B0 Variations
GLMRobustGLMReal−Beta0=0.01
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Significant variable
0 10 20 30 40 50 60 70 800
0.2
0.4
0.6
0.8
1onsets
0 10 20 30 40 50 60 70 80−2
0
2
4
6Explanatory Signal (HRF)
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
Y = X1 + β2X2 + · · ·+ β6X6 +N(0, 1)
Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
0 10 20 30 40 50 60 70 80−5
0
5
10Points on the time where the signal is contaminated
0 10 20 30 40 50 60 70 80−5
0
5
10Signal contaminated
◮ Normal (0, σ2), σ2 varying from 1 to 100.
Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
0 10 20 30 40 50 60 70 80−5
0
5
10Points on the time where the signal is contaminated
0 10 20 30 40 50 60 70 80−5
0
5
10Signal contaminated
◮ Normal (µ, σ2), both varying from 1 to 100.
Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
0 10 20 30 40 50 60 70 80−5
0
5
10Points on the time where the signal is contaminated
0 10 20 30 40 50 60 70 80−5
0
5
10
15Signal contaminated
◮ Normal (µ, 1), µ varying from 1 to 100.
Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80 90 100−10
0
10
20
30T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−1
0
1
2
3B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−10
−5
0
5
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80 90 100−10
0
10
20
30T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−1
0
1
2
3B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−5
0
5
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80 90 100−10
0
10
20
30T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 1000.8
1
1.2
1.4
1.6B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−2
0
2
4B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
No Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80 90 100−2
−1
0
1
2T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−2
−1
0
1
2B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−10
−5
0
5
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
No Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80 90 100−2
−1
0
1
2T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−1
0
1
2B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−5
0
5
10
15B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
No Significant variable and Contaminating a 5%
0 10 20 30 40 50 60 70 80 90 100−2
−1
0
1
2T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−0.2
0
0.2
0.4
0.6B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−2
0
2
4B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
0 10 20 30 40 50 60 70 80−5
0
5
10Points on the time where the signal is contaminated
0 10 20 30 40 50 60 70 80−20
−10
0
10Signal contaminated
◮ Normal (0, σ2), σ2 varying from 1 to 100.
Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
0 10 20 30 40 50 60 70 80−5
0
5
10Points on the time where the signal is contaminated
0 10 20 30 40 50 60 70 80−5
0
5
10
15Signal contaminated
◮ Normal (µ, σ2), both varying from 1 to 100.
Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80−5
0
5
10Explained Signal (YfMRI)
0 10 20 30 40 50 60 70 80−5
0
5
10Points on the time where the signal is contaminated
0 10 20 30 40 50 60 70 80−5
0
5
10
15Signal contaminated
◮ Normal (µ, 1), µ varying from 1 to 100.
Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80 90 100−10
0
10
20
30T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−5
0
5B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−10
−5
0
5
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80 90 100−10
0
10
20
30T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−20
−10
0
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80 90 100−10
0
10
20
30T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 1000
2
4
6B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−6
−4
−2
0
2B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
No Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80 90 100−4
−2
0
2
4T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−2
0
2
4B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−10
−5
0
5
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (0, σ2), σ2 varying from 1 to 100.◮ X-axis represents σ2 values varying from 1 to 100.
No Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80 90 100−2
0
2
4T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 100−2
0
2
4
6B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−20
−10
0
10B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, σ2), both varying from 1 to 100.◮ X-axis represents [µ; σ2] values both varying from 1 to 100.
No Significant variable and Contaminating a 10%
0 10 20 30 40 50 60 70 80 90 100−2
0
2
4T−value Variations
GLMRobustGLMMahaGLM
TrimingR2
T−value=2
0 10 20 30 40 50 60 70 80 90 1000
2
4
6B1 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta1
0 10 20 30 40 50 60 70 80 90 100−6
−4
−2
0
2B0 Variations
GLMRobustGLMMahaGLM
TrimingR2
Real−Beta0
◮ Normal (µ, 1), µ varying from 1 to 100.◮ X-axis represents µ values varying from 1 to 100.
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Non-parametric estimation for Σ
Pearson correlation coefficientThe Pearson correlation matrix is related to the covariancematrix by
ρp(i, j) =C(i, j)
√
C(i, i)C(j, j)(1)
Our methodology basically changes in (1) the Pearsoncorrelation coefficient ρp(i, j) by a non parametric correlationcoefficient as:
Non-parametric estimation for Σ
Pearson correlation coefficientThe Pearson correlation matrix is related to the covariancematrix by
ρp(i, j) =C(i, j)
√
C(i, i)C(j, j)(1)
Our methodology basically changes in (1) the Pearsoncorrelation coefficient ρp(i, j) by a non parametric correlationcoefficient as:
◮ Kendall ρk(i, j)
◮ Spearman ρs(i, j)
Non-parametric estimation for Σ
Pearson correlation coefficientThe Pearson correlation matrix is related to the covariancematrix by
ρp(i, j) =C(i, j)
√
C(i, i)C(j, j)(1)
Our methodology basically changes in (1) the Pearsoncorrelation coefficient ρp(i, j) by a non parametric correlationcoefficient as:
◮ Kendall ρk(i, j)
◮ Spearman ρs(i, j)
◮ Robust version ρr(i, j)
Kendall
ρk(i, j) =2 ∗ (Nc −Nd)
N(N − 1)
Spearman
ρs(i, j) = 1−6∑
d2in(n2
− 1)
Robust version
ρr(i, j) =Cr(i, j)
√
Cr(i, i)Cr(j, j)
where,
Cr(i, j) = median{(xi − median(xi))(xi − median(xi))}
Non-parametric estimation for Σ
0 0.2 0.4 0.6 0.8 1−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
Figure: ρp = 0.81, ρk = ρs = ρr = 1
Non-parametric estimation for Σ
−4 −2 0 2 4 6 8 10 12−4
−2
0
2
4
6
8
10
12
Figure: ρp = 0.86, ρk = 0.15, ρs = 0.23, ρr = 0.005
Non-parametric estimation for Σ
−4 −2 0 2 4 6 8 10 12−4
−3
−2
−1
0
1
2
3
4
Figure: ρp = 0.61, ρk = 0.77, ρs = 0.89, ρr = 0.83
Non-parametric estimation for Σ
−3 −2 −1 0 1 2 3−3
−2
−1
0
1
2
3
Figure: ρp = 0.94, ρk = 0.85, ρs = 0.93, ρr = 0.92
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Conclusions
◮ We have introduced the fMRI problem and the importance ofusing robust statistical techniques to study the brain
Conclusions
◮ We have introduced the fMRI problem and the importance ofusing robust statistical techniques to study the brain
◮ A fast review of different approaches to estimate thecovariance matrix has been made
Conclusions
◮ We have introduced the fMRI problem and the importance ofusing robust statistical techniques to study the brain
◮ A fast review of different approaches to estimate thecovariance matrix has been made
◮ The GLM is quite sensitive in the presence of outliers
Conclusions
◮ We have introduced the fMRI problem and the importance ofusing robust statistical techniques to study the brain
◮ A fast review of different approaches to estimate thecovariance matrix has been made
◮ The GLM is quite sensitive in the presence of outliers◮ The fMRI based on GLM has very hight false discovery rate.
Conclusions
◮ We have introduced the fMRI problem and the importance ofusing robust statistical techniques to study the brain
◮ A fast review of different approaches to estimate thecovariance matrix has been made
◮ The GLM is quite sensitive in the presence of outliers◮ The fMRI based on GLM has very hight false discovery rate.◮ The estimations introduced here improve the performance
of GLM
Robustness:a fast review
Central Region: bivariate case
Central Region: functional case
General Linear Model (GLM)
Robust GLM, with an explanatory variableContaminating a 5%Contaminating a 10%
Robust GLM, Multiavarite caseContaminating a 5%Contaminating a 10%
Non parametric estimators
Conclusions
Research Lines
Research lines
◮ Find good estimations for the variance and covariancematrix and its properties
Research lines
◮ Find good estimations for the variance and covariancematrix and its properties
◮ Implement the estimations to the fMRI problem
Research lines
◮ Find good estimations for the variance and covariancematrix and its properties
◮ Implement the estimations to the fMRI problem◮ Find some criterium to know the optimal cut-off in the
robust estimations
Research lines
◮ Find good estimations for the variance and covariancematrix and its properties
◮ Implement the estimations to the fMRI problem◮ Find some criterium to know the optimal cut-off in the
robust estimations◮ To design estimators with a good computational
performance in high dimensional data and big data
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