robust and nonparametric statistical tools for big data in ... · example table: data from n(0,1)....
TRANSCRIPT
![Page 1: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/1.jpg)
Robust and Nonparametric StatisticalTools for Big Data in Neuroscience
Henry Laniado.Seminar
PhD in Mathematical Engineering
August 18, 2017
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Motivation: fMRI problem
Figure: Functional Magnetic Resonance
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Motivation: fMRI problem
Figure: Stimulus
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Motivation: fMRI problem
Figure: 900000 Voxels
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Motivation: fMRI problemThe theoretical activity is compared with that observed in e achvoxel.
Figure: Activation and non-activation
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Motivation: fMRI problemThe theoretical activity is compared with that observed in e achvoxel.
Figure: Activation and non-activation
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Motivation: fMRI problem
Figure: Activation and non-activation
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Motivation: fMRI problem
Figure: Activation and non-activation
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Motivation: fMRI problem
Figure: Identification
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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
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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.
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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
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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
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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
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Population: unknown parameters. θ =∫
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Population: unknown parameters. θ =∫
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Population: unknown parameters. θ =∫
Sample: estimators. θ̂ =∑
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Population: unknown parameters. θ =∫
Sample: estimators. θ̂ =∑
µ =
∫
f(x)dx the estimator X̄ =
∑
xi
n
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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
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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
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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
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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
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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
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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
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Depth
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Depth
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Depth
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Depth
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Depth
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Dependent uniform distribution in dimension 2
0 0.2 0.4 0.6 0.8 10
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1Central Region
Figure: Random Tukey Depth
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Normal distribution in dimension 3
−4−2
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−2
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Central Region
−4 −3 −2 −1 0 1 2 3−4
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Projection on X1 and X2
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Projection on X1 and X3
−4 −3 −2 −1 0 1 2 3−4
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Projection on X3 and X2
Figure: Random Tukey Depth
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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Figure: Data
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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Figure: Median
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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Figure: Central Region 10%
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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Figure: Central Region 20%
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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Figure: Central Region 30%
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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Figure: Central Region 40%
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Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
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0.6
0.8
1
Figure: Central Region 50%
![Page 39: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/39.jpg)
Central Region: univariate case
−4 −3 −2 −1 0 1 2 3 4 5−1
−0.5
0
0.5
1
−4 −3 −2 −1 0 1 2 3 4 5
1
Values
Col
umn
Num
ber
Figure: Box plot
![Page 40: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/40.jpg)
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
![Page 41: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/41.jpg)
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
−2
−1
0
1
2
3
4
Figure: Data
![Page 42: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/42.jpg)
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
−2
−1
0
1
2
3
4
Figure: deepest point
![Page 43: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/43.jpg)
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
−2
−1
0
1
2
3
4
Figure: Central Region 10%
![Page 44: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/44.jpg)
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
−2
−1
0
1
2
3
4
Figure: Central Region 30%
![Page 45: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/45.jpg)
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
−2
−1
0
1
2
3
4
Figure: Central Region 40%
![Page 46: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/46.jpg)
Central Region: bivariate case
−4 −3 −2 −1 0 1 2 3 4−3
−2
−1
0
1
2
3
4
Figure: Central Region 50%
![Page 47: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/47.jpg)
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
![Page 48: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/48.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Data
![Page 49: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/49.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: deepest curve
![Page 50: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/50.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Central Region 10%
![Page 51: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/51.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Central Region 20%
![Page 52: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/52.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Central Region 20%
![Page 53: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/53.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Central Region 50%
![Page 54: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/54.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Central Region , Extremes
![Page 55: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/55.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Data
![Page 56: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/56.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Contaminated Data
![Page 57: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/57.jpg)
Central Region: functional case
0 10 20 30 40 50 60 70−2
0
2
4
6
8
10
12
Figure: Central Region , Extremes
![Page 58: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/58.jpg)
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
![Page 59: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/59.jpg)
General Linear Model
0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
y
![Page 60: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/60.jpg)
General Linear Model
0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
y
![Page 61: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/61.jpg)
General Linear Model
0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x
y
![Page 62: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/62.jpg)
General Linear Model
0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x
y
![Page 63: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/63.jpg)
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
![Page 64: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/64.jpg)
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)
![Page 65: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/65.jpg)
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.
![Page 66: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/66.jpg)
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.
![Page 67: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/67.jpg)
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.
![Page 68: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/68.jpg)
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.
![Page 69: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/69.jpg)
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.
![Page 70: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/70.jpg)
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.
![Page 71: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/71.jpg)
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.
![Page 72: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/72.jpg)
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.
![Page 73: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/73.jpg)
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.
![Page 74: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/74.jpg)
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.
![Page 75: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/75.jpg)
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.
![Page 76: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/76.jpg)
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.
![Page 77: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/77.jpg)
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.
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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.
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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.
![Page 80: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/80.jpg)
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.
![Page 81: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/81.jpg)
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.
![Page 82: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/82.jpg)
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.
![Page 83: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/83.jpg)
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
![Page 84: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/84.jpg)
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)
![Page 85: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/85.jpg)
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.
![Page 86: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/86.jpg)
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.
![Page 87: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/87.jpg)
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.
![Page 88: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/88.jpg)
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.
![Page 89: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/89.jpg)
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.
![Page 90: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/90.jpg)
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.
![Page 91: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/91.jpg)
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.
![Page 92: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/92.jpg)
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.
![Page 93: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/93.jpg)
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.
![Page 94: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/94.jpg)
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.
![Page 95: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/95.jpg)
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.
![Page 96: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/96.jpg)
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.
![Page 97: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/97.jpg)
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.
![Page 98: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/98.jpg)
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.
![Page 99: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/99.jpg)
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.
![Page 100: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/100.jpg)
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.
![Page 101: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/101.jpg)
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.
![Page 102: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/102.jpg)
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.
![Page 103: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/103.jpg)
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
![Page 104: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/104.jpg)
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:
![Page 105: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/105.jpg)
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)
![Page 106: Robust and Nonparametric Statistical Tools for Big Data in ... · 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](https://reader034.vdocuments.site/reader034/viewer/2022051912/6003459bfde28667e65d0f28/html5/thumbnails/106.jpg)
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)
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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))}
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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
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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
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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
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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
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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
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Conclusions
◮ We have introduced the fMRI problem and the importance ofusing robust statistical techniques to study the brain
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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
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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
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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.
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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
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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
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Research lines
◮ Find good estimations for the variance and covariancematrix and its properties
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Research lines
◮ Find good estimations for the variance and covariancematrix and its properties
◮ Implement the estimations to the fMRI problem
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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
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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