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Kernel Methods for De-noising with Neuroimaging Application

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Presentation on theme: "Kernel Methods for De-noising with Neuroimaging Application"— Presentation transcript:

1 Kernel Methods for De-noising with Neuroimaging Application
Trine Julie Abrahamsen September 8, 2009

2 Overview Introduction Kernel Principal Component Analysis
The Pre-image Problem Analysis of Cimbi Data Conclusions

3 Introduction Objective Kernel PCA for de-noising
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data – Conclusions Introduction Kernel PCA for de-noising The pre-image problem is a key aspect in achieving good results Objective The over all aim of this project is two-fold • To investigate the pre-image problem • Apply kernel methods for de-noising on neuroimaging data

4 Introducing Kernels The idea of kernel methods Often
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Introducing Kernels The idea of kernel methods Often Definition of kernel function: The Gaussian kernel Mika et al. 1999 Schölkopf et al. 2001

5 Kernel Principal Component Analysis
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Kernel Principal Component Analysis Linear PCA is performed in feature space. Thus, the first PC can be found as the normal direction, v1 , by All solutions must lie in the span of the training images, hence, The projection of onto the i’th PC can be found as While the projection onto the subspace spanned by the first q PCs is given by Schölkopf et al. 1998

6 Kernel PCA - Illustration
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Kernel PCA - Illustration PCA Kernel PCA

7 Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions
The pre-image problem = reconstruction of point in input space from feature space point Ill-posed due to properties of the -map. Relax search to find approximate pre-image Common methods seek to minimize the feature space distance where Mika et al. 1999 Schölkopf et al. 1999

8 Overview of Current Estimation Schemes
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Overview of Current Estimation Schemes Mika et al. Kwok & Tsang Dambreville et al.

9 Input Space Distance Regularization
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Input Space Distance Regularization Which is equivalent to minimizing For RBF kernels the cost function (which should be maximized) reduces to

10 Experiments on the USPS Data Set
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Experiments on the USPS Data Set USPS digits Gaussian noise added Mika et al Input space regularization Hull 1994

11 Experiments on the USPS Data Set
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Experiments on the USPS Data Set Evaluating the stability by confidence intervals on the mean squared error. Kwok & Tsang (2004) Mika et al. (1999) Dambreville et al. (2006) Input Space Reg.

12 Introducing the Cimbi Analysis
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Introducing the Cimbi Analysis De-noising for refining statistical significance D = 14 regions representing the frontolimbic area. Serotonin receptor binding potential quantified from PET scans Neuroticism, Anxiety, and Vulnerability N = 129. Initial experiments on training and test sets Frøkjaer 2008

13 Applying Kernel PCA De-noising
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Applying Kernel PCA De-noising Vulnerability Original De-noised Neuroticism p=0.11 p=0.057* Anxiety p=0.063* p=0.015** Vulnerability p=0.039** p=0.018**

14 Learning curves 100 subsets sampled without replacement
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Learning curves 100 subsets sampled without replacement Data set sizes 5,10,…,125,129 c chosen as 5th percentile and q chosen so 65% of variance is described Dambreville et al.

15 Conclusions Future Work
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Conclusions Adding input space distance regularization stabilizes the pre-image with limited sacrifice in terms of de-noising efficiency For the Cimbi data Dambreville et al.’s method proved very efficient When working on all 129 subjects a remarkable decrease in p-value could be achieved for both neuroticism, vulnerability, and anxiety Derive guidelines for choosing the regularization parameter λ Introduce other types of regularization Investigate other applications of kernel PCA (e.g., outlier detection) Improve performance by varying the kernel and its parameters Include non-linear adjustment for age and gender Future Work

16 References

17 Kernel Methods for De-noising with Neuroimaging Application
Trine Julie Abrahamsen September 8, 2009

18 Many local minima with almost equal value
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Many local minima with almost equal value

19 Distance distortions for non-linear kernels
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Distance distortions for non-linear kernels

20 Permutation Test Non-parametric test Re-arrange trait score on BP
Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Permutation Test Non-parametric test Re-arrange trait score on BP p-value is found as the proportion of sampled permutations where the correlation is greater or equal to the correlation found on the original data.

21 Experiments on the USPS Data Set
Introduction - Kernel PCA - The Pre-Image Problem - Analysis of Cimbi Data - Conclusions Experiments on the USPS Data Set

22 Introduction - Kernel PCA - The Pre-Image Problem - Analysis of Cimbi Data - Conclusions
Bootstrap Resampling Log(p) after de-noising Log(p) before de-noising


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