Kernel Methods for De-noising with Neuroimaging Application

1. Kernel Methods for De-noising withNeuroimagingApplication Trine Julie Abrahamsen September 8, 2009

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

3. Introduction Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data – Conclusions • Kernel PCA for de-noising • The pre-image problem is a keyaspect in achievinggoodresults 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. IntroducingKernels Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions 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 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’thPC 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 PCA Kernel PCA

7. The Pre-Image Problem 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 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 Which is equivalent to minimizing For RBF kernels the costfunction (whichshouldbemaximized) reduces to

10. Experiments on the USPS Data Set Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions USPS digitsGaussiannoiseadded Mika et al. Input spaceregularization Hull 1994

11. Experiments on the USPS Data Set Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions 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 CimbiAnalysis Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions • 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 • www.cimbi.org Frøkjaer 2008

13. ApplyingKernel PCA De-noising Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Vulnerability

14. Learningcurves Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions • 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 Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - 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.’smethodprovedveryefficient • 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 • Introduceother types of regularization • Investigateotherapplications of kernel PCA (e.g., outlierdetection) • Improve performance by varying the kernel and its parameters • Includenon-linearadjustment for age and gender Future Work

16. References

17. Kernel Methods for De-noising withNeuroimagingApplication 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

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

20. Permutation Test Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions • Non-parametric test • Re-arrangetrait 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

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