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Kernel Methods for De-noising with Neuroimaging Application

Kernel Methods for De-noising with Neuroimaging Application. Trine Julie Abrahamsen M.Sc. defense, August 10, 2009. Overview. Introduction Kernel Principal Component Analysis The Pre-image Problem Analysis of Cimbi Data Conclusions. Introduction.

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Kernel Methods for De-noising with Neuroimaging Application

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  1. Kernel Methods for De-noising withNeuroimaging Application Trine Julie Abrahamsen M.Sc. defense, August 10, 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 til de-noising • The pre-image problem is a key aspect in achieving good results Thesis 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 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

  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’th PC can be found as While the projection onto the subspace spanned by the first q PCs is given by

  6. 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

  7. 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.

  8. 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 cost function (which should be maximized) reduces to

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

  10. Experiments on the USPS Data Set Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions • : Kwok & Tsang • : Mika et al. • : Dambreville et al. • : Input Space Reg.

  11. Introducing the Cimbi Analysis 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 binding potential quantified from PET scans • Neuroticism, Anxiety, and Vulnerability • N = 129. • Initial experiments on training and test sets • www.cimbi.org

  12. Comparing Pre-image Estimators Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions Vulnerability

  13. Learning curves 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.

  14. Conclusions Introduction - Kernel PCA - The Pre-image Problem - Analysis of Cimbi Data - Conclusions • Several of the current pre-image algorithms were shown to suffer from instabilities • Adding input space distance regularization stabilizes the pre-image with limited sacrifice in terms of de-noising efficiency • Sole limitation lies within tuning λ • 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 • The results could not be verified by bootstrap resampling • Introduce other types of regularization • Investigate other applications of kernel PCA (e.g., outlier detection) • Better performance by varying the kernel and its parameters

  15. References

  16. Kernel Methods for De-noising withNeuroimagingApplication Trine Julie Abrahamsen M.Sc. defense, August 11, 2009

  17. Limitations of the Current Approaches Introduction - Kernel PCA - The Pre-Image Problem - Analysis of Cimbi Data - Conclusions

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

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

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