Conditional Fuzzy C Means

1. Conditional Fuzzy C Means A fuzzy clustering approach for mining event-related dynamics Christos N. Zigkolis

2. Contents • The problem • Our approach • Fuzzy Clustering • Conditional Fuzzy Clustering • Graph-Theoretic Visualization techniques • The experiments and the datasets • Applications • Future Work • Conclusions Aristotle University of Thessaloniki

3. The problem Visualizing the variability of MEG responses understanding the single-trial variability Describe the single-trial (EEG) variability in the presence of artifacts make single-trial analysis robust, robust prototyping Aristotle University of Thessaloniki

4. Our approach CONDITIONAL criteria grades content constraints FUZZY 0 or 1 [0, 1] partial membership CLUSTERING creating clusters Aristotle University of Thessaloniki

5. Clustering Fuzzy Clustering Every Pattern to only one cluster Every Pattern to every cluster with partial membership Aristotle University of Thessaloniki

6. Fuzzy C Means XdataNxp U membership matrix Centroids Objective function CONTINUE STOP Iterative procedure Aristotle University of Thessaloniki

7. FCM 2D Example compact groups spurious patterns FCM sensitivity to noisy data Aristotle University of Thessaloniki

8. Conditional Fuzzy Clustering The presence of Condition(s) mark Pattern Condition(s) Aristotle University of Thessaloniki

9. Conditional Fuzzy C Means scaled to [0, 1] <Xdata, F>  CFCM  <U, Centroids> F affects the computations of U matrix and consequently the centroids. Aristotle University of Thessaloniki

10. FCM VS CFCM FCM CFCM uij uij uij uij Fk VS uij uij uij uij Aristotle University of Thessaloniki

11. Graph-theoretic Visualization Techniques Topology Representing Graphs Build a graph G [C x C] Topological relations between prototypes Gij corresponding to the strength of connection between prototypes Oi and Oj • Computation of the graph G • For each pattern find the nearest prototypes and increase the corresponding values in G matrix • Simple elementwise thresholding  Adjacency Matrix A A: a link connects two nearby prototypes only when they are natural neighbors over the manifold Aristotle University of Thessaloniki

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13. Graph-theoretic Visualization Techniques Compute the G graph via CFCM results Apply CFCM algorithm: (O, U) = CFCM(X, Fk, C) Build Compute G = U’.U’T FCG: Fuzzy Connectivity Graph Aristotle University of Thessaloniki

14. Graph-theoretic Visualization Techniques Minimal Spanning Tree MST-ordering 5 4 7 6 3 root 2 1 Aristotle University of Thessaloniki

15. Minimal Spanning Tree with MST-ordering Aristotle University of Thessaloniki

16. Graph-theoretic Visualization Techniques Locality Preserving Projections Rp Rr r<p Dimensionality Reduction technique Linear approach ≠ MDS, LE, ISOMAP • generalized eigenvector problem • use of FCG matrix • select the first r eigenvectors and tabulate them (Apxr matrix) P = [pij]Cxr = OA Alternative to PCA: different criteria, direct entrance of a new point into the subspace Aristotle University of Thessaloniki

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18. The experiments Magnetoencephalography Electroencephalography + 197 single trials + control recording 110 single trials Online outlier rejection Aristotle University of Thessaloniki

19. The datasets Feature Extraction MEG EEG pT μV msec msec X_data [197 x p1], p:number of features X_data [110 x p2], p:number of features Aristotle University of Thessaloniki

20. Applications (MEG) Exploit the background noise for better clustering MEG single trials + Spontaneous activity as a auxiliary set of signals Exploit the distances to extract the grades Aristotle University of Thessaloniki

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22. Applications (EEG) Robust Prototyping Elongate the possible outliers from the clustering procedure Find the distances from the nearest neighbors and compute the grades for every pattern Aristotle University of Thessaloniki

23. FCM Aristotle University of Thessaloniki

24. CFCM Aristotle University of Thessaloniki

25. Future Work Knowledge-Based Clustering Algorithms • conditional fuzzy clustering wavelet transform • horizontal collaborative clustering wavelet transform Aristotle University of Thessaloniki

26. Conclusions Through the proposed methodology • exploit the presence of noisy data • elongate the outliers from the clustering procedure Graph-Theoretic Visualization Techniques • study the variability of brain signals • study the relationships between clustering results Paper submitted “Using Conditional FCM to mine event-related dynamics” Aristotle University of Thessaloniki