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

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

3. 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

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

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

6. Aristotle University of Thessaloniki 6 Fuzzy C Means U membership matrixCentroidsObjective function Iterative procedure CONTINUE STOP Xdata Nxp

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

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

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

10. Aristotle University of Thessaloniki 10 FCM VS CFCM FCM u ij CFCM u ij FkFk VS

11. Aristotle University of Thessaloniki 11 Graph-theoretic Visualization Techniques Topology Representing Graphs Build a graph G [C x C]Topological relations between prototypes G ij corresponding to the strength of connection between prototypes O i and O j 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

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

14. Aristotle University of Thessaloniki 14 Graph-theoretic Visualization Techniques Minimal Spanning TreeMST-ordering 12 3 4 5 6 7 root

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

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

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

19. Aristotle University of Thessaloniki 19 The datasets Feature Extraction MEG EEG X_data [197 x p 1 ], p:number of featuresX_data [110 x p 2 ], p:number of features pT msec μVμV

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

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22. Aristotle University of Thessaloniki 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

23. Aristotle University of Thessaloniki 23 FCM

24. Aristotle University of Thessaloniki 24 CFCM

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

26. 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”