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Conditional Fuzzy C Means A fuzzy clustering approach for mining event-related dynamics Christos N. Zigkolis Contents • The problem • Our approach • Fuzzy Clustering • Conditional Fuzzy Clustering • Graph-Theoretic Visualization techniques • The experiments and the datasets • Applications • Future Work • Conclusions 2 Aristotle University of Thessaloniki 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 3 Aristotle University of Thessaloniki Our approach criteria CONDITIONAL grades content constraints 0 or 1 FUZZY partial membership CLUSTERING creating clusters 4 Aristotle University of Thessaloniki [0, 1] Fuzzy Clustering Clustering Every one cluster EveryPattern Patternto toonly every cluster with partial membership 5 Aristotle University of Thessaloniki Fuzzy C Means XdataNxp U membership matrix u11 u1 N uC1 uCN Centroids V [V1 ,V2 ,...,VC ] Objective function | Q(t ) Q(t 1) | CONTINUE STOP Iterative procedure 6 Aristotle University of Thessaloniki FCM 2D Example compact groups spurious patterns FCM sensitivity to noisy data 7 Aristotle University of Thessaloniki Conditional Fuzzy Clustering The presence of Condition(s) mark Pattern 8 Condition(s) Aristotle University of Thessaloniki Conditional Fuzzy C Means F [ f1 , f 2 ,..., f N ] scaled to [0, 1] <Xdata, F> CFCM <U, Centroids> F affects the computations of U matrix and consequently the centroids. 9 Aristotle University of Thessaloniki FCM VS CFCM FCM uij uij uij uij 10 uij Fk VS uij CFCM uij Aristotle University of Thessaloniki uij 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 11 Aristotle University of Thessaloniki 12 Aristotle University of Thessaloniki Graph-theoretic Visualization Techniques Compute the G graph via CFCM results Apply CFCM algorithm: (O, U) = CFCM(X, Fk, C) Build U [u ] ' ' ij CxN Compute G = U’.U’T 13 , suchthat u uij . (uij ) ' ij FCG: Fuzzy Connectivity Graph Aristotle University of Thessaloniki Graph-theoretic Visualization Techniques Minimal Spanning Tree MST-ordering 5 7 4 6 3 2 14 1 root Aristotle University of Thessaloniki Minimal Spanning Tree with MST-ordering 15 Aristotle University of Thessaloniki Graph-theoretic Visualization Techniques Locality Preserving Projections Dimensionality Reduction technique Rp Rr r<p 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 16 Aristotle University of Thessaloniki 17 Aristotle University of Thessaloniki The experiments Magnetoencephalography 18 Electroencephalography + 197 single trials 110 single trials + control recording Online outlier rejection Aristotle University of Thessaloniki The datasets Feature Extraction MEG EEG pT μV msec X_data [197 x p1], p:number of features 19 msec X_data [110 x p2], p:number of features Aristotle University of Thessaloniki Applications (MEG) Exploit the background noise for better clustering MEG single trials 20 + Spontaneous activity as a auxiliary set of signals Aristotle University of Thessaloniki Exploit the distances to extract the grades 21 Aristotle University of Thessaloniki 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 22 Aristotle University of Thessaloniki FCM 23 Aristotle University of Thessaloniki CFCM 24 Aristotle University of Thessaloniki Future Work Knowledge-Based Clustering Algorithms • conditional fuzzy clustering wavelet transform • horizontal collaborative clustering wavelet transform 25 Aristotle University of Thessaloniki 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” 26 Aristotle University of Thessaloniki