conditional fuzzy c means

Conditional Fuzzy C Means

A fuzzy clustering approach for mining event-related dynamics

Christos N. Zigkolis

Aristotle University of Thessaloniki2

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 Thessaloniki3

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

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Our approach

CLUSTERING

FUZZY

CONDITIONAL

creating clusters

0 or 1 [0, 1]partial membership

content constraints

criteria grades

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Every Pattern to only one clusterEvery Pattern to every cluster with partial membership

ClusteringFuzzy Clustering

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Fuzzy C Means

CNC

N

uu

uu

1

111],...,,[ 21 CVVVV

U membership matrix Centroids Objective function

|)1()(| tQtQ

Iterative procedure

CONTINUE STOP

XdataNxp

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FCM 2D Example

compact groupsspurious patterns

FCM sensitivity to noisy data

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Conditional Fuzzy ClusteringThe presence of Condition(s)

Condition(s)Pattern

mark

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Conditional Fuzzy C Means

scaled to [0, 1]

],...,,[ 21 NfffF

<Xdata, F> CFCM <U, Centroids>

F affects the computations of U matrix and consequently the centroids.

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FCM VS CFCM

FCM

uij

uij

uij

uijCFCM

uij

uij

uij

uij

FkVS

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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

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Graph-theoretic Visualization Techniques

Compute the G graph via CFCM results

Apply CFCM algorithm: (O, U) = CFCM(X, Fk, C)

Build )(.,][ ''' ijijijCxNij uuuthatsuchuU

Compute G = U’.U’T FCG: Fuzzy Connectivity Graph

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Graph-theoretic Visualization Techniques

Minimal Spanning Tree MST-ordering

12

3

4

5

67

root

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Minimal Spanning Tree with MST-ordering

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Graph-theoretic Visualization Techniques

Locality Preserving Projections

Dimensionality Reduction technique Rp Rr r<pLinear 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 (Apxr matrix)

P = [pij]Cxr = OA

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Aristotle University of Thessaloniki18

The experiments

Magnetoencephalography Electroencephalography

+ 197 single trials

+ control recording

110 single trials

Online outlier rejection

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The datasetsFeature Extraction

MEG EEG

X_data [197 x p1], p:number of features X_data [110 x p2], p:number of features

pT

msec msec

μV

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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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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

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FCM

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CFCM

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Future WorkKnowledge-Based Clustering Algorithms

• horizontal collaborative clustering

wavelet transform• conditional fuzzy clustering

wavelet transform

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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”