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

Aristotle University of Thessaloniki4

Our approach

CLUSTERING

FUZZY

CONDITIONAL

creating clusters

0 or 1 [0, 1]partial membership

content constraints

criteria grades

Aristotle University of Thessaloniki5

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

ClusteringFuzzy Clustering

Aristotle University of Thessaloniki6

Fuzzy C Means

CNC

N

uu

uu

1

111],...,,[ 21 CVVVV

U membership matrix Centroids Objective function

|)1()(| tQtQ

Iterative procedure

CONTINUE STOP

XdataNxp

Aristotle University of Thessaloniki7

FCM 2D Example

compact groupsspurious patterns

FCM sensitivity to noisy data

Aristotle University of Thessaloniki8

Conditional Fuzzy ClusteringThe presence of Condition(s)

Condition(s)Pattern

mark

Aristotle University of Thessaloniki9

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.

Aristotle University of Thessaloniki10

FCM VS CFCM

FCM

uij

uij

uij

uijCFCM

uij

uij

uij

uij

FkVS

Aristotle University of Thessaloniki11

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 Thessaloniki12

Aristotle University of Thessaloniki13

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

Aristotle University of Thessaloniki14

Graph-theoretic Visualization Techniques

Minimal Spanning Tree MST-ordering

12

3

4

5

67

root

Aristotle University of Thessaloniki15

Minimal Spanning Tree with MST-ordering

Aristotle University of Thessaloniki16

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

Aristotle University of Thessaloniki17

Aristotle University of Thessaloniki18

The experiments

Magnetoencephalography Electroencephalography

+ 197 single trials

+ control recording

110 single trials

Online outlier rejection

Aristotle University of Thessaloniki19

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

Aristotle University of Thessaloniki20

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

Aristotle University of Thessaloniki21

Aristotle University of Thessaloniki22

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 Thessaloniki23

FCM

Aristotle University of Thessaloniki24

CFCM

Aristotle University of Thessaloniki25

Future WorkKnowledge-Based Clustering Algorithms

• horizontal collaborative clustering

wavelet transform• conditional fuzzy clustering

wavelet transform

Aristotle University of Thessaloniki26

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”