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Transcript
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
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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
criteria
CONDITIONAL
grades
content constraints
0 or 1
FUZZY
partial membership
CLUSTERING
creating clusters
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[0, 1]
Fuzzy Clustering
Clustering
Every
one
cluster
EveryPattern
Patternto
toonly
every
cluster
with partial membership
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Fuzzy C Means
XdataNxp
U membership matrix
 u11  u1N 
    


uC1  uCN 
Centroids
V  [V1 ,V2 ,...,VC ]
Objective function
| Q(t )  Q(t  1) | 
CONTINUE STOP
Iterative procedure
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FCM 2D Example
compact groups
spurious patterns
FCM sensitivity
to noisy data
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Conditional Fuzzy Clustering
The presence of Condition(s)
mark
Pattern
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Condition(s)
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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.
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FCM VS CFCM
FCM
uij
uij
uij
uij
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uij
Fk
VS
uij
CFCM
uij
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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
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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
U '  [uij' ]CxN , such that uij'  uij . (uij  )
Compute G = U’.U’T
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FCG: Fuzzy Connectivity Graph
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Graph-theoretic
Visualization Techniques
Minimal Spanning Tree
MST-ordering
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7
4
6
3
2
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1
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<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
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The experiments
Magnetoencephalography
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Electroencephalography
+ 197 single trials
110 single trials
+ control recording
Online outlier rejection
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The datasets
Feature Extraction
MEG
EEG
pT
μV
msec
X_data [197 x p1], p:number of features
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msec
X_data [110 x p2], p:number of features
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Applications (MEG)
Exploit the background noise for better clustering
MEG single trials
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+ Spontaneous activity as a auxiliary set of signals
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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 Work
Knowledge-Based Clustering Algorithms
• conditional fuzzy clustering
wavelet transform
• horizontal collaborative 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”
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