Zigkolis C FCM Brain Responses
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Conditional Fuzzy C Means
A fuzzy clustering approach for mining event-related dynamics
Christos N. Zigkolis
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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
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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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Aristotle University of Thessaloniki4
Our approach
CLUSTERING
FUZZY
CONDITIONAL
creating clusters
0 or 1 [0, 1]
partial membership
content constraints
criteria grades
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Aristotle University of Thessaloniki5
Every Pattern to only one clusterEvery Pattern to every cluster with partial membership
ClusteringFuzzy Clustering
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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
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Aristotle University of Thessaloniki7
FCM 2D Example
compact groupsspurious patterns
FCM sensitivity to noisy data
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Aristotle University of Thessaloniki8
Conditional Fuzzy Clustering
The presence of Condition(s)
Condition(s)Pattern
mark
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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.
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Aristotle University of Thessaloniki10
FCM VS CFCM
FCM
uij
uij
uij
uijCFCM
uij
uij
uij
uij
FkVS
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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 O i 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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Aristotle University of Thessaloniki12
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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
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Aristotle University of Thessaloniki14
Graph-theoretic Visualization Techniques
Minimal Spanning Tree MST-ordering
12
3
4
5
67
root
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Aristotle University of Thessaloniki15
Minimal Spanning Tree with MST-ordering
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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
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Aristotle University of Thessaloniki17
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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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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
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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
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Aristotle University of Thessaloniki21
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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
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Aristotle University of Thessaloniki23
FCM
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Aristotle University of Thessaloniki24
CFCM
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Aristotle University of Thessaloniki25
Future Work
Knowledge-Based Clustering Algorithms
• horizontal collaborative clustering
wavelet transform• conditional fuzzy clustering
wavelet transform
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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”