Nonlinear approach for clustering magnetoencephalographic curves

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Nonlinear approach for clustering
magnetoencephalographic curves

• Sebino Stramaglia
• Center of Innovative Technologies for Signal
  Detection and Processing
• Physics Dept. University of Bari, Italy

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Summary

• SEF after somatosensory stimulation.
• Clustering.
• Classification of SEF curves.
• Results.

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Brain and its rhythms.

• Teta band 3.5Hz 7.5Hz
• Alfa band 7.5Hz 12.5Hz
• Beta1 band 12.5Hz 22Hz
• Beta2 band 22Hz 32Hz
• Gamma band 32Hz 60Hz

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Magnetoencephalography

• measuring neuromagnetic fields
  50-500 fT
• superconducting SQUID sensors
• Temporal resolution 1 ms
• multisensor recordings
• MEG analysis mostly involves brain cortex
• SEF Somatosensory Evoked magnetic Field

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Experiment of somatosensory stimulation

• Median nerve has been stimulated 300 times by an
  electric pulse the time between consecutive
  stimuli has been randomly chosen in 1.3sec.,
  2.4sec.. The response has been recorded on the
  opposite hemisphere.

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The MEG signal
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Extracting the SEF averaging
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SEF spatial map, current dipole
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EXPERIMENT OF SOMATOSENSORIAL STIMULATION(I)

• Analysis morphology of SEF curves

• Experimental evidence high variability of SEF
  in different subjects

recognition of alterations of morphology
(lesions of cortex)

• Aim discovering morphological classes

• Tools unsupervised learning algorithms
  clustering algorithms

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EXPERIMENT OF SOMATOSENSORIAL STIMULATION(II)

• Data set made of 37 healthy subjects
• Electrical Stimolations of the median nerve at
  the wrist level, 0.2 ms long.
• Recordings by a magnetoencephalographer with 28
  channels

Definition of a similarity index
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Clustering

• Aim partitioning N objects in K classes, so that
  similar objects belong to the same class.
• Problem the transitive property does not hold
  for similarity!
• K assigned or to be inferred from data?
• Resolution (scale) at which data are seen?
  Hierarchy by varying the scale.

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Centroid Methods - K-means

• Start with random position of K centroids.
• Iterate until centroids are stable
• Assign points to centroids
• Move centroids to centerof assign points

Iteration 0
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Centroid Methods - K-means

• Start with random position of K centroids.
• Iteratre until centroids are stable
• Assign points to centroids
• Move centroids to centerof assign points

Iteration 1
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Centroid Methods - K-means

• Start with random position of K centroids.
• Iteratre until centroids are stable
• Assign points to centroids
• Move centroids to centerof assign points

Iteration 1
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Centroid Methods - K-means

• Start with random position of K centroids.
• Iteratre until centroids are stable
• Assign points to centroids
• Move centroids to centerof assign points

Iteration 3
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Centroid Methods - K-means

• Result depends on initial centroids position
• Fast algorithm compute distances from data
  points to centroids
• No way to choose K.
• Breaks long clusters

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Agglomerative Hierarchical Clustering
Need to define the distance between thenew
cluster and the other clusters. Single Linkage
distance between closest pair. Complete
Linkage distance between farthest pair. Average
Linkage average distance between all pairs
or distance between
cluster centers
Distance between joined clusters
The dendrogram induces a linear ordering of the
data points
Dendrogram
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Agglomerative Hierarchical Clustering

• Results depend on distance update method
• Single Linkage elongated clusters
• Complete Linkage sphere-like clusters
• Greedy iterative process
• NOT robust against noise
• No inherent measure to choose the clusters

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Goal revealing genuine structures hidden in
data

• The deceit of randomness in a data-set
  structures may arise as realizations of
  randomness.
• The risk is that in noisy and high dimensional
  data one always finds what he is looking for !!
• Retarded Learning (Van den Broeck, Engel, Opper).

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Statistical Physics for Clustering

• Temperature controls the order and the structures
  arising from interactions, since it determines
  the effective scale at which the system is
  sensed.
• Robustness of the statistical properties, w.r.t.
  details of interactions
• Varying the temperature, the transition from one
  structure to the another is abrupt.

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Physics and CLUSTERING

• Algorithms rooted on Physical approaches
• Deterministic Annealing (Rose, Gurewitz)
  Physical Review Letters (1990)
• Coupled Chaotic Maps (L. Angelini et al.)
  Physical Review Letters (2000)
• Superparamagnetic Clustering (E. Domany)
  Physical Review E (1998)

• Applications
• PA e HEP particle classification
• Biology classification of DNA sequences
• Econophysics analysis of financial indexes
• Humanitarian demining. Many others.

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Super-Paramagnetic Clustering (SPC) M.Blatt,
S.Weisman and E.Domany (1996) Neural Computation

• The idea behind SPC is based on the physical
  properties dilute magnets.
• Calculating correlation between magnet
  orientations at different temperatures (T).

TLow
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Super-Paramagnetic Clustering (SPC) M.Blatt,
S.Weisman and E.Domany (1996) Neural Computation

• The idea behind SPC is based on the physical
  properties dilute magnets.
• Calculating correlation between magnet
  orientations at different temperatures (T).

THigh
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Super-Paramagnetic Clustering (SPC) M.Blatt,
S.Weisman and E.Domany (1996) Neural Computation

• The idea behind SPC is based on the physical
  properties dilute magnets.
• Calculating correlation between magnet
  orientations at different temperatures (T).

TIntermediate
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Super-Paramagnetic Clustering (SPC)

• The algorithm simulates the magnets behavior at a
  range of temperatures and calculates their
  correlation
• The temperature (T) controls the resolution
• Example N4800 points in D2

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Super-Paramagnetic Clustering (SPC)

• The algorithm simulates the magnets behavior at a
  range of temperatures and calculates their
  correlation
• The temperature (T) controls the resolution
• Example N4800 points in D2

T 1 All points are in ONE cluster
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Super-Paramagnetic Clustering (SPC)

• The algorithm simulates the magnets behavior at a
  range of temperatures and calculates their
  correlation
• The temperature (T) controls the resolution
• Example N4800 points in D2

T 20 Three clusters emerge
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Super-Paramagnetic Clustering (SPC)

• The algorithm simulates the magnets behavior at a
  range of temperatures and calculates their
  correlation
• The temperature (T) controls the resolution
• Example N4800 points in D2

T 30 Substructures appear
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Super-Paramagnetic Clustering (SPC)

• The algorithm simulates the magnets behavior at a
  range of temperatures and calculates their
  correlation
• The temperature (T) controls the resolution
• Example N4800 points in D2

T ? Each points is its own cluster
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Output of SPC
A function ?(T) that peaks when stable clusters
break
Size of largest clusters as function of T
Dendrogram
Stable clusters live for large ?T
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Choosing a value for T
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Advantages of SPC

• Scans all resolutions (T)
• Robust against noise and initialization
  -calculates collective correlations.
• Identifies natural (?) and stable clusters (?T)
• No need to pre-specify number of clusters
• Clusters can be any shape

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Clustering SEF curves COMPLETE LINKAGE
Complete Linkage
No sound criterion to choose the optimal partition
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Clustering SEF curves Application of SPC
SPC
There is not a partition which is stable w.r.t.
the temperature.
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PRE-PROCESSING MULTIDIMENSIONAL SCALING

MDS
dij n?n
N points xi ??d

• reduction of complexity
• data visualization (d2,3)

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Multidimensional scaling (1)

• The dissimilarity measure allows to state that
  two SEF curves are more similar than two other
  curves, the exact value of the dissimilarity is a
  detail.

• We need an approach whose output is invariant
  under arbitrary variations of the dissimilarity
  matrix leaving unchanged the ranking of the
  matrix elements.

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Multidimensional scaling (2)

• MDS find N points, in an Euclidean d-dimensional
  space, whose interpoint distances match the
  dissimilarity matrix.

• We only require a monotone relationship between
  dissimilarities and distances in the
  configurations.

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MULTIDIMENSIONAL SCALING (3)
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REGRESSIONE MONOTONA
Monotone Regression to find reference distances
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APPLICATION OF MDS
MDS
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APPLICATION OF MDS COMPLETE LINKAGE RESULTS
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OUR APPROACH APPLICATION OF MDS AND SPC
MDS
SPC
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OUR PARTITION BY MDS AND SPC
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CONCLUSIONS

• Use of MDS e SPC revealed the natural
  morphological classes in the data set.
• This approach can be applied in other situations
  where only a fuzzy definition of dissimilarity is
  at hand.
• From the physiological point of view, it has been
  found that anomalouos SEF curves do not belong
  to any of the morphological classes we found.