Transcript ViSOM

ViSOM－A Novel Method for
Multivariate Data Projection and
Structure Visualization
Advisor : Dr. Hsu
Graduate : Sheng-Hsuan Wang
Author : Hujun Yin
Outline
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Motivation
Objective
Introduction
Data Projection Methods
ViSOM
Experimental Results
Conclusion
Personal opinion
Review
Motivation
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In SOM, the structures of the data
clusters may not be apparent and their
shapes are often distorted.
Objective
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In this paper, a visualization-induced
SOM(ViSOM) is proposed to overcome
these shortcomings.
Introduction
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The linear principal component
analysis(PCA)－dimension reduction.
Sammon mapping－nonlinear, minimize.
Neural networks－can learn complex
nonlinear relationships of variables.
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Ex:Self-Organization Maps(SOMs)
Introduction
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When the SOM is used for visualization,
the inter-neuron distances are not
directly visible or measurable on the
map.－using a coloring scheme such as
U-matrix.
Introduction
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The ViSOM projects as does the SOM,
but constrains the lateral contraction
force and regularizes the inter-neuron
distance to a parameter that defines
and controls the resolution of the map.
Data Projection Methods
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PCA
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PCA is a classic linear data analysis method
aiming at finding orthogonal principal
directions from a set of data, along which
the data exhibit the largest variances.
m
min
T
2
[
x

(
q
x
)
q
]
  j j
x
(1)
j 1
x  [ x1 , x2 ,...xn ]T ,{q j , j  1,2,...m, m  n}
Data Projection Methods
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Sammon Mapping
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A traditional subject related to dimension
reduction and data projection is
multidimensional scaling(MDS).
A general fitness function, stress
2
[
d

f
(

)]
 ij
ij
S
i, j
d
i, j
2
i, j
(2)
Data Projection Methods
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Sammon Mapping
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The Sammon's mapping maps data points
to the output space by minimizing the
distance difference between data points in
the input and output spaces.
1
S Sammon 

d
 i, j
i j

i j
[d i, j  d i , j ]2
d

i, j
(3)
Data Projection Methods
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SOM
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The SOM is an unsupervised learning
algorithm that uses a finite grid of neurons
to map or frame the input space.
min
  (c, k )( w
x c x  c
k
c
 x)
2
(4)
ViSOM
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ViSOM Structure and Derivation
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The ViSOM uses a similar grid structure of
neurons as does the SOM.
A winning neuron v can be found according
to its distance to the input, i.e.,
v  arg min || x(t )  wc ||
c
(5)
ViSOM
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Then the SOM updates the weight of the
winning neuron according to
wv (t  1)  wv (t )   (t )[ x(t )  wv (t )]
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(6)
The weight of the neurons in a
neighborhood of the winner are updated
by
wk (t  1)  wk (t )   (t ) (v, k , t )[ x(t )  wk (t )]
(7)
ViSOM
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Decomposition of the SOM updating force
Fkx  x(t )  wk (t )  [ x(t )  wv (t )]  [ wv (t )  wk (t )]
 Fvx  Fkv
(8)
ViSOM
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ViSOM Algorithm
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Find the winner from (5).
Update the winner according to (6).
Update the neighborhood according to
wk (t  1)  wk (t )   (t ) (v, k , t )
(d vk   vk  )
([ x(t )  wv (t )]  [ wv (t )  wk (t )]
)
 vk 
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(9)
Refresh the map by randomly choosing the
weights of the neurons.(optional)
ViSOM
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The rigidity of the map is controlled by
the ultimate size,  f , of the
neighborhood.
The resolution parameter  depends
on the size of the map, data variance
and required resolution of the map.
Experimental Results
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Two Illustrative Data Sets
Experimental Results
Experimental Results
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Iris Data Set
Conclusion
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In this paper, a new mapping method,
ViSOM, is proposed for visualization and
projection of high-dimensional data.
Personal Opinion
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This method can be used in our lab’s
SOM program to improve the quality of
clustering.
Review
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Data Projection Methods
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PCA
Sammon Mapping
SOM
ViSOM