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Transcript
A New Gravitational Clustering
Algorithm
Jonatan Gomez, Dipankar Dasgupta, Olfa Nasraoui
Outline
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Introduction
Background
Proposed Algorithm
Analysis
Introduction
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Many clustering techniques rely on the assumption that a
data set follows a certain distribution and is free of noise
Given noise, several techniques (k-means, fuzzy k-means)
based on a least squares estimate are spoiled
Most clustering algorithms require the number of clusters
to be specified
The authors propose a novel, robust, unsupervised
clustering technique based on Newton’s Law of
Gravitation, and Newton’s second law of motion
Introduction
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Gravitational concepts have been applied to cluster
visualization and analysis before
Properties of Wright’s Gravitational Clustering [2]:
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New position of a particle is found using remaining particles
When two particles are close they merge
Maximum movement of particles per iteration is capped
Algorithm terminates when only one particle remains
Improvements over Wright:
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Speed, robustness, and determining number of clusters
Background
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Newton’s Laws of Motion
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If acceleration is constant:
Background

Newton’s Law of Gravitation
Background
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Optimal Disjoint Set Union-Find Structure
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A disjoint set Union-Find structure supports three operators:
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MAKESET(X)
FIND(X)
UNION(X,Y)
Time complexity of any sequence of m Union and Find
operations on n elements is at most O(m+n) in practice
Proposed Algorithm
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Ideas behind applying gravitational law:
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A data point exerts a higher gravitational force on other data
points in the same cluster than on data points not in the same
cluster. Thus, points in a cluster move toward the center of the
cluster.
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If a point is a noise point, the gravitational forces acting on it
will be so small the point will be immobile. Thus, noise points
won’t be assigned to any cluster
Proposed Algorithm
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Simplified equation used to move point x according to
gravitational field of point y
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Velocity considered to be zero at all points in time
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Reduce G after each iteration to prevent the “big crunch”
Proposed Algorithm
Proposed Algorithm
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Use threshold to extract valid clusters which have at
least a minimum number of points
Proposed Algorithm
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Similarities to Agglomerative Hierarchical Clustering
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Differences from Agglomerative Hierarchical Clustering
Proposed Algorithm
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Comparison to Wright [2]
Experiments
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Synthetic data
Experiments
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Results (over 10 trials)
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Parameters: M = 500, G = 7x10-6, ∆G = 0.01, ε = 10-4
k-Means and Fuzzy k-Means given 150 iterations
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Experiments
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Clusters found by the G-algorithm
Experiments
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Clusters found by the G-algorithm (noise removed)
Experiments
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Movement of points over iterations
Experiments
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Scalability (average of 50 trails for each percentage)
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Do not need to use entire data set to get good
results
Experiments
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Sensitivity to α
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Use α = 0.03
Experiments
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Sensitivity to G
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To big => one cluster
To small => no clusters
No universal value => depends on data set
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Experiments
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Sensitivity to ∆(G)
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To big => no clusters
To small => one cluster
Best value ~0.01 based on experiments
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Experiments
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Sensitivity to ε
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To big => one cluster
Experiments
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Real data set
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Intrusion detection benchmark data set
42 attributes, 33 numerical, N = 492,021
2 classes – no intrusion (19.3%) and intrusion (80.7%)
Use only the numerical attributes
Use only 1% of the data (chosen randomly)
Parameter settings
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G = 1x10-4 (based on testing)
∆(G) = 0.01
α = 0.03
ε = 1x10-6
M = 100
Experiments
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Clustering-Classification Strategy
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Assign to each cluster the class with more training data
records assigned to that cluster
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Given an unknown data point, the data point is assigned to
the closest cluster (the center of the clusters is used to
compute the distance)
Experiments
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Real data set results (over 100 trials)
Conclusions / Future Work
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Successfully determines the number of clusters in
noisy data sets
Can be used to pre-process data by removing noise
Three of four parameters can be set to constant
values
Future Work:
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Determine method to automatically set G
Extend to different distance metrics
References
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[1] J. Gomez, D. Dasgupta, and O. Nasraoui, “A New Gravitational
Clustering Algorithm,” In Proc. of the SIAM Int. Conf. on Data Mining,
2003.
[2] W. E. Wright, “Gravitational Clustering,” Pattern Recognition, 9:151166, Pergamon Press, 1977.