ClusteringDEC

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Clustering Using Differential Evolution
Navdha Sah
Thesis Adviser: Dr. Thang N. Bui
Department of Math & Computer Science
Penn State Harrisburg
Spring 2016

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Background DEC Algorithm Results Conclusion
Outline
1 Background
2 DEC Algorithm
3 Results
4 Conclusion

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Clustering?

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Clustering?
What is clustering?

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Clustering?
What is clustering?
The problem of partitioning a collection of objects into groups.

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Clustering?
What is clustering?
Objects in the same group are similar.

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Clustering?
What is clustering?
Objects in different groups are dissimilar.

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Clustering?
What is clustering?
Why clustering?

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Clustering?
What is clustering?
Why clustering?
Useful in ﬁnding natural groupings within a data set

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Clustering?
What is clustering?
Why clustering?
Useful in ﬁnding natural groupings within a data set
Useful in analysis, description and utilization of valuable information
hidden within groups

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Example
What is “natural” grouping?

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Example
Grouping by shapes?

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Example
Grouping by colors?

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Example
The Ground Truth!

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Example
Data Set Grouping by Shape
Grouping by Color The Ground Truth

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What is a clustering?

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Data set

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Data set
Let X = {X1, X2, . . . , Xn} be a set of n data items, where Xi ∈ IRf
, i = 1, . . . , n,
that is, Xi = (Xi1, . . . , Xif ), where Xij ’s are called features of Xi .

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Data set
, i = 1, . . . , n,
Clustering

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Data set
, i = 1, . . . , n,
Clustering
A clustering of X is then a partition C = (C1, . . . , Ck ) of X such that:

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Data set
, i = 1, . . . , n,
Clustering
∪k
i=1 Ci = X,

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Data set
, i = 1, . . . , n,
Clustering
∪k
i=1 Ci = X,
Ci ∩ Cj = ϕ ∀i, j ∈ {1, 2, . . . , k}, i ̸= j,

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Data set
, i = 1, . . . , n,
Clustering
∪k
i=1 Ci = X,
Ci ∩ Cj = ϕ ∀i, j ∈ {1, 2, . . . , k}, i ̸= j,
Ci ̸= ϕ ∀i ∈ {1, 2, . . . , k}.

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How is clustering measured?

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Need to measure similarity or dissimilarity between items.

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Need to measure similarity or dissimilarity between items.
Similarity can be measured using distance measures.

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Distance Measures

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Distance Measures
Euclidean distance
d(Xi , Xj ) =
f∑
p=1
(Xip − Xjp)2 ,
where Xi and Xj are two f-dimensional items.
It is a special case of the Minkowsky metric (α = 2)
d(Xi , Xj ) =


f∑
p=1
(Xip − Xjp)α


1/α
.

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Distance Measures
MinMax distance
d(Xi , Xj ) =
f∑
p=1
|Xip − Xjp|
MAXp − MINp
,
where Xip and Xjp are the pth
feature values for items Xi and Xj , and
MAXp and MINp are the maximum and minimum values for the pth
feature across the data set, that is,
MAXp = max{Xip | i = 1, . . . , n},
MINp = min{Xip | i = 1, . . . , n}.

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Clustering Validity Indexes

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Used to analyze the quality of a clustering solution on a
quantitative basis.

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quantitative basis.
A good clustering would be one with compact and distinct
clusters.

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quantitative basis.
clusters.
A validity index helps capture the notion of compactness
and separation in a clustering solution.

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quantitative basis.
clusters.
A validity index helps capture the notion of compactness
and separation in a clustering solution.
Need the distance measure and the clustering to calculate
the validity index.

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DB Index
FDB(C, D)

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DB Index
FDB(C, D)
A clustering

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DB Index
FDB(C, D)
A clustering
Distance function

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Di
A clustering
Distance function

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Di
A clustering
Distance function
Number of clusters

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Di
Di = max
0≤i,j≤K
i̸=j
{
Si + Sj
Mij
}
A clustering
Distance function
Number of clusters

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Di
Si =

 1
|Ci|
∑
X∈Ci
D(X, ωi)q


1
q
Di = max
0≤i,j≤K
i̸=j
{
Si + Sj
Mij
}
A clustering
Distance function
Number of clusters

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Di average distance of items from centroid
Si =

 1
|Ci|
∑
X∈Ci
D(X, ωi)q


1
q
Di = max
0≤i,j≤K
i̸=j
{
Si + Sj
Mij
}
A clustering
Distance function
Number of clusters

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Si =

 1
|Ci|
∑
X∈Ci
D(X, ωi)q


1
q
Mij = D(ωi, ωj), i ̸= j Di = max
0≤i,j≤K
i̸=j
{
Si + Sj
Mij
}
A clustering
Distance function
Number of clusters

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DB Index
FDB(C, D) =
1
K
K∑
i=1
Si =

 1
|Ci|
∑
X∈Ci
D(X, ωi)q


1
q
Mij = D(ωi, ωj), i ̸= j Di = max
0≤i,j≤K
i̸=j
{
Si + Sj
Mij
}
A clustering
Distance function
Number of clusters
distance between clusters i and j

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CS Index
FCS(C, D) =
1
K
K∑
i=1
[
1
|Ci |
∑
Xi ∈Ci
max
Xj ∈Ci
{
D(Xi, Xj)
}
]
1
K
K∑
i=1
[
min
j∈K,j̸=i
{
D(ωi, ωj)
}
]

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CS Index
FCS(C, D) =
1
K
K∑
i=1
[
1
|Ci |
∑
Xi ∈Ci
max
Xj ∈Ci
{
D(Xi, Xj)
}
]
1
K
K∑
i=1
[
min
j∈K,j̸=i
{
D(ωi, ωj)
}
]
A clustering

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CS Index
FCS(C, D) =
1
K
K∑
i=1
[
1
|Ci |
∑
Xi ∈Ci
max
Xj ∈Ci
{
D(Xi, Xj)
}
]
1
K
K∑
i=1
[
min
j∈K,j̸=i
{
D(ωi, ωj)
}
]
A clustering
Distance function

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CS Index
FCS(C, D) =
1
K
K∑
i=1
[
1
|Ci |
∑
Xi ∈Ci
max
Xj ∈Ci
{
D(Xi, Xj)
}
]
1
K
K∑
i=1
[
min
j∈K,j̸=i
{
D(ωi, ωj)
}
]
A clustering
Distance function
Number of clusters

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.
CS Index
average distance of furthest away items in Ci
FCS(C, D) =
1
K
K∑
i=1
[
1
|Ci |
∑
Xi ∈Ci
max
Xj ∈Ci
{
D(Xi, Xj)
}
]
1
K
K∑
i=1
[
min
j∈K,j̸=i
{
D(ωi, ωj)
}
]
A clustering
Distance function
Number of clusters

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.
CS Index
average distance of furthest away items in Ci
FCS(C, D) =
1
K
K∑
i=1
[
1
|Ci |
∑
Xi ∈Ci
max
Xj ∈Ci
{
D(Xi, Xj)
}
]
1
K
K∑
i=1
[
min
j∈K,j̸=i
{
D(ωi, ωj)
}
]
average minimum distance between closest centroids
A clustering
Distance function
Number of clusters

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.
PBM Index
FPBM(C, D) =
1
K
×
E
EK
× DK

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PBM Index
FPBM(C, D) =
1
K
×
E
EK
× DK
A clustering

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PBM Index
FPBM(C, D) =
1
K
×
E
EK
× DK
A clustering
Distance function

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PBM Index
FPBM(C, D) =
1
K
×
E
EK
× DK
A clustering
Distance function
Number of clusters

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PBM Index
E =
n∑
i=1
D(Xi, ωx )
FPBM(C, D) =
1
K
×
E
EK
× DK
A clustering
Distance function
Number of clusters

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PBM Index
E =
n∑
i=1
D(Xi, ωx )
FPBM(C, D) =
1
K
×
E
EK
× DK
A clustering
Distance function
Number of clusters
sum of distance of all items from global centroid

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PBM Index
E =
n∑
i=1
D(Xi, ωx )
FPBM(C, D) =
1
K
×
E
EK
× DK
EK =
K∑
k=1
∑
X∈Ck
D(X, ωk )
A clustering
Distance function
Number of clusters

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.
PBM Index
E =
n∑
i=1
D(Xi, ωx )
FPBM(C, D) =
1
K
×
E
EK
× DK
EK =
K∑
k=1
∑
X∈Ck
D(X, ωk )
A clustering
Distance function
Number of clusters
sum of distance of all items from their centroid

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PBM Index
E =
n∑
i=1
D(Xi, ωx )
FPBM(C, D) =
1
K
×
E
EK
× DK
DK = max
0≤i,j≤K
i̸=j
{D(ωi, ωj)}
EK =
K∑
k=1
∑
X∈Ck
D(X, ωk )
A clustering
Distance function
Number of clusters

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PBM Index
E =
n∑
i=1
D(Xi, ωx )
FPBM(C, D) =
1
K
×
E
EK
× DK
DK = max
0≤i,j≤K
i̸=j
{D(ωi, ωj)}
EK =
K∑
k=1
∑
X∈Ck
D(X, ωk )
A clustering
Distance function
Number of clusters
max distance between centroids

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The Clustering Problem

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Deﬁnition

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Deﬁnition
Input: Data set X = {X1, . . . , Xn} where Xi ∈ IRf
, a distance function
D and a validity index F.

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Deﬁnition
Output: A clustering C = {C1, . . . , Ck } of X such that F(C, D) is opti-
mized over all possible clusterings of X.

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Deﬁnition
Output: A clustering C = {C1, . . . , Ck } of X such that F(C, D) is opti-
mized over all possible clusterings of X.
The problem is known to be NP-Hard.

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Applications

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Applications
Marketing

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Applications
Marketing
Medical sciences

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Applications
Marketing
Medical sciences
Engineering

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Applications
Marketing
Medical sciences
Engineering
Recommender systems

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Applications
Marketing
Medical sciences
Engineering
Recommender systems
Community detection

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Related Work

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Related Work
K-Means algorithm

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Related Work
K-Means algorithm
Evolution Strategy (ES) algorithms

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Related Work
K-Means algorithm
Particle Swarm Optimization (PSO) algorithms

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Related Work
K-Means algorithm
Genetic Algorithms (GA)

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Related Work
K-Means algorithm
Genetic Algorithms (GA)
Differential Evolution (DE) algorithms

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Differential Evolution

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Similar to genetic algorithms.

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Population based search methodology.

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A population of candidate solutions is maintained and
evolved.

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A population of candidate solutions is maintained and
evolved.
Differs from Genetic Algorithms in terms of offspring
creation.

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Differential Evolution Clustering (DEC)

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DEC has two main stages:

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Initialization stage

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Evolution stage

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Evolution stage
In the initialization stage, a population of individuals is generated.

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Evolution stage
In the evolution stage, the population evolves through a number of
cycles.

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Evolution stage
cycles.
Each cycle has two phases:

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Evolution stage
cycles.
Exploration phase

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Evolution stage
cycles.
Exploration phase
Exploitation phase

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Evolution stage
cycles.
Exploration phase
Exploitation phase
Best individual is returned at the end.

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DEC Algorithm
P ← Initialization

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DEC Algorithm
Repeat
Until some criteria are met

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DEC Algorithm
Repeat
while g ≤ number of generations per cycle
end while
Cycle

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DEC Algorithm
Repeat
for each individual Ip in P
end for
end while
Cycle
Generation

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DEC Algorithm
Repeat
Perform crossover to generate offspring Ic
end for
end while
Cycle
Generation

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DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Cycle
Generation

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DEC Algorithm
Repeat
Local optimize Ic
Compare Ic and Ip and keep one of them in P
end for
end while
Cycle
Generation

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DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Cycle
Generation
Perturb P

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DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Exploration
[<70%
generations]
Perturb P

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DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Exploitation
[≥70%
generations]
Perturb P

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DEC Algorithm
D ← Euclidean distance /*distance measure*/
Repeat
Local optimize Ic
end for
end while
Perturb P
D ← MinMax distance [≥ 70% cycles]

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DEC Algorithm
D ← Euclidean distance
Repeat
Local optimize Ic
end for
end while
Perturb P
D ← MinMax distance
Return the best clustering found

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Initialization
DEC Algorithm
Repeat
Local optimize Ic
Compare Ic and Ip and keep better individual in P
end for
end while
Perturb P
If exploitation phase, D ← MinMax distance

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Encoding

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Encoding
Each individual I is a triple,
I = (Ω, T , C)

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Encoding
I = (Ω, T , C)
where Ω = {ω1, . . . , ωK } are the centroids,

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Encoding
I = (Ω, T , C)
T = {t1, . . . , tK }, 0 ≤ ti ≤ 1 , are the thresholds, and

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Encoding
I = (Ω, T , C)
T = {t1, . . . , tK }, 0 ≤ ti ≤ 1 , are the thresholds, and
C is the clustering for the set of centroids.

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Encoding
1 2 3 K − 1 K
(ω1, t1) (ω2, t2) (ω3, t3) · · · (ωK−1, tK−1) (ωK , tK )

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Encoding
1 2 3 K − 1 K
(ω1, t1) (ω2, t2) (ω3, t3) · · · (ωK−1, tK−1) (ωK , tK )
C1 C2 ϕ
CK−1
ϕ

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Encoding
1 2 3 K − 1 K
(ω1, t1) (ω2, t2) (ω3, t3) · · · (ωK−1, tK−1) (ωK , tK )
C1 C2 ϕ
CK−1
ϕ
Inactive centroid Active centroid

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Setup

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Setup
Create a population P of N individuals, where

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Setup
N/3 individuals have 90% − 100% active centroids

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Setup
N/3 individuals have up to 70% − 80% active centroids

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Setup
N/3 individuals have up to 50% active centroids

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Setup
A clustering is then computed by assigning items to their
closest active centroids.

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Setup
A clustering is then computed by assigning items to their
closest active centroids.
Centroids are recomputed periodically.

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Fitness of Individual

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Fitness

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Fitness
Each individual in the population is assigned a ﬁtness determined by the
validity index calculated for its clustering. The validity indexes used are:

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Fitness
DB index,

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Fitness
DB index,
CS index, or

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Fitness
DB index,
CS index, or
PBM index

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Evolution stage
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Crossover
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Crossover

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Crossover
Idea

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Crossover
Idea
For each individual Ip (parent) in the population, an offspring Ic is
generated using crossover as follows:

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Crossover
Idea
Ic ← (Ωc, Tc, Cc)

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Crossover
Idea
Randomly select three unique individuals Ix , Iy , Iz (donors) from the
population.

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Crossover
Idea
Randomly select three unique individuals Ix , Iy , Iz (donors) from the
population.
The threshold vector Tc is computed using:
tc
i =
{
tx
i + σt (t
y
i
− tz
i ) if URandom(0, 1) < η
tp otherwise,
where η ∈ [0, 1] is the crossover probability, σt is the scaling factor and
URandom(0, 1) denotes a random number selected from [0, 1].

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Crossover
Idea
The centroid vector, Ωc is computed in a similar fashion,
ωi =
{
ωx
i + σω(g)(ωy
i − ωz
i ) if URandom(0, 1) < η
ωp
i otherwise,
but with a different scaling factor, σω(g), that changes based on
exploration and exploitation phase.

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Crossover
Example of Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .

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Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .
centroid boundary

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Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .
Exploration
From Ip
Offspring ωc
11
. . . ωc
1f

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Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .
Exploration
From Ip From Ix , Iy , Iz
Offspring ωc
11
. . . ωc
1f
ωc
21
. . . ωc
2f

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Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .
Exploration
From Ip From Ix , Iy , Iz From Ix , Iy , Iz
Offspring . . . .ωc
11
. . . ωc
1f
ωc
21
. . . ωc
2f
ωc
i1
. . . ωc
if

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Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .
Exploration
From Ip From Ix , Iy , Iz From Ix , Iy , Iz From Ip
11
. . . ωc
1f
ωc
21
. . . ωc
2f
ωc
i1
. . . ωc
if
ωc
K1
. . . ωc
Kf

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Crossover
Parent Ip ωp
11
. . . ωp
1f
ωp
21
. . . ωp
2f
ωp
i1
. . . ωp
if
ωp
K1
. . . ωp
Kf
. . . .
Donor Ix ωx
11
. . . ωx
1f
ωx
21
. . . ωx
2f
ωx
i1
. . . ωx
if
ωx
K1
. . . ωx
Kf
. . . .
Donor Iy ωy
11
. . . ωy
1f
ωy
21
. . . ωy
2f
ωy
i1
. . . ωy
if
ωy
K1
. . . ωy
Kf
. . . .
Donor Iz ωz
11
. . . ωz
1f
ωz
21
. . . ωz
2f
ωz
i1
. . . ωz
if
ωz
K1
. . . ωz
Kf
. . . .
From Ip From Ix , Iy , Iz From Ix , Iy , Iz From Ip
Exploitation
different σ for each feature f
11
. . . ωc
1f
ωc
21
. . . ωc
2f
ωc
i1
. . . ωc
if
ωc
K1
. . . ωc
Kf

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Local Optimization
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Local Optimization

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Local Optimization
Objective

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Local Optimization
Objective
Improve the quality of the offspring after crossover.

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Local Optimization
Objective
How

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Local Optimization
Objective
How
The three techniques implemented for local optimization are:

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Local Optimization
Objective
How
BreakUp

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Local Optimization
Objective
How
BreakUp
Breaking up a large cluster into smaller clusters, where a cluster
with more than 40% items of the data set is considered large.

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Local Optimization
Objective
How
BreakUp
Merge

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Local Optimization
Objective
How
BreakUp
Merge
Merging two close clusters to see if ﬁtness can be improved.

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Local Optimization
Objective
How
BreakUp
Merge
Scatter

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Local Optimization
Objective
How
BreakUp
Merge
Scatter
Redistributing items of a cluster to other clusters to see if ﬁtness
can be improved.

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Local Optimization
Exploration phase

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Local Optimization
Exploration phase
BreakUp

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Local Optimization
Exploration phase
BreakUp
Exploitation phase

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Local Optimization
Exploration phase
BreakUp
Exploitation phase
BreakUp

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Local Optimization
Exploration phase
BreakUp
Exploitation phase
BreakUp
Merge

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Local Optimization
Exploration phase
BreakUp
Exploitation phase
BreakUp
Merge
Scatter

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BreakUp : A clustering of an individual

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BreakUp : Largest cluster
Largest Cluster

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Largest Cluster
marks the closest item selected

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Largest Cluster
marks the closest item selected
marks the centroid for the red items.

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Merge

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Merge
Objective

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Merge
Objective
The aim is to improve the clustering by recombining clusters that are similar.

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Merge
Objective
How

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Merge
Objective
How
Find closest clusters Ci and Cj in offspring I.

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Merge
Objective
How
If on combining Ci and Cj , ﬁtness of I improves, merge and form one
bigger cluster.

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Merge
Objective
How
bigger cluster.
Else, search for next closest pair of clusters.

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Merge
Objective
How
bigger cluster.
Else, search for next closest pair of clusters.
Repeat till k pairs of clusters have been merged, or all pairs have been
considered.

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Scatter

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Scatter
Objective

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Scatter
Objective
The aim is to get rid of clusters that might affect the ﬁtness adversely.

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Scatter
Objective
How

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Scatter
Objective
How
Find the smallest cluster C in I.

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Scatter
Objective
How
If ﬁtness of I improves on redistributing items from C to other clusters,
do it.

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Scatter
Objective
How
do it.
Else, search for the next smallest cluster.

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Scatter
Objective
How
do it.
Else, search for the next smallest cluster.
Repeat till k clusters have been scattered, or every cluster has been
considered.

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Replacement
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Replacement
If ﬁtness of Ic is better than Ip, replace Ip with Ic in the
population.

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Replacement
If ﬁtness of Ic is better than Ip, replace Ip with Ic in the
population.
Else, accept Ic with a probability µ, where µ decreases with
time.

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Perturb
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Perturb population

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Perturb population
Idea

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Perturb population
Idea
In order to prevent premature convergence and avoid getting stuck in local
optima, the population is perturbed at the end of every cycle.

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Perturb population
Idea
How

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Perturb population
Idea
How
Randomly select a number of individuals (not the best) to be modiﬁed
from the current population.

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Perturb population
Idea
How
Tweak the centroids and threshold values for the selected individual
with a small probability.

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Perturb population
Idea
How
Tweak the centroids and threshold values for the selected individual
with a small probability.
Recompute the clustering.

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Change Distance Measure
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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In Euclidean distance, features with larger magnitudes
dominate the distance calculation.

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In Euclidean distance, features with larger magnitudes
dominate the distance calculation.
This problem is addressed by using MinMax distance.

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Termination condition
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Termination Criteria

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The algorithm stops when one of the following conditions is
met:
2500 generations have passed, or

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The algorithm stops when one of the following conditions is
met:
2500 generations have passed, or
70% cycles have passed and ﬁtness of the best offspring
has not improved in the last 100 generations.

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Return Best
DEC Algorithm
Repeat
Local optimize Ic
end for
end while
Perturb P

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Notes
An individual in DEC can have at most K number of
clusters.

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Notes
clusters.
A clustering solution in DEC needs to have at least two
clusters.

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Notes
clusters.
clusters.
The user can set the minimum number of clusters a
solution must have.

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Notes
clusters.
clusters.
The user can set the minimum number of clusters a
solution must have.
The minimum number of clusters set affects the quality of
the solution obtained.

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Implementation Details
Implemented in C++ and compiled using g++ and the -O3
optimization ﬂag.

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Implementation Details
Implemented in C++ and compiled using g++ and the -O3
optimization ﬂag.
Testing machines: Intel Core i5-3570 CPU at 3.40 GHz
and 32 GB of RAM with Ubuntu 14.04 OS.

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Experimental Setup
Data sets Size # of features # of clusters
Cancer 683 9 2
Cleveland 297 13 5
Dermatology 358 34 6
E.coli 336 7 8
Glass 214 9 6
Haberman 306 3 2
Heart 270 13 2
Ionosphere 351 33 2
Iris 150 4 3
M.Libras 360 90 15
Pendigits 3497 16 10
Pima 768 8 2
Sonar 208 60 2
Vehicle 846 18 4
Vowel 990 13 11
Wine 178 13 3
Yeast 1484 8 10
Tested with a total of 17
data sets.

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Experimental Setup
Cancer 683 9 2
Cleveland 297 13 5
E.coli 336 7 8
Glass 214 9 6
Haberman 306 3 2
Heart 270 13 2
Ionosphere 351 33 2
Iris 150 4 3
M.Libras 360 90 15
Pima 768 8 2
Sonar 208 60 2
Vehicle 846 18 4
Vowel 990 13 11
Wine 178 13 3
Yeast 1484 8 10
data sets.
Data sets selected based
on:

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Experimental Setup
Cancer 683 9 2
Cleveland 297 13 5
E.coli 336 7 8
Glass 214 9 6
Haberman 306 3 2
Heart 270 13 2
Ionosphere 351 33 2
Iris 150 4 3
M.Libras 360 90 15
Pima 768 8 2
Sonar 208 60 2
Vehicle 846 18 4
Vowel 990 13 11
Wine 178 13 3
Yeast 1484 8 10
data sets.
on:
Number of items [150
to 3500 items],

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Experimental Setup
Cancer 683 9 2
Cleveland 297 13 5
E.coli 336 7 8
Glass 214 9 6
Haberman 306 3 2
Heart 270 13 2
Ionosphere 351 33 2
Iris 150 4 3
M.Libras 360 90 15
Pima 768 8 2
Sonar 208 60 2
Vehicle 846 18 4
Vowel 990 13 11
Wine 178 13 3
Yeast 1484 8 10
data sets.
on:
to 3500 items],
Number of
classiﬁcations [2 to 15
classes],

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Experimental Setup
Cancer 683 9 2
Cleveland 297 13 5
E.coli 336 7 8
Glass 214 9 6
Haberman 306 3 2
Heart 270 13 2
Ionosphere 351 33 2
Iris 150 4 3
M.Libras 360 90 15
Pima 768 8 2
Sonar 208 60 2
Vehicle 846 18 4
Vowel 990 13 11
Wine 178 13 3
Yeast 1484 8 10
data sets.
on:
to 3500 items],
Number of
classes],
Number of features [3
to 90 features]

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Experimental Setup
Cancer 683 9 2
Cleveland 297 13 5
E.coli 336 7 8
Glass 214 9 6
Haberman 306 3 2
Heart 270 13 2
Ionosphere 351 33 2
Iris 150 4 3
M.Libras 360 90 15
Pima 768 8 2
Sonar 208 60 2
Vehicle 846 18 4
Vowel 990 13 11
Wine 178 13 3
Yeast 1484 8 10
data sets.
on:
to 3500 items],
Number of
classes],
Number of features [3
to 90 features]
30 runs for each data set.

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Results

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Results
Results for DEC were collected to analyze:
the quality of the solution based on different validity
indexes,

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Results
indexes,
the impact of distance measures on the performance of the
algorithm,

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Results
indexes,
algorithm,
different methods for computing clustering,

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Results
indexes,
algorithm,
the effect of minimum number of clusters (Kmin), and

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Results
indexes,
algorithm,
the running time of the algorithm.

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Results
Results for DEC algorithm were collected to analyze:
indexes,
algorithm,
the effect of minimum number of clusters (Kmin),and

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Effect of Validity Index
Wine
Ground Truth
Class 1 Class 2 Class 3
59 items 71 items 48 items
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Using DB Index

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Wine
Ground Truth
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Using DB Index
Wine
Ground Truth
Cluster 1 36 0 0
36 items
Cluster 2 23 63 0
86 items
Cluster 3 0 8 48
56 items
Using CS Index

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Wine
Ground Truth
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Using DB Index
Wine
Ground Truth
Cluster 1 36 0 0
36 items
Cluster 2 23 63 0
86 items
Cluster 3 0 8 48
56 items
Using CS Index
Wine
Ground Truth
Cluster 1 49 6 0
55 items
Cluster 2 4 13 1
18 items
Cluster 3 6 52 47
105 items
Using PBM Index

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Effect of distance measure
Wine
Ground Truth
Cluster 1 17 1 0
18 items
Cluster 2 14 70 48
132 items
Cluster 3 28 0 0
28 items
Euclidean distance (0.47)

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Wine
Ground Truth
Cluster 1 17 1 0
18 items
Cluster 2 14 70 48
132 items
Cluster 3 28 0 0
28 items
Wine
Ground Truth
Cluster 1 47 0 0
47 items
Cluster 2 12 69 0
81 items
Cluster 3 0 2 48
50 items
MinMax distance (1.65)

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Wine
Ground Truth
Cluster 1 17 1 0
18 items
Cluster 2 14 70 48
132 items
Cluster 3 28 0 0
28 items
Wine
Ground Truth
Cluster 1 47 0 0
47 items
Cluster 2 12 69 0
81 items
Cluster 3 0 2 48
50 items
MinMax distance (1.65)
Wine
Ground Truth
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Euclidean & MinMax (1.12)

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Results
Results for DEC algorithm were collected to analyze:
indexes,
algorithm,

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Effect of clustering method

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Different clusterings can be obtained for a given set of
centroids.

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centroids.
DEC uses the following two methods to recompute
clustering for the ﬁnal set of centroids, Ω:

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centroids.
The clustering is recomputed using one iteration of
K-means algorithm on Ω (CA1).

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centroids.
The clustering is recomputed using one iteration of
K-means algorithm on Ω (CA1).
Same as above but centroids are recomputed as well
(CA2).

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GTK
Ground Truth
Cluster 1 50 3 1
54 items
Cluster 2 0 49 17
66 items
Cluster 3 9 19 30
58 items
GT Using CA1 (1.14)

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GTK
Ground Truth
Cluster 1 50 3 1
54 items
Cluster 2 0 49 17
66 items
Cluster 3 9 19 30
58 items
GT Using CA1 (1.14)
Wine
Ground Truth
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Using DEC (1.12)

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GTK
Ground Truth
Cluster 1 50 3 1
54 items
Cluster 2 0 49 17
66 items
Cluster 3 9 19 30
58 items
GT Using CA1 (1.14)
Wine
Ground Truth
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Using DEC (1.12)
Wine
Ground Truth
Cluster 1 50 4 2
56 items
Cluster 2 0 43 8
51 items
Cluster 3 9 24 38
71 items
DEC Using CA1 (1.80)

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GTK
Ground Truth
Cluster 1 50 3 1
54 items
Cluster 2 0 49 17
66 items
Cluster 3 9 19 30
58 items
GT Using CA1 (1.14)
Wine
Ground Truth
Cluster 1 51 4 0
55 items
Cluster 2 8 64 0
72 items
Cluster 3 0 3 48
51 items
Using DEC (1.12)
Wine
Ground Truth
Cluster 1 50 4 2
56 items
Cluster 2 0 43 8
51 items
Cluster 3 9 24 38
71 items
Wine
Ground Truth
Cluster 1 46 1 0
47 items
Cluster 2 0 50 17
67 items
Cluster 3 13 20 31
64 items

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Effect of Kmin
2 4 6 8 10
1
2
3
4
Wine-GT
Wine-GTk
Glass-GT
Glass-GTk
E.coli-GT
E.coli-GTk
Minimum Number of Clusters Required (Kmin)
DBValidityIndex
E.coli-DB
Wine-DB
Glass-DB
Better ﬁtness with
lower value of Kmin.

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Effect of Kmin
2 4 6 8 10
0
1
2
3
4
Wine-GT
Wine-GTk
Glass-GT
Glass-GTk
E.coli-GTE.coli-GTk
CSValidityIndex
E.coli-CS
Wine-CS
Glass-CS
Worse ﬁtness for
ground truth
shows that external
factors
might have been
used to
determine the
clustering.

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Effect of Kmin
2 4 6 8 10
0
200
400
Wine-GT
Wine-GTk
Glass-GT Glass-GTk
E.coli-GT
E.coli-GTk
PBMValidityIndex
E.coli-PBM
Wine-PBM
Glass-PBM
Worse ﬁtness for
ground truth
shows that external
factors
might have been
used to
determine the
clustering.
DEC converges to
Kmin
clusters due to this
characteristic of the
validity index.

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Running Time

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Running Time
The DEC algorithm has a running time of O(n2).

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Running Time
DEC takes longer to ﬁnish for CS Index as compared to DB
and PBM indexes.

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Running Time
and PBM indexes.
Example approximate run times of DEC:

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Running Time
and PBM indexes.
Example approximate run times of DEC:
size = 200, f ∈ [2, 15]: 200 seconds
size = 300, f > 30: 600 seconds
size = 700, f ∈ [2, 15]: upto 30 minutes
size ≥ 1000, f ∈ [2, 15]: upto 5 hours

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DEC vs Existing Algorithms
Data Set Algorithm
CS DB
# of clusters Index # of clusters Index
Cancer
DEC(2) 2.00 1.13 2.82 0.58
ACDE 2.00 0.45 2.05 0.52
DCPSO 2.25 0.48 2.50 0.57
GCUK 2.00 0.61 2.50 0.63
Classical DE 2.25 0.89 2.10 0.51
Average Link 2.00 0.90 2.00 0.76
Glass
DEC(6) 6.00 0.30 6.00 1.02
DEC(2) 2.00 0.10 2.00 0.84
ACDE 6.05 0.33 6.05 1.01
DCPSO 5.95 0.76 5.95 1.51
GCUK 5.85 1.47 5.85 1.83
Classical DE 5.60 0.78 5.60 1.66
Average Link 6.00 1.02 6.00 1.85

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Data Set Algorithm
CS DB
Cancer
DEC(2) 2.00 1.13 2.82 0.58
ACDE 2.00 0.45 2.05 0.52
DCPSO 2.25 0.48 2.50 0.57
GCUK 2.00 0.61 2.50 0.63
Classical DE 2.25 0.89 2.10 0.51
Average Link 2.00 0.90 2.00 0.76
Glass
DEC(6) 6.00 0.30 6.00 1.02
DEC(2) 2.00 0.10 2.00 0.84
ACDE 6.05 0.33 6.05 1.01
DCPSO 5.95 0.76 5.95 1.51
GCUK 5.85 1.47 5.85 1.83
Classical DE 5.60 0.78 5.60 1.66
Average Link 6.00 1.02 6.00 1.85
DEC(2) represents the results
when Kmin is set to 2

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Data Set Algorithm
CS DB
Cancer
DEC(2) 2.00 1.13 2.82 0.58
ACDE 2.00 0.45 2.05 0.52
DCPSO 2.25 0.48 2.50 0.57
GCUK 2.00 0.61 2.50 0.63
Classical DE 2.25 0.89 2.10 0.51
Average Link 2.00 0.90 2.00 0.76
Glass
DEC(6) 6.00 0.30 6.00 1.02
DEC(2) 2.00 0.10 2.00 0.84
ACDE 6.05 0.33 6.05 1.01
DCPSO 5.95 0.76 5.95 1.51
GCUK 5.85 1.47 5.85 1.83
Classical DE 5.60 0.78 5.60 1.66
Average Link 6.00 1.02 6.00 1.85
DEC(k) represents the results
when Kmin is set to known
number of clusters (ground
truth)

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Data Set Algorithm
CS DB
Iris
DEC(3) 3.00 0.60 3.00 0.56
DEC(2) 3.00 0.60 2.00 0.42
ACDE 3.25 0.66 3.05 0.46
DCPSO 2.23 0.73 2.25 0.69
GCUK 2.35 0.72 2.30 0.73
Classical DE 2.50 0.76 2.50 0.58
Average Link 3.00 0.78 3.00 0.84
Wine
DEC(3) 3.00 0.94 3.00 1.12
DEC(2) 2.00 0.70 2.00 0.96
ACDE 3.25 0.92 3.25 3.04
DCPSO 3.05 1.87 3.05 4.34
GCUK 2.95 1.58 2.95 5.34
Classical DE 3.50 1.79 3.50 3.39
Average Link 3.00 1.89 3.00 5.72

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Summary
The clustering returned by DEC is affected by:

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Summary
the distance measure,

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Summary
the validity index,

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Summary
the validity index,
the value of Kmin, and

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Summary
the validity index,
the value of Kmin, and
the method for computing the clustering.

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DEC on Iris

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DEC on Iris
2 2.5 3 3.5 4 4.5
2
4
6
Sepal Width (y)
PetalLength(z)

ClusteringDEC

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