Clustering Using Shared Reference Points Algorithm Based On a Sound Data Model

Mohamed A. Abbas & Amin A. Shoukry
International Journal of Data Engineering (IJDE), Volume (3) : Issue (2) : 2012 28
Clustering Using Shared Reference Points Algorithm
Based On a Sound Data Model
Mohamed A. Abbas mohamed.alyabbas@gmail.com
Graduate student with the College of Computing and
Information Technology
Arab Academy for Science and Technology
Alexandria, P.O. Box 1029, Egypt.
Amin A. Shoukry amin.shoukry@ejust.edu.eg
Department of Computer Science and Engineering
Egypt-Japan University of Science and Technology
New Borg El-Arab City, P.O. Box 179, Egypt
Abstract
A novel clustering algorithm CSHARP is presented for the purpose of finding clusters of
arbitrary shapes and arbitrary densities in high dimensional feature spaces. It can be
considered as a variation of the Shared Nearest Neighbor algorithm (SNN), in which each
sample data point votes for the points in its k-nearest neighborhood. Sets of points sharing a
common mutual nearest neighbor are considered as dense regions/ blocks. These blocks are
the seeds from which clusters may grow up. Therefore, CSHARP is not a point-to-point
clustering algorithm. Rather, it is a block-to-block clustering technique. Much of its
advantages come from these facts: Noise points and outliers correspond to blocks of small
sizes, and homogeneous blocks highly overlap. The proposed technique is less likely to
merge clusters of different densities or different homogeneity. The algorithm has been applied
to a variety of low and high dimensional data sets with superior results over existing
techniques such as DBScan, K-means, Chameleon, Mitosis and Spectral Clustering. The
quality of its results as well as its time complexity, rank it at the front of these techniques.
Keywords: Shared Nearest Neighbors, Mutual Neighbors, Spatial Data, High Dimensional
Data, Time Series, Cluster Validation.
1. INTRODUCTION
The present paper is a modified version of [25]. The modification includes the incorporation of
a new measure of cluster homogeneity (section 2.2) which has been used in defining a strict
order for cluster’s propagation in the proposed algorithm (section 3). Also, new validation
indexes; as well as new data sets; have been adopted for the purpose of comparing the
proposed algorithm with the previous techniques (section 4).
Clustering of data is an important step in data analysis. The main goal of clustering is to
divide data objects into well separated groups so that objects lying in the same group are
more similar to one another than to objects in other groups.
Given a set = of data objects (sample points) to be clustered, where is the
i-th object. is an -dimensional column vector. consisting of
measured attributes ( is the dimensionality of the feature space). The Jarvis-Patrick
clustering technique [12], needs a measure of the distance between two objects and two
integers: and . is the size of the neighborhood list, and is the number of common
neighbors. This method works as follows:
Determine the -nearest neighbors for each object in the set to be clustered. Two objects are
placed in the same cluster if they are contained in each other’s nearest neighbors list:
K-NBj and K-NBi, where K-NBj and K-NBi denote the K-nearest neighbors of data points
and , respectively. (1)
They have at least nearest neighbors in common: . (2)

As stated in [15], the principle of K-NB consistency of a cluster states that for any data object
in a cluster its K-Nearest Neighbors should also be in the same cluster. The principle of K-
Mutual Nearest-Neighbor consistency (K-MNB consistency) states that for any data object in
a cluster its K-Mutual Nearest-Neighbors should also be in the same cluster. The principle of
cluster K-MNB consistency is stronger than the cluster K-NB consistency concept, and it is
also a more strict representation of the natural grouping in the definition of clustering. In the
present work, the concept of K-MNB is used in developing a new clustering algorithm,
CSHARP, (Clustering using SHAred Reference Points). CSHARP is a density based
clustering algorithm in which dense regions are identified using mutual nearest neighborhood.
Then, sets of points sharing a common mutual nearest neighbor are considered instead of
points themselves in an agglomerative process.
1.1 Proposed Technique
Specifically, the technique proposed in this paper:
• Determines; for every data point; the largest possible set of points satisfying condition
(1) only (the cardinality of this set lies in the range . For a point , this set is
called its Reference-List and is denoted RL . Point is considered as the
"Representative Point" (or "Reference Point") of this Reference-List. Such sets of
points are the seeds from which clusters may grow up.
• Avoids an early commitment to condition (2), and, instead, proceeds directly from
point(s) to set(s) relation(s) (the Reference-Lists); instead of point(s) to point(s)
relation(s) (as required by conditions (1) and (2), above). Therefore, data in CSHARP
is processed as blocks of points not as individual points. Reference lists constitute a
key concept in the proposed algorithm. To preserve K-MNB consistency, CSHARP
processes data as blocks of tiny groups of Reference-Lists, on which clusters are
built, agglomeratively.
• Allows clusters to be linked if their Reference- Lists share a number of points .
Thus, parameter controls the extent to which mutual neighborhood consistency is
satisfied. Allowing clusters to grow in a chain is similar; though not identical; to the
reachability relation in DBScan algorithm [5]. Reference-Lists of size (a selected
threshold) are either considered as noise or may merge with other clusters as will be
explained in section 2.
Several experiments conducted on a variety of data sets show the efficiency of the proposed
technique for the detection of clusters of different sizes, shapes and densities; whether in low
or high dimensional feature spaces; in the presence of noise and outliers. Although the
motivations behind the algorithm are quite heuristic, the experimental work showed the
validity of the proposed approach.
1.2 Overview of Related Algorithms
Many clustering techniques have been developed based on different concepts. Several
approaches utilize the concept of cluster center or centroid, other methods build clusters
based on the density of the objects, and a lot of methods represent the data objects as
vertices of graphs where edges represent the similarity between these objects.
Centroid based algorithms represent each cluster by using the centre of gravity of its
instances. The most well-known centroid algorithm is the K-means [11]. The K-means method
partitions a data set into k subsets such that all points in a given subset are close to the same
centre. K-means then computes the new centers by taking the mean of all data points
belonging to each cluster. The operation is iterated until there is no change in the centers
locations. The result strongly depends on the initial guess of centroids, besides, it does not
perform well on data with outliers or with clusters of different sizes or non globular shapes.
The key idea of density-based clustering is that for each instance of a cluster, a neighborhood
of a given radius has to contain at least a minimum number of instances. One of the most well
known density-based clustering algorithms is the DBScan [5]. In DBScan the density
associated with an object is obtained by counting the number of objects in a region of a
specified radius, , around the object. An object with density greater than or equal to a
specified threshold, MinPts, is treated as a core (dense), otherwise it is considered as a non-
core (sparse) object. Non-core objects that do not have a core object within the specified

radius are discarded as noise. Clusters are formed around core objects by finding sets of
density connected objects that are maximal with respect to density-reachability. While
DBScan can find clusters of arbitrary sizes and shapes, it cannot handle data containing
clusters of different densities, since it uses global density parameters, MinPts and , which
specify only the lowest possible density of any cluster.
Chameleon [13] and [4] algorithms attempt to obtain clusters with variable sizes, shapes
and densities based on K-nearest neighbor graphs. Chameleon finds the clusters in a data
set by using a two-phase algorithm. In the first phase, it generates a K-nearest neighbor
graph that contains links between a point and its K-nearest neighbors. Then it uses a graph
partitioning algorithm to cluster the data items into a large number of relatively small sub-
clusters. During the second phase, it uses an agglomerative hierarchical clustering algorithm
to find the genuine clusters by repeatedly combining together these sub-clusters. The Shared
Nearest Neighbors clustering algorithm, [4] uses K-nearest neighbor approach for
density estimation. It constructs a K-nearest neighbor graph in which each data object
corresponds to a node which is connected to the nodes corresponding to its K-nearest
neighbors. From the K-nearest neighbor graph a shared nearest neighbor graph is
constructed, in which edges exist only between data objects that have each other in their
nearest neighbor lists. A weight is assigned to each edge based on the number and ordering
of shared neighbors. Clusters are obtained by removing all edges from the shared nearest
neighbor graph that have a weight below a certain threshold .
A recent clustering algorithm, Mitosis [22], is proposed for finding clusters of arbitrary shapes
and arbitrary densities in high dimensional data. Unlike previous algorithms, it uses a dynamic
model that combines both local and global distance measures. The model is depicted in the
proposed dynamic range neighborhood, and the proposed clustering criteria which use
distance relatedness to merge patterns together. Mitosis uses two main parameters and .
Parameter , controlling the neighborhood size, is the main parameter which decides the
lower bound on the number of clusters that can be obtained. The value of , should be varied
in an incremental fashion so as not to deteriorate the speed of the algorithm. It can be
selected just above the value of , and increased by small steps, to avoid unnecessary large
neighborhood sizes. Parameter controls the degree of merging patterns/clusters together,
within the limits of the neighborhood decided by . Increasing values of , for the same
value, means decreasing the number of clusters obtained, and vice versa.
In recent years, spectral clustering [20] has become one of the most popular modern
clustering techniques. It starts from a similarity matrix between data objects, modifies it to a
sparse matrix, then computes its Laplacian matrix "L". The first k eigenvectors of L constitute
the first k columns of a matrix . K-means algorithm is applied then on the row vectors of a
normalized version of matrix (where each original data object is assigned to the same
cluster to which the corresponding row vector in is assigned to). Spectral clustering is
simple to implement and can be solved efficiently on standard linear algebra software.
Additionally, it is more effective in finding clusters than some traditional algorithms such as K-
means. However, it suffers from a scalability problem discussed in [17]. It cannot successfully
cluster data sets that contain structures at different scales of size and density. To overcome
these limitations, a novel spectral clustering algorithm [2] is proposed for computing spectral
clustering using a sparse similarity matrix.
1.3 Outline of the Paper
The rest of this paper is organized as follows. Section 2 describes our approach for the
definition of similarity and density (or reference points structure), which is the key concept to
our clustering algorithm. Section 3 describes the algorithm itself followed by a logical model of
the data entities it manipulates, a graph-based interpretation of its effect and its time
complexity analysis. An anatomy of the proposed algorithm with respect to related algorithms
is discussed next. Section 4 presents the data sets used in the experiments conducted to
evaluate the performance of the algorithm using well known cluster validation indexes.
Section 5 presents a short conclusion and possible future work.

2. BASIC DEFINITIONS
In this section we describe the basic concepts used in the proposed algorithm.
2.1 Reference-List and Reference-Block
As shown in Figure 1, although the Euclidean distance is a symmetric metric, from the
perspective (and considering =4), point is in 4-NBB, however, point is not in 4-NBA.
Given a set of points , let = be the ordered list of points
according to their distances from a point , and let K-NB be the K-
nearest neighbors of . This represents the list of points that point refers to. If K-NB ,
then is referred by . Let RB denote the set of points having in their K-Nearest
Neighborhood. Then, K-NB RB represents a set of dense points called a Reference-
List, RL , associated with point . is called the representative point of RL . Each
representative point with its Reference-List, RL constitute a Reference-Block. Points in a
Reference-Block are shown in Figure 2.
FIGURE 1: Concept of mutual neighboring: is in 4-NBB, however, is not in 4-NBA.
FIGURE 2: A Reference-List (points within circle), associated with a representative point “A” ). All
together they constitute a Reference-block.
There are three possible types of representative points:
• Strong Points (or Reference Points), representing blocks of size greater than a pre-
defined threshold parameter .
• Noise points, representing empty Reference-Lists. These points will be excluded
initially from final clusters.
• Weak points which are neither strong points nor noise points. These points may be
merged with another existing clusters if they are members of another strong points.

2.2 Homogeneity Factor
A homogeneity measure is proposed here for the ordering of the cluster’s propagation
process. While CSHARP[25] performs this process according to the cardinality of the
reference lists (i.e their densities), the algorithm proposed here, considers both the density
and homogeneity of the blocks that will be granted priority to propagate first.
Given RL = {q1 , q2 , …qc}, a Reference List associated with point pi having cardinality
c = | RL | > (i.e. corresponding to a strong reference point), its homogeneity factor αi is
computed as:
αi = (3)
Where is the average distance between pi and its associated reference list points qj,
computed as:
= , (4)
and is the maximum distance between pi and its associated reference list points qj,
computed as:
,
(5)
Figure 3 shows three cases of strong reference lists with their corresponding homogeneity
factors. Note that the odd case of a reference list with cardinality c=1 in which αi=1 is
excluded, since it does not correspond to a strong reference list. In the experiments described
in section four, T is always greater than 2.
FIGURE 3: Three reference lists with different homogeneity factors (a) 0.98, (b) 0.80 and (c) 0.55
2.3 Cluster’s Propagation
Given two clusters such that ; where is a chosen parameter, called merge-
parameter, then the two clusters can be merged together. Hence, measures the minimum
link strength required between two clusters. In CSHARP data is processed as blocks of points
(reference-blocks) not as individual points. Two clusters are merged if they share a number of
points equal to or greater than some threshold M; as explained in Figure 4.
A b
FIGURE 4: Two steps of Cluster propagation with M = 2 (the merging parameter) for (a) Ck =
such that M and (b) Cu = Ck Cl such that M.

To illustrate the process of clusters propagation, Chameleon’s data set DS5 is used. DS5
consists of 8000 spatial data points. All genuine clusters were detected at the parameters
setting( =24, =18, =6). The number of strong points obtained were 5122.
Several snapshots of the clustering process are shown in Figure 5. Figure 5, also, illustrates
how clusters propagate agglomeratively, and simultaneously, in CSHARP.
a b c
D e f
g h i
FIGURE 5: Nine snapshots of Modified CSHARP cluster propagation for the data set Chameleon DS5
at different iterations (a) 10, (b) 25, (c) 100, (d) 500, (e) 1000, (f) 2000, (g) 3000, (h) 4000 and (i) 5122.

3. MODIFIED CSHARP ALGORITHM
Figure 6 describes the modified CSHARP algorithm. Next, a logical data model and a graph-
based interpretation of the algorithm are given. Finally, the time complexity of this algorithm is
analyzed in section 3.2.
Input: Data points ;
{size of the neighborhood of a point};
{threshold parameter for the size of a reference list} and
{merge parameter for the size of the intersection between two reference blocks}.
Output: set of generated clusters.
1: Construct similarity matrix .
{Construct the Refer-To-List, K-NB for each point }
2: K-NB
{construct the Referred-By-List, RB for each point .
3:
4: K-NB
5: for all do
6: If K-NB then
7: RB
8: end if
9: end for
10: end for
{From K-NB and RB , Construct the reference Lists RL
11: for all do
12: RL = K-NB RB .
13: end for
14: Form a sorted (in a descending order) list L. L = is a representative point , based on
the densities (i.e. RL ). Exclude the weak points from list L (those points for which
|RL ).
15: Sort the new list according to the homogeneity factors αi’s of the Reference-Lists RL .
{Building Clusters}
16: {Initialize i}
17: = RL
18: label point as belonging to cluster .
19: label each point in RL as belonging to cluster .
20: While do
21: {increment i}
22: = RL
23: label point as belonging to cluster .
24: label each point in RL as belonging to cluster .
25: , {where -1} then
26:
27: Update labels, by marking each point in as belonging to cluster
28: end if
29: end while
FIGURE 6: CSHARP Clustering Algorithm
Note that although a reference list associated with a weak point does not participate in

Cluster’s growing, a weak point may itself belong to a Reference list of another strong point.
This implies that this weak point is not considered as noise.
3.1 Logical Data Model and Graph-based Interpretation
Figure 7 depicts the conceptual data model underlying the proposed algorithm. Two entities
are shown; namely, the data sample and Reference-List associated with a data sample
entities; as well as a one one-to-many" relation between them and one "many-to-many"
relation associated with each of them.
• The relation “Has as member in its K-NB list” is not symmetric.
• The relation “Has as member” between a Reference List and its members is
symmetric.
• The relation “Overlaps” is symmetric but not transitive.
• The entity "Reference List Associated with a Data sample"; say , is obtained as the
intersection of two sets: the set of points having in their K-Nearest Neighborhood
and the K-Nearest Neighborhood of itself.
FIGURE 7: Conceptual data model.
Consider the following weighted graph , where a vertex in this graph corresponds to a
Reference-List and an edge corresponds to an Overlaps relation between two Reference
Lists. The weight of an edge corresponds to the cardinality of the intersection set between two
Reference lists. Now, the effect of the CSHARP algorithm can be viewed as follows: Edges
with weights less than the threshold are removed. This decomposes the graph into
independent components. The remaining connected vertices (i.e. vertices connected by
edges having weights greater than ) are combined, the union of the data samples belonging
to their reference lists correspond to the obtained clusters.
3.2 Time Complexity
The time complexity for computing the similarity matrix is O(N
2
), where N is the number of
data points. This can be reduced to O(N log N), by using a data structure such as a k-d tree
[1] or an R-tree[6]. The space complexity for t computing distances between samples is
O(NF) where N is the number of data points and F is the number of features (or dimensions).
The time complexity of the algorithm can be analyzed as follows:
• line 2, finding k-nearest neighbors(refer-to-list): k iterations through all N data points are
needed, hence it has a complexity of O(NK)
• lines 3-10, finding referred-by-list: it has, also, a complexity of O(NK)
• lines 11-13, finding reference-points: for each data point, its k-nearest neighbors are
searched for mutual neighborhood. As a binary search is adopted; its complexity is of O(K
log K), thus, the overall complexity of this step is O(NK log k).
• Line 14, computing homogeneity factor procedure: has a complexity of ), as k
iterations are needed for each of the N data points .

• line 15, sorting data sets procedure: sorting has a complexity of O(N log N), as binary sort
is used.
• lines 25-28, clusters overlapping procedure: detecting overlapping among clusters has a
linear complexity of O(K),since we iterate through reference-blocks of processed points
and detect previously labeled points.
• lines 20-29, clusters propagation procedure: this is the main procedure. It has a
complexity of O(NK) , since we iterate through all N data points, running clusters
overlapping procedure for each data point. Accordingly, the overall complexity is O(NK
log N) .
The overall time complexity for the modified CSHARP algorithm is O(N log N) where N is the
number of data points and K is the number of nearest neighbors. Modified CSHARP has a
space complexity of O(NK), where N is the number of data points and K is the number of
nearest neighbors used
3.3 Anatomy of CSHARP vs. Jarvis-Patrick and SNN Algorithms
• In SNN, similarity between two points p and q is computed as the number of nearest
neighbors they share. In contrast, CSHARP's similarity is computed as the number of
reference points two blocks share.
• In contrast to Jarvis-Patrick and SNN, CSHARP is a block-to-block rather than point-to-
point clustering.
• Jarvis-Patrick and SNN work with k-nearest neighborhood which corresponds to a non-
symmetric relationship between data points. On the other hand, CSHARP relies on a
symmetric relation between any point and its reference list points.
• Jarvis-Patrick and SNN use static k-nearest neighbor lists, while CSHARP starts with
static k-nearest neighbor lists then turns them into dynamic Reference Lists.
• In Jarvis-Patrick and SNN, the association between points is one-to-one, while it is one-to
many in CSHARP.
• Both SNN and CSHARP propagate clusters simultaneously and agglomeratively, while
Jarvis-Patrick builds a similarity graph then decomposes it by removing the weakest links.
4. EXPERIMENTAL RESULTS
4.1 Datasets Used
4.1.1. 2-d Chameleon’s DS5 data set
DS5 consists of 8,000 points. It is a mixture of clusters with different densities used by
Chameleon algorithm to illustrate its efficiency in obtaining arbitrary density clusters. The aim
is to classify the data into 8 clusters of different densities.
4.1.2. Eight Low Dimensional Datasets
• Iris data set: This is a well known database in the pattern recognition literature. The
data set contains 3 classes of 50 instances each, where each class refers to a type of
iris plant. One class is linearly separable from the others 2; the latter are NOT linearly
separable from each other [16].
• The Synthetic Control Charts (SCC) data set; obtained from UCR repository [14]; it
includes 600 patterns, each of 60 dimensions (time points).
• The pen digit character recognition data from the UCI repository [16], consists of
10992 patterns, each of 16 dimensions. Features (dimensions) are used to describe
the bitmaps of the digit characters. The aim is to properly classify the digit characters
to 10 classes from 0 to 9.
• Libras movement data set: The dataset contains 15 classes of 24 instances each,
where each class references to a hand movement type. It consists of 360 patterns,
each of 91 dimensions.
• The Breast Cancer Wisconsin Diagnostic data from the UCI repository, consists of
569 patterns, each of 30 dimensions. The aim is to classify the data into two
diagnosis (malignant or benign).
• SPECT heart data set: The dataset describes diagnosing of cardiac Single Proton

Emission Computed Tomography (SPECT) images. It consists of 267 patterns, each
of 22 dimensions. Each of the patients is classified into two categories: normal and
abnormal
• The Protein localization data; obtained from the UCI repository; is the Ecoli data set
with 336 proteins of seven dimensions. The aim is to properly classify it to eight
classes.
• The Protein Localization Sites, obtained from the UCI repository, consists of 1484
patterns, each of 8 dimensions. This database contains information about a set of
Yeast cells. The task is to determine the localization site of each cell by partitioning it
into 10 classes of varying distribution.
4.1.3. Two High Dimensional Datasets
• Corel image features data set: This dataset contains image features extracted from a
Corel image collection. 2074 images of 144 dimensions were selected in this
experiment according to criteria discussed in [2].
• Arcene data set which consists of 200 patterns each of 10,000 features. The task is
to distinguish cancer versus normal patterns from mass-spectrometric data. This is a
two-class classification problem with continuous input variables. Arcene has been
part of the NIPS 2003 feature selection challenge.
4.2 Cluster Validation
As described in [21], for a given data set, a clustering algorithm can always produce a
partitioning whether or not a particular structure in the data really exists. Different clustering
approaches usually yield different results. Even for the same algorithm, the selection of a
parameter or the presentation order of the input patterns may affect the final results.
Therefore, effective evaluation criteria are critically important to provide users with a degree of
confidence in the obtained clustering results. These assessments should be objective and
have no preferences to any algorithm. V-measure [24], Purity and Entropy [23] are used for
the purpose for clustering validation. Noise is taken into consideration in the validation
process. This reduces the indexes values than if a noise-free validation process is adopted.
However, this presents a more accurate assessment.
4.3 Results and Performance Evaluation
To compare the results obtained by the Modified CSHARP with those obtained by other
algorithms, the nine data sets presented above have been used.
4.3.1. Chameleon’s DS5 Data Set.
Five algorithms are compared: K-means, DBScan, SNN, Chameleon, and mitosis, in addition
to Modified CSHARP.
Due to the presence of different densities in the DS5 data set, DBScan either identifies the
lower density cluster but merges the two neighboring higher density ones, or do not identify
the lower density cluster, but identifies the higher density ones. DBScan at the parameters
setting ( = 10 and MinPts = 3) can only identify six rather than eight clusters. Similarly, SNN
merges two clusters together due to its static nature derived from DBScan. Both DBScan and
SNN , (Figure 8b, and d) were unable to obtain a good clustering solution. Modified CSHARP
obtained the genuine clusters at the parameters setting given in Table 1b for K in the range
[23...25], Figure 8f.
Mitosis has been able to obtain the genuine clusters at parameter settings ( = 2.15 and =
2.5) (Figure 8e). Mitosis results were obtained after discarding outliers. Clusters of sizes less
than 1% of the data size are identified as outliers. While Mitosis uses a static model for
discarding noise, CSHARP [25] focuses on the detection of chains of dense connected
regions (defined by strong points) possibly including non-dense regions ; represented by
weak points; as explained in section 2.1. Hence, in CSHARP any point not taken into
consideration during cluster’s propagation is considered as noise. Table 2 gives the numbers
and percentages of strong, weak and noise points found in all the investigated data sets.

TABLE 1: Modified CSHARP’s range of parameters setting for DS5 data set.
4.3.2. Eight Dimensional Data
Five data sets have been used to compare the results obtained by Modified CSHARP to
the ground truth as well as to the results obtained by DBScan, K-means, Mitosis and
Spectral clustering (as a state-of- the art technique [2]) algorithms. Chameleon and SNN
have not been included in these experiments, due to the difficulty of adjusting their
parameters. The Euclidean distance has been adopted as a metric for all data sets.
Numerous experiments (100 experiments at least per algorithm per dataset) have been
done on each algorithm to obtain its best indexes’ values. V-Measure, Purity, and Entropy
have been used to evaluate all the above clustering algorithms.
TABLE 2: Number of strong, weak and noise points found in the tested data sets and their percentages
;at given parameters settings; relative to the size of the corresponding data set.
M
23 [3...5] 7
23 19 [6,7]
23 22 [6…9]
24 [15…17] 7
24 18 6
24 22 [6...9]
24 23 [4...9]
25 [16…18] [7,8]
25 22 [6...9]
25 23 [5...9]
Dataset size Strong
Points
Weak
Points
Noise
Points
Chameleon DS5
(K=24, T=18 and M=6)
8000 6399
(79.98%)
1601
(20.02%)
225
(2.81%)
Synthetic Control Charts
(K=14, T=7 and M=4)
600 479
(79.83%)
121
(20.17%)
1
(0.17%)
Pen Digit Data
(K=31, T=14 and M=9)
10992 8732
(79.44%)
2260
(20.56%)
253
(2.30%)
Breast Cancer Diagnostic
(K=41, T=22 and M=14)
569 492
(86.47%)
77
(13.53%)
15
(2.64%)
Ecoli
(K= 22, T=11 and M=8)
336 240
(71.43%)
96
(28.57%)
25
(7.44%)
Yeast
(K= 44, T=25 and M=15)
1484 848
(57.14%)
636
(42.86%)
134
(9.03%)
Arcene
(K= 11, T=3 and M=1)
200 180
(90.0%)
20
(10.0%)
4
(2.0%)
SPECT Heart
(K= 30, T=7 and M=4)
267 222
(83.15%)
45
(16.85%)
18
(6.74%)
Libras Movement
(K= 10, T=7 and M=3)
360 173
(48.06%)
187
(51.94%)
46
(12.78%)
Iris
(K= 24, T=8 and M=9)
360 138
(92.00%)
12
(8.00%)
0
(0.00%)
Corel
(K= 43, T=21 and M=13)
2074 1230
(59.31%)
844
(40.69%)
120
(5.78%)

a b c
d e f
FIGURE 8: Results for the DS5 data set. Clusters are identified by a specific color, and a specific
marker for: (a) K-means clusters; (b) DBScan clusters; (c) Chameleon clusters; (d) SNN clusters; (e)
Mitosis clusters; and (f) Modified CSHARP clusters.
• Iris data set: Modified CSHARP and spectral clustering performed better for this data set,
reached the highest indexes for F-measure and Purity and lowest index for entropy as
shown in Figure 10a. The concept of relatedness fails with iris data set, since it consists
of three classes, two of them are not linearly separable; thus, Due to the merging of
these two clusters, Mitosis obtained the lowest indexes for this data set. Mitosis failed to
detect the original three clusters and detected only two clusters. On the contrary, Modified
CSHARP and Spectral clustering detected the three original clusters, Figure 11.
• Time series data: For SCC, The original six clusters are shown in Figure 9. All algorithms
(with the exception of CSHARP) failed to discover the cluster labeled normal, due to its
relatively low density. DBScan using the parameters setting ( = 50 and MinPts = 3)
detected the cyclic cluster. However, it merged the Up-Shift, and the Up-Trend clusters
together, as well as the Down-Shift and the Down-trend clusters. K-means, at
merged patterns from different clusters together. The values of the V-Measure, Purity and
Entropy are shown in Figure 10b. Due to the discovery of the Normal low dense cluster;
at the parameters setting ( =11, =5 and =3); CSHARP reached the maximum values
for both V-measure and Purity indexes and the lowest for Entropy index, Figure 12.
FIGURE 9: Time series clusters (mean ± standard deviation) of SCC data original classes.

• Pen Digits data set: CSHARP gave good solution and gave a very high Purity and low
Entropy relative to the other solutions as shown in Figure 10c. DBScan and K-means
combined several characters in one cluster. For instance, K-means combined 5 with (8,
9) and 1 with (2, 7), DBScan combined (1,2) and (5,9) and Mitosis combined 1 with (2, 7)
and (5,9).
• Disease diagnosis data set: For breast cancer diagnosis, CSHARP obtained the
maximum values for all the indexes at the parameters setting (K=50, T=8 and M=5).
Figure 13a gives the corresponding validity indexes values.
• SPECT heart and Libras Movement : Modified CSHARP performed better for all indexes
as shown in Figures 13b and c respectively.
• Ecoli and Yeast: The values of the indexes are given in Figures 13e and f, respectively
showing the superiority of CSHARP over all other algorithms.
a
b
c
FIGURE 10: V-Measure, Purity and Entropy for: (a) Iris data set, and (b) SCC time series, and (c) pen digit
recognition data set.

a b c
FIGURE 11: Plotting Iris data set using PCA for (a) reference solution, (b) Mitosis, and (c) Modified CSHARP
and Spectral clustering.
a b
c d
FIGURE 12: Plotting SCC time series data set using PCA for (a) reference solution, (b) Mitosis, (c)
Spectral clustering, and (d) Modified CSHARP.

a
b
c
d
e
FIGURE 13: V-Measure, Purity and Entropy for: (a) Breast cancer Wisconsin diagnostic data set, (b)
SPECT heart data set, and (c) Libras movement data set.(d) Ecoli data set, and (e) Yeast data set.

4.3.3. High Dimensional Data
To investigate the performance of the Modified CSHARP when applied to high dimensional
data sets, two data sets were used from the 2003 Nips Feature extraction challenge [7]. Both
CSHARP and spectral clustering obtained competitive results as shown in Figures 14a and b
for Corel and Arcene data sets, respectively, with CSHARP performing slightly better for all
indexes. Spectral clustering algorithm implemented by [2] uses two parameters, "K" for the
number of K-nearest neighbors; used to generate a sparse symmetric distance matrix and
value used in similarity function S,
where .
In this section, the efficiency of Modified CHARP is recorded when compared with well-
known clustering algorithms such as K-means, DBScan, Chameleon, Mitosis, and Spectral
Clustering. V-measure, Purity, and Entropy are used showing the superiority of CSHARP
over the other tested algorithms for almost all used data sets .
Modified CSHARP succeeded to overcome the limitations that the other algorithms suffer
from. It can deal with classes of different densities whereas DBScan cannot, deal with
arbitrary shapes whereas K-means cannot, deal with arbitrary cluster’s sizes where spectral
clustering cannot, and deal with interlaced (overlapped) clusters where Mitosis cannot.
Moreover, it can scale easily whereas Chameleon cannot. Therefore, it can be said that the
proposed technique is less likely to merge clusters of different densities or different
homogeneities as indicated by the obtained results. Next, the performance of Modified
CSHARP relative to the other algorithms is investigated.
4.4 Speed Performance
The speed of Modified CSHARP has been compared to the speed of CSHARP, Chameleon
and DBScan as shown in Figure. 15, using the DS5 data set, after dividing it into data
subsets, each of size 1000 patterns. The subsets are added incrementally, and the speed of
the algorithm is recorded for each increment. The time considered is the time required for
running the core clustering algorithms, excluding the time for obtaining the similarity matrix
a
b
FIGURE 14: V-Measure, Purity and Entropy for: (a) Corel data set and (b) Arcene data set.

between the sample points. The time is measured in seconds. The average running times are
computed for DBScan, K-means, and Modified CSHARP for 10 runs, with standard deviation
listed in Table 3. Chameleon was tested only once for each increment. K-means was iterated
100 times for each experiment to provide better convergence.
The adopted algorithms as well as the proposed Modified CSHARP algorithm have been
executed on a machine with the following configuration: 3.00 GHz processor, 1.0 GB RAM,
and running Linux operating system (Ubuntu 10.04 LTS).
TABLE 3: Standard Deviation for 10 Runs on Modified CSHARP, DBScan, and K-means on
Chameleon’s DS5.
Data Size 1000 2000 3000 4000 5000 6000 7000 8000
K-means 0.020 0.008 0.017 0.015 0.021 0.018 0.023 0.043
DBScan 0.008 0.012 0.017 0.025 0.028 0.021 0.017 0.034
Modified
CSHARP
0.013 0.013 0.014 0.016 0.016 0.019 0.025 0.037
a
b
c
FIGURE. 15: Speed of CSHARP and Modified CSHARP using Chameleon’s DS5 data set, compared to (a)
DBScan and K-means (b) DBScan, Kmeans, and Chameleon (c) DBScan.

5. CONCLUSION
In this paper, a modified version of the novel shared nearest neighbors clustering algorithm,
CSHARP[25] has been presented. Density and homogeneity are combined in a new
homogeneity factor used to order the merging of blocks of points . It can find clusters of
varying shapes, sizes and densities; even in the presence of noise and outliers and in high
dimensional spaces as well. It is based on the idea of letting each data point vote for its K-
nearest neighbors and adopting the points with the highest votes as clusters’ seeds. Two
clusters can be merged if their link strength is sufficient. Any data point not belonging to a
cluster is considered as noise. The results of our experimental study on several data sets are
encouraging. A wide range of possible parameters settings yield satisfactory solutions using
the validation indexes adopted in [8], [9] and [10]. Modified CSHARP solutions have been
found, in general, superior to those obtained by DBScan, K-means and Mitosis and
competitive with spectral clustering algorithm adopted in[2].
More work is needed to introduce a procedure for parameters selection. Also, we intend to
investigate the behavior of our clustering approach on other types of data. Moreover, we
intend to parallelize our algorithm as its clustering propagation is inherently parallel, as has
been shown in section 2.2. Finally, we have made the algorithm publicly available to other
researchers at http://www.csharpclustering.com.
ACKNOWLEDGMENT
Authors would like to thank Dr. Noha A. Yousri for several useful discussions. Noha A. Yousri
is with the Department of Computer and System Engineering, Faculty of Engineering,
Alexandria University, Alexandria, Egypt.

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