Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 3, Issue 3 , Marc h -2012 1

ISSN 2229-5518

Single Link clustering on data sets

Ajaya Kushwaha, Manojeet Roy

Abs tractCluster analysis itself is not one specif ic algorithm, but the general task to be solved. It can be achieved by various algorithms that diff er signif icantly in their notion of w hat constitutes a cluster and how to eff iciently f ind them. Popular notions of clusters include groups w ith low distances among the cluster members, dense areas of the data space, intervals or particular statistical distributions. Most data-mining methods assume data is in the f orm of a f eature-vector (asingle relational table) and cannot handle multi-relational data.

Inductive logic programming is a f orm of relational data mining that discovers rules in _f irst-order logic f rom multi-relational data. This paper discusses the application of SLINK to learning patterns f or link discovery . Clustering is among the oldest techniques used in data mining applications. Typical imple mentations of the hierarchical agglomerative clustering methods (HACM) require an amount of O( N2)-space

w hen there are N data objects, making such algorithms impractical f or problems involving large datasets. The w ell-know n clustering algorithm RNN-CL INK requires only O(N)-space but O(N3)-time in the w orst case, although the average time appears to be O(N2 log N).

Inde x Termsclustering, nearest neighbor, reciprocal nearest neighbor, complete link, probabilistic analysis.

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LUSTER analysis, or clustering, is a multivariate statisti- cal technique which identifies groupings of the data ob- jects based on the inter-object similarities computed by a
chosen distance metric.. Clustering methods can be divided
into two types: partitioned and hierarchical. The pa rtitioned
clustering methods start with a single group that contains all the objects, then proceed to divide the objects into a fixed number of clusters. Many early clustering applications used partitional clustering due to its computational efficiency when large datasets need to be processed. However, partitional clustering methods require prior specification of the number of clusters as an input parameter. Also, the partitions ob- tained are often strongly dependent on the order in which the objects are processed.7. The hierarchical agglomerative cluster- ing methods (HACM) attempt to cluster the objects into a h i- erarchy of nested clusters. Starting with the individual objects as separate clusters, the HACMs successively merge the most similar pair of the clusters into a higher level cluster, until a single cluster is left as the root of the hierarchy. HACMs are also known as unsupervised classification, because their abi l- ity in automatically discovering the inter-object similarity pat- terns. Clustering techniques have been applied in a variety of engineering and scientific fields such as biology, psychology, computer vision, data compression, information retrieval, and more recently, data mining.[3 4 8 10] .
The various technical aspect work on

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Author name is Mr. Ajay Kushwaha, Reader c.s.e Deptt.RCET

,bhilaiM.C.A , Mtech(CS),PhD (CSE) pursuing from CSVTU

,Chhattisgarh Research area – MANET



Co-Author name is Mr. Manojeet Roy, Computer Science De- partment , CSVTU University/ Rungta College of Engineering And Technology College/ Rungta Group of colleges Organization , City Bhilai Country India, Phone/ Mobile 09993649781., (e-mail:


2.1 Review Stage

Detailed DATA CLUSTERING is considered an interesting approach for finding similarities in data and putting similar data into groups. Clustering partitions a data set into several groups such that the similarity within a group is larger than that among groups [1]. The idea of data
Grouping, or clustering, is simple in its nature and is close to
the human way of thinking; whenever we are presented with
a large amount of data, we usually tend to summarize this huge number of data into a small number of groups or catego- ries in order to further facilitate its analysis. Moreover, most of the data collected in many problems seem to have some inh e- rent properties that lend themselves to natural groupings. Nevertheless, finding these groupings or trying to categorize the data is not a simple task for humans unless the data is of low dimensionality (two or three dimensions at maximum.) This is why some methods in soft computing have been pro- posed to solve this kind of problem. Those methods are called
―Data Clustering Methods‖ and they are the subject of this
paper. Clustering algorithms are used extensively not only to organize and categorize data, but are also useful for data com- pression and model construction. By finding similarities in data, one can represent similar data with fewer symbols for example. Also if we can find groups of data, we can build a model of the problem based on those groupings.
Clustering techniques are broadly divided into hierarchical
clustering and partitional clustering

2.2 Clu stering Algorithm s

The community of users has played lot emphasis on develop- ing fast algorithms for clustering large data sets [13]. Cluster- ing is a technique by which similar objects are grouped to-

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Inte rnatio nal Jo urnal o f Sc ie ntific & Eng inee ring Re se arc h Vo lume 3, Issue 3 ,

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gether. Clustering algorithms can be classified into several categories such partitioning-based clustering, hierarchical a l- gorithms,
density based clustering and grid based clustering.
Now a day‘s huge amount of data is gathered from remote
sensing, medical data, geographic information system, envi- ronment etc. So everyday we are left with enormous amount of data that requires proper analysis. Data mining is one of the emerging fields that are used for proper decision -making and utilizing these resources for analysis of data. They are several researches focused on medical decision making [14]

2.3 A. Hierarchical Clustering

―Hierarchical clustering builds a cluster hierarchy or, in other words, a tree of clusters, also known as a dendrogram. Every cluster node contains child clusters; sibling clusters pa rtition the points covered by their common parent. Such an approach allows exploring data on different levels of granula rity ―. Hierarchical clustering methods are categorized into aggl o- merative (bottom-up) and divisive (top-down) [Jain and Dubes 1988; Kaufman and Rousseeuw 1990]. An agglomera- tive clustering starts with one-point (singleton clusters. It then recursively merges two or more most appropriate clusters . Fig. 2 provides a simple example of hierarchical clustering

2.4 Figure

Agglomerative Divis ive

P, Q, R, S, T

P ,Q
S, T R


Instead of a clustering structure a partitional clustering algo- rithm obtains a single partition of the data [2]. They generate a single partition of the data to recover natural groups pres ent in the data. The proximity matrix among the objects is re- quired by hierarchical clustering techniques. The partitional techniques expect data in the form of a pattern matrix. Parti- tioning techniques are used frequently in engineering applica- tions where single partitions are important. Partitional cluster- ing methods are especially appropriate for the efficient repre- sentation and compression of large databases [10]. The algo- rithm is typically run multiple times with different starting states. The best configuration obtained from all the runs is used as the output clustering provides a simple example of partitional clustering. Fig. 3 Splitting of a large cluster by Pa r- titional Algorithm [4] Dubes and Jain (1976) emphasize the distinction between clustering methods and clustering algo- rithms. The K-means is
the simplest and most commonly used algorithm employing a
squared error criterion [McQueen 1967]. The K-means algo-
rithm is popular because it is easy to implement. Its time com- plexity is O (n), where n is the number of partitions. The algo- rithm is sensitive to the selection of initial partition. It may converge to a local minimum of the criterion function value if the initial partition is not properly chosen

SLINK Algorithm

In cluster analysis, single linkage, nearest neigh- bour or shortest distance is a method of calculating distances
between clusters in hierarchical clustering. In single linkage, the distance between two clusters is computed as the distance be- tween the two closest elements in the two clusters.

Mathematically, the linkage function – the distance D(X,Y) be- tween clusters X and Y – is described by the expression


clusters. It then recursively merges two or more most appropriate clusters. Fig. 2 provides a simple example of hierarchical clustering.
Hierarchical Clustering
where X and Y are any two sets of elements considered as clusters, and d(x,y) denotes the distance between the two ele- ments x and y.
A drawback of this method is the so-called chaining phenome-

non: clusters may be forced together due to single elements being

close to each other, even though many of the elements in each cluster may be very distant to each other.


The following algorithm is an agglomerative scheme that eras- es rows and columns in a proximity matrix as old clusters are merged into new ones. The proximity ma- trix D contains all distances d(i,j). The clustering are assigned

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sequence numbers 0,1,......, (n − 1) and L(k) is the level of the kth clustering. A cluster with sequence number m is denoted (m) and the proximity between clusters (r) and (s) is de- noted d[(r),(s)].
The algorithm is composed of the following steps:
1. Begin with the disjoint clustering having level L(0) = 0 and sequence number m = 0.
2. Find the most similar pair of clusters in the current
clustering, say pair (r), (s), according to d[(r),(s)] = min d[(i),(j)] where the minimum is over all pairs of clusters in the current clustering.
3. Increment the sequence number: m = m + 1. Merge clusters (r) and (s) into a single cluster to form the next clustering m. Set the level of this clustering to L(m) = d[(r),(s)]
4. Update the proximity matrix, D, by deleting the rows
and columns corresponding to clusters (r) and (s) and
adding a row and column corresponding to the newly formed cluster. The proximity between the new clus- ter, denoted (r,s) and old cluster (k) is defined as d[(k), (r,s)] = min d[(k),(r)], d[(k),(s)].
5. If all objects are in one cluster, stop. Else, go to step 2.


As mentioned earlier, data clustering is concerned with the parti- tioning of a data set into several groups such that the similarity within a group is larger than that among groups. This
implies that the data set to be partitioned has to have an inherent
grouping to some extent; otherwise if the data is uniformly distri-
buted, trying to find clusters of data will fail, or will lead
to artificially introduced partitions. Another problem that may
arise is the overlapping of data groups. Overlapping groupings
sometimes reduce the efficiency of the clustering method, and this reduction is proportional to the amount of overlap
between groupings.
Usually the techniques presented in this paper are used in con- junction with other sophisticated neural or fuzzy models. In particular, most of these techniques can be used as preprocessors for determining the initial locations for radial basis functions or fuzzy ifthen rules. The common approach of all the clustering techniques presented here is to find cluster centers that will represent each cluster. A cluster center is a way to tell where the heart of each cluster is located, so that later when presented
with an input vector, the system can tell which cluster this vector
belongs to by measuring a similarity metric between the input vector and al the cluster centers, and determining which cluster
is the nearest or most similar one.Some of the clustering tech-
niques rely on
knowing the number of clusters apriori. In that case the algorithm
tries to partition the data into the given number of clusters. K-
means and Fuzzy C-means clustering are of that type. In other casesit is not necessary to have the number of clusters
known from the beginning; instead the algorithm starts by find-
ing the first large cluster, and then goes to find the second, and so on. Mountain and Subtractive clustering are of that type. In both cases a problem of known cluster numbers can be applied; how-
ever if the number of clusters is not known, K-means and Fuzzy
C-means clustering cannot be used.
Another aspect of clustering algorithms is their ability to be im-
plemented in on-line or offline mode. On-line clustering is a process in which each input vector is used to update the
cluster centers according to this vector position. The system in
this case learns where the cluster centers are by introducing new input every time. In off-line mode, the system is presented with a training data set, which is used to find the cluster centers by ana- lyzing all the input vectors in the training set. Once the cluster centers are found they are fixed, and they are used later to classify new input vectors. The techniques presented here are of the off- line type. A brief overview of the four techniques is presented here. Full detailed discussion will follow in the next section.
The first technique is K-means clustering [6] (or Hard C-means
clustering, as compared to Fuzzy C-means clustering.) This tech- nique has been applied to a variety of areas, including
image and speech data compression, [11,12] data preprocessing
for system modeling using radial basis function networks, and task decomposition in heterogeneous neural network architec- tures [6]. This algorithm relies on finding cluster
centers by trying to minimize a cost function of
dissimilarity (or distance) measure. The second technique is
Fuzzy C-means clustering, which was proposed by Bezdek in
1973 [5] as an improvement over earlier Hard C means clustering. In this technique each data point belongs to a cluster to a degree specified by a membership grade. As in K-means clustering, Fuzzy C-means clustering relies on minimizing a cost function of dissimilarity measure. The third technique is Mountain cluster- ing, proposed by Yager and Filev [5]. This technique
builds calculates a mountain function (density function) at every possible position in the data space, and chooses the position with the greatest density value as the center of the first cluster. It
then destructs the effect of the first cluster mountain function and finds the second cluster center. This process is repeated until the desired number of clusters have been found.
The fourth technique is Subtractive clustering, proposed by Chiu [5]. This technique is similar to mountain clustering, except that instead of calculating the density function at every possible posi- tion in the data space, it uses the positions of the data points to calculate the density function, thus reducing the number of calculations significantly.


They are several series of experiments performed in this sec- tion to determine relevant pattern detection for medical dia g- nosis.

Clinical Database II

In this study, the data was taken from SEER datasets which has record of cancer patients from the year 1975 – 2001 from Ref.[11]. The data was again classified into two groups that arespatial and non spatial dataset. Spatial dataset consists of location collected include remotely sensed images, geograph i- cal information with spatial attributes such as location, digital sky survey data, mobile phone usage data, and medical data. The five major cancer areas such as lung, kidney, throat, sto- mach and liver were experimented. After this data mining

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algorithms were applied on the data sets such as K-means, SOM and Hierarchical

clustering technique

The K-means method is an efficient technique for clustering large data sets and is used to determine the size of each clus-
ter. After this the HAC (hierarchical agglomerative cluster- ing), is used on our datasets in which we have used tree based partition method in which the results has shown a tree stru c- ture and the gap between the nodes has been highlighted in the Table 3. The HAC has proved to have for better results than other clustering methods. The principal component ana l- ysis technique has been used to visualize the data. The X, Y coordinates identify the point location of objects. The coord i- nates were used and the clusters were determined by appro- priate attribute value. The mean and standard deviation of each cluster was determined hey were interesting facts that suggests that number of cancer cases that occur during the time interval. Table 1 represents the size of each cluster deter- mined by Kmeans clustering technique for dataset 2 .In Table
2 shows the
number trials generated for the cluster determination.
Table 1. Result of K-means



















Table 2 Number of trials

Number of trials















6.5 Theorems and Proofs

Table 3 presents HAC (hierarchical agglomerative clustering) in which the cluster were determined with appropriate size. Table 4 represents the best cluster selection in which the gap is defined as the space in between the clusters

Table 3. HAC Cluster size

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Figure. - Dendrogram

Above Figure represents the dendogram in which the dataset has been partitioned into three clusters with the K-means. The HAC clustering algorithm is applied on K-means to generate the den- dogram. In a dendogram, the elements are grouped together in one cluster when they have the closest values of all elements available. In the diagram the cluster2 and cluster 3 are combined. The subdivisions of clusters are then analyzed
This paper focuses on clustering algorithms such as HAC and
KMeans in which, HAC is applied on K-means to determine the number of clusters. The quality of cluster is improved, if HAC is applied on K-means. The paper has referenced and discussed the issues on the specified algorithms for the data analysis. The anal- ysis does not include missing records. The application can be used to demonstrate how data mining technique can be com- bined with medical data sets and can be effectively demonstrated in modifying the clinical research.
This study clearly shows that data mining techniques are promis-
ing for clinical datasets. Our future work will be related to miss- ing values and applying various algorithms for the fast impl e- mentation of records. In addition, the research would be focusing on spatial data clustering to develop a new spatial data mining algorithm.


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