1. Introduction

The choice of feature types and measurement levels depends on data type. For this reason, many clustering methods have been developed. According to clustering strategies, these methods can be classified as hierarchical clustering [1–3], partitional clustering [4, 5], artificial system clustering [6], kernel-based clustering and sequential data clustering. This chapter examines some popular partitional clustering techniques and algorithms.

> © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>J</sup>ð Þ¼ <sup>Γ</sup>; <sup>Μ</sup> <sup>X</sup>

mi <sup>¼</sup> <sup>1</sup> Ni P N i¼1

is determined.

another set.

as in (3).

3. K-means algorithm

where Γ is a partition matrix of γij and defined as;

γijxj is the sample mean for the i

K

X N

γij xj � mi � � � � <sup>2</sup> <sup>¼</sup> <sup>X</sup> K

i¼1

<sup>γ</sup>ij <sup>¼</sup> <sup>1</sup> if xj <sup>∈</sup> cluster i <sup>0</sup> otherwise �

The partition minimizes the sum-square-error (SSE). When the SSE regarded is minimum, minimum variance partition will be achieved. As a result of this calculation, optimum cluster

The K-Means clusters were first developed by Mac Queen [8]. In the K-Means clusters, clusters are formed using Euclidean distance. In the K-Means algorithm, unsupervised learning is used

In K-Means clustering, k cluster centers are created from the selected data set. It is then placed at the nearest cluster using Euclidean distance. New cluster centers are formed according to the results of the clustering. From the calculations of the clustering, the cluster center is recalculated. The arithmetic average is used as the calculation method, and the new cluster center is determined. All samples are reclassified according to the new center. This process is repeated until it is determined that the samples in the set have not passed to

The partitioning of the k pieces of data x is represented by the minimization of the J parameter

X x∈ ck

If the data are classified in a cluster near the center of the nearest cluster, the J value will be the minimum. If the data x are classified in the kth number cluster, the value can be optimized by changing the weighting value of wx to obtain the minimum J value. dist xð Þ ; ok is the notation which represents the distance function. In this formula, x represents the pixel data, ok is the

wxdist xð Þ ; ok

!

<sup>J</sup> <sup>¼</sup> min <sup>X</sup>

center of the cluster. k sets are shown as in (4).

k

X N

γij xj � mi � �<sup>T</sup>

th number cluster corresponding to Ni objects.

xj � mi

http://dx.doi.org/10.5772/intechopen.75836

� � (1)

Partitional Clustering

(2)

21

(3)

j¼1

j¼1

M is the cluster prototype or centroid matrix and mi defined as;

and k classes are created which minimize the error function [9].

i¼1

Figure 1. Clustering techniques: (a) data set; (b) partitional clustering; and (c) hierarchical clustering.

In contrast to hierarchical clustering methods, partitional clustering aims successive clusters using some iterative processes. Partitional clustering assigns a set of data points into k-clusters by using iterative processes. In these processes, n data are classified into k-clusters. The predefined criterion function J assigns the datum into k th number set according to the maximization and minimization calculation in k sets.

Figure 1 represents the hierarchical clustering and partitional clustering. In addition, hierarchical clustering, all sub-clusters defined in another sub-cluster shown in Figure 1. Figure 1a represents the raw data, Figure 1b shows the partitional clustering and Figure 1c represents the hierarchical clustering. In hierarchical clustering, raw data are firstly clustered in some subgroups (three-clustered shape). After that procedure, subgroups hierarchically defined in two green clusters. Last procedure includes all these clusters which are defined in the union set.

This chapter starts with an introduction to clustering criteria and continues with K-Means algorithm, different fuzzy clustering techniques and genetic algorithm-based clustering.
