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A Performance of the Scattered Averaging Technique based on the Dataset for the Cluster Center Initialization

Clustering is one of the primary functions in data mining explorations and statistical data analysis which widely used in various fields. There are two types of the clustering algorithms which try to optimize certain objective function, i.e. the hierarchical and partitional clustering. This study focuses on the achievement of the best cluster results of the hard and soft clustering (K-Mean, FCM, and SOM clustering). The validation index called GOS (Global Optimum Solution) used to evaluate the cluster results. GOS index defined as a ratio of the distance variance within a cluster to the distance variance between clusters. The aim of this study is to produce the best GOS index through the use of the proposed method called the scattered averaging technique based on datasets for the cluster center initialization. The cluster results of each algorithm are also compared to determine the best GOS index between them. By using the annual rainfall data as the dataset, the results of this study showed that the proposed method significantly improved KMean clustering ability to achieve the global optimum solution with a performance ratio of 69.05% of the total performance of the three algorithms. The next best clustering algorithm is SOM clustering (24.65%) followed by FCM clustering (6.30%). In addition, the results of this study also showed that the three clustering algorithms achieve their best global optimum solution at the number of even clusters.


Introduction
Clustering is one of the primary functions in data mining explorations and statistical data analysis which widely used in various fields. The purpose of clustering is to group datasets into groups or clusters based on the relationships between them. There are various algorithms for clustering, depend on the relationship of between the data in datasets and how the clusters are formed efficiently. The types of the relationship between data commonly are the distance of between data, connectivity between data, the density of the data space, the interval or the distribution of certain statistics, and the other definition of their relationships. From the standpoint of how to achieve efficient cluster formation, clustering regarded as a multi-purpose optimization problem.
There are two types of the clustering algorithms which try to optimize certain objective function, i.e. the hierarchical and partitional clustering. Hierarchical clustering builds clusters using dendrogram based on agglomerative or divisive approaches. This algorithm has been used to measure stock market liquidity in [1][2][3]. Partitional clustering algorithms attempt to partition the datasets directly into a given number of clusters where each cluster characterized by a cluster representative or vector prototype (a collection of the cluster center called centroid). The most common interdata relationship used in partitional clustering is the shortest distance. The most common optimization goal is until there is no change in cluster membership during the grouping process. The K-Mean Clustering is the most primitive, simple and effective method. The relationship of between its datasets and the center of its cluster is the shortest distance. This algorithm has been used to cluster the earthquake epicenter in [4], to mining the attitude of social network users in [5], and mapping of image and video in [6]. The uniform effects of this algorithm have studied in [7]. The Genetic Algorithm has also been used to optimize K-Mean for GPS data clustering in [8].
K-Mean clustering is classified as hard clustering because each data pattern occupies only one cluster. The cluster membership stated by 1 and vice versa expressed by 0. Unlike the hard clustering, each data pattern of soft clustering has a certain degree of membership on each cluster expressed in the interval {0...1}. One type of this algorithm is Fuzzy C-Mean (FCM). The FCM has been used to perform the meta-analysis of cell-specific transcriptomic data discovers versatile viral responsive genes in [9]. This algorithm also used to segment an image in [10,11] and classify the mental task in [12]. Combined FCM and PCA have been studied to modify the multiple generalized regression neural network in [13]. In-depth analyses of the convergence of between the K-Mean and FCM algorithms have studied in [14]. Other clustering types are model-based clustering [15][16][17][18], density-based clustering [19,20], and spatial clustering [21]. SOM (Self-Organizing Maps) is one of Artificial Neural Network architecture used for clustering that works with one type of machine learning that is unsupervised learning. SOM has been used for geospatial analysis of extreme weather events in [22] and reduce the complexity of terrestrial lidar data in [23]. The asymmetric SOM compare to the asymmetric K-Mean have studied in [24]. Several other studies have also been carried out related to the application of SOM clustering [25][26][27][28].
The validation index measure how well the general goal of clustering which is consists of two parameters, namely the similarity within clusters and separability between clusters. The clustering results can be evaluated by using SC (Silhouette coefficient) [29][30][31], PBM (Pakhira-Bandyopadhyay-Maulik) [32], SSE (Sum of Squared Errors), and other techniques. SC allows for the evaluation of a given assignment for a particular observation. PBM measure the goodness of clustering on different partitions of a given dataset and describe a cluster validity index of a cluster solution. SSE is the most popular evaluation of distance which only needs to consider the cohesion of clusters to evaluate the cluster quality [33]. WB index is one of the clustering results evaluation based on SSW (sum of squares within a cluster) used to measure the similarity within a cluster and SSB SSW (sum of squares between clusters) used to measure the separability between clusters [34].
Various optimization techniques related to clustering activities with different aims and objectives have also been studied as in [35][36][37][38]. Optimization of cluster center initialization is one approach that has been widely used to obtain a good validation index measurement of clustering activities. Several studies related to this have been carried out in [20,[39][40][41][42].
This study focuses on the achievement of the best cluster results of the hard and soft clustering (K-Mean, FCM, and SOM clustering). The validation index called GOS (Global Optimum Solution) used to evaluate the cluster results. GOS index defined as a ratio of the distance variance within a cluster to the distance variance between clusters. The aim of this study is to produce the best GOS index through the use of the proposed method called the scattered averaging technique based on datasets for the cluster center initialization. The cluster results of each algorithm are also compared to determine the best GOS index between them.

A. The scattered averaging technique
One of the simplest techniques to measure pattern the dataset to be grouped can be mapped into the N-dimensional Cartesian coordinates, where N is the number of data attributes. The clustering process performed on N-dimensional Cartesian coordinates that act as the data space. The scattered averaging technique is a proposed technique for cluster central initialization where the initial cluster center (called the initial centroid) placed scattered in the data space based on its dataset. This technique is expected to ensure the achievement of global optimum solutions regardless of how many iterations the process requires. Suppose there are 6 data has two attributes (X and Y) as follows: Firstly, the dataset is sorted ascending by all its attributes (by X, then by Y) as follows: If the dataset will be grouped into two clusters, then the initial cluster center as follows: If the dataset will be grouped into three clusters then the initial cluster center as follows: If D is a dataset, M is the number of data, and k is the number of clusters desired then the cluster center initialization using the scattered averaging technique is expressed by: where   ,: ci is the i th cluster center initialization, and (:) is the index notation of the number of data attributes. This technique will be applied to the K-Mean algorithm, FCM, and SOM clustering to know its performance through measurement of GOS index of cluster result of each algorithm.

B. The Global Optimum Solution (GOS)
A centroid is the cluster center. The best GOS achieved if the distance variance within a cluster as minimum as possible and the distance variance between clusters as maximum as possible. The GOS illustrated in Fig. 2. Suppose the data as follows: where M is the number of data. The measurement of data distribution to the average data is expressed by: If   V a r m em b er represents the average of the distance variance between the cluster members against its centroid and   V a r cen tro id represents the distance variance of between their centroids, then the global optimum solution is reached if it satisfies: The GOS index is expressed by: The formula shows that the smaller the   V a r m em b er , the higher the similarity between cluster members in a cluster. Besides, the larger the   V a r cen tro id , the higher the separation between clusters. Finally, the higher the GOS index, the better the clustering results.

C. K-Mean clustering
K-Mean clustering is an iterative process that partitions a dataset into a number of K clusters. K-Mean clustering is classified as hard clustering because the cluster membership stated by 1 and vice versa expressed by 0. The initial cluster center (called centroid) used as the starting point of the partition based on the shortest distance between the cluster center against each data in the dataset. The iteration process is done continuously until no change of cluster membership in each cluster. The distance function used is Euclidean distance [43][44][45][46]. The K-Mean clustering algorithm is shown in Fig. 3.
The degree of membership of each data in each cluster is expressed by: where q is the fuzzification parameter (fuzzyfier, q>1). The generation of new centroids using the COA (Center of Area) formula expressed by: The iteration process terminated if it satisfies: In principle, the intra-layer is a layer that represents the neighbouring distance function between the input layer weights with the output layer weights that map data from the input layer into the cluster membership in the output layer [45]. The neighbouring distance function is expressed by:  (11) If there is still a change in cluster membership, then the intra-layer weighted is updated using the following formula: where  is learning rate within the range {0.1 ... 0.9}. In this case, the SOM clustering algorithm can achieve its best GOS index by setting the appropriate learning rate value. The SOM clustering algorithm is shown in Fig. 6.

Result and Discussions
In this study, the dataset grouped into 3 and 4 clusters using K-Mean, FCM, and SOM clustering. By ignoring the number of iterative processes, the visualization of the optimum global solution of each algorithm is shown in Fig. 7, Fig.  8, and Fig. 9.  Table 3. The dataset in this table has been grouped into 3, 4, 5, and 6 clusters. It was intended to review the results of clustering based on the number of odd and even clusters. Table 3 has shown that cluster center initialization using the scattered averaging technique based on the dataset used has improved K-Mean clustering ability to achieve the global optimum solution with a performance ratio of 69.05% of the total performance of the three algorithms. This result is evidenced by the average GOS Index of the K-Mean clustering (3.2145) is greater than the others (FCM = 0.2935, SOM = 1.1478). The next best clustering algorithm is SOM clustering (24.65%) followed by FCM clustering (6.30%). In addition, Table 3 also shown that the three clustering algorithms achieve their best global optimum solution at the number of even clusters.