A hybrid data clustering approach based on improved cat swarm optimization and K -harmonic mean algorithm

Abstract

Clustering is a process to discover unseen patterns in a given set of objects. Objects belonging to the same pattern are homogenous in nature while they are heterogeneous in other patterns. In this paper, a hybrid data clustering algorithm comprising of improved cat swarm optimization (CSO) and K-harmonic means (KHM) is proposed to solve the clustering problem. The proposed algorithm exhibits strengths of both the mentioned algorithms, it is named as improved CSOKHM (ICSOKHM). The performance of the proposed algorithm is evaluated using seven datasets and is compared with existing algorithms like KHM, PSO, PSOKHM, ACA, ACAKHM, GSAKHM and CSO. The experimental results demonstrate that the proposed algorithm not only improves the convergence speed of CSO algorithm but also prevents KHM algorithm from running into local optima.

Keywords

Cat swarm optimization data clustering gravitational search algorithm particle swarm optimization

1. Introduction

Clustering is a powerful technique in pattern recognition, data mining and machine learning paradigm. It is an NP complete problem. It is used to discover hidden patterns, knowledge and information from a dataset using a criterion function which were previously unknown [4]. In clustering, a dataset is divided into K disjoint groups and data belongs to a group having more similar characteristics when compared to data resides in other groups. K-means is one of the oldest and most widely accepted clustering algorithm which is applied for obtaining the optimal cluster centers [15]. It is simple, fast and efficient algorithm, but suffers with initialization and local optima problems [20]. To overcome these problems and also to improve its efficacy, many hybrid versions of K-means algorithm have also been reported in the literature [7]. Zhang et al. [35] developed K-harmonic means (KHM) algorithm for data clustering using average harmonic mean to obtain the new cluster centers. KHM algorithm showed good results in comparison to the K-means algorithm, but the local optima problem remained unsolved. In recent years, a number of algorithms based on swarms, insects and natural phenomena have been developed to solve clustering problem. For example, an artificial bee colony (ABC) [10], ant colony optimization (ACO) [28], genetic algorithm (GA) [17], particle swarm optimization (PSO) [1], cat swarm optimization (CSO) [25], teacher learning based optimization (TLBO) [24,27], black hole optimization (BH) [6], gravitational search algorithm (GSA) [36] and charge system search algorithm (CSS) [11,12]. The above algorithms can be classified into swarm based algorithms, biological algorithms and basic science algorithms. Although, these algorithms have shown good potential over traditional algorithms, but they also suffer with several problems. For instance, GA suffers from population diversity problem and the quality of solutions in GA depends on the mutation and cross-over probability [23]. In ACO algorithm, convergence time is uncertain and the probability distribution function is also changed in each iteration [16]. PSO algorithm possesses weak exploitation properties and sometimes gets stuck in local optima [22]. In case of ABC algorithm, the performance of the algorithm depends on the dimension of the problem; increase in the dimension of the problem results in a decrease of the convergence speed [9]. In GSA algorithm, premature convergence can occur due to its memory-less nature [26].

Cat swarm optimization (CSO) is a latest and state of art algorithm developed by Chu et al. [2], by observing the behavior of cats. CSO is the first algorithm which reports on the behavior of cats in the literature. It has been applied in many domains, and giving remarkable results [13,14,19,21,30,31]. The advantage of the CSO algorithm is its good exploration property.

In this paper, a hybrid data clustering algorithm is proposed based on the Improved CSO and KHM algorithms, to prevent the KHM from running into local optima, and also to enhance the convergence speed of the CSO. To hybridize the CSO with KHM, few modifications need to be made to improve the original CSO. The performance of the proposed Improved CSOKHM (ICSOKHM) algorithm is tested on several benchmark datasets from the UCI repository. The experimental results prove that the proposed algorithm is more efficient, accurate and precise than others.

The rest of the paper is organized as follows. Sections 2 and 3 cover the discussion on KHM and CSO algorithms. Section 4 first presents the improvement of the CSO algorithm, followed by the improved CSOKHM algorithm for clustering. Investigational results of the proposed algorithm are discussed in Section 5 and the paper is concluded in Section 6.

2. K-harmonic algorithm

KHM is one of the popular techniques that has been applied in the clustering domain. It is insensitive to initialization issues due to inbuilt boosting function [35]. KHM is a partition based iterative algorithm which is applied to split the data into groups of pre-specified clusters. Data in the same group have similar characteristics when compared to data in other groups. KHM provides faster convergence than the K-means. But, it is also faced the same local optima problem [34]. In KHM, harmonic mean serves as the distance measure which can be described in Eq. (1), $KHM (X, C) = \sum_{i = 1}^{n} \frac{k}{\sum_{j = 1}^{k} \frac{1}{∥ (x_{i} - x_{j}) ∥^{p}}},$ (1) where $x_{i}$ represents the ith data vector of set $X \in (x_{1}, x_{2}, x_{3}, \dots, x_{n})$ , $c_{j}$ is the jth cluster center of set $C \in (c_{1}, c_{2}, \dots, c_{k})$ , “p” is power of the objective function and $“ p ” > 2$ .

The main steps of the KHM algorithm for clustering is listed as below.

Define the initial cluster centers ( $c_{j}$ ) in D-dimensional search space; $j = 1, 2, \dots, k$ and $d = 1, 2, \dots, N$ .

Compute the value of objective function using Eq. (1).

For each data instance $x_{i}$ :

Calculate its membership function $m (c_{j} / x_{i})$ from all cluster centers using given equation $m (c_{j} ∖ x_{i}) = \frac{∥ x_{i} - c_{j} ∥^{(- p - 2)}}{\sum_{j = 1}^{k} ∥ x_{i} - c_{j} ∥^{(- p - 2)}} .$ (2) This function is used to define the data vector $x_{i}$ belonging to cluster center $c_{j}$ .

Calculate the weight $w (x_{i})$ using following equation $m (c_{j} ∖ x_{i}) = \frac{\sum_{j = 1}^{k} ∥ x_{i} - c_{j} ∥^{(- p - 2)}}{(\sum_{j = 1}^{k} ∥ x_{i} - c_{j} ∥^{(- p - 2)})} .$ (3) This function is used to determine the impact of data vector $x_{i}$ on the cluster centers.

For each cluster center, recompute the cluster centers from all data vectors $x_{i}$ with the help of membership and weight functions $c_{j} = \frac{\sum_{i = 1}^{n} m (c_{j} / x_{i}) \times w (x_{i}) x_{i}}{\sum_{i = 1}^{n} m (c_{j} / x_{i}) \times w (x_{i})} .$ (4)

Repeat the steps 2–4, until the cluster centers do not change further.

Assign the data vector $x_{i}$ to cluster j with the biggest $m (c_{j} ∖ x_{i})$ .

3. Cat Swarm Optimization algorithm (CSO)

Chu and Tasi developed CSO algorithm in 2007, inspired from the behavior of cats [2]. The behavior of cats is measured in two states – in acting state and in resting state. These states are expressed in CSO algorithm as seeking mode and tracing mode. In CSO algorithm, the positions of the cats represent the possible solution sets. The detailed description of the CSO algorithm is given below.

3.1. Seeking mode

Seeking mode of the CSO algorithm describes the resting state of cats. In this mode, cats are always in alert position. In seeking mode, a cat continuously changes its position to achieve better position. The position of a cat changes according to its fitness function. The fitness function of CSO algorithm is determined using Eq. (5). Seeking mode of the CSO algorithm acts as global search for the solution. The following terms are involved in the seeking mode of the CSO algorithm.

Seeking Memory Pool (SMP): It indicates the number of copies of a cat produced in seeking mode.

Seeking Range of selected Dimension (SRD): It is the maximum difference between new and old values of the dimensions selected for mutation.

Counts of Dimension to Change (CDC): It represents the number of dimensions to be mutated, ${Fit}_{i} = \frac{∥ {FS}_{j} - {FS}_{\max} ∥}{{FS}_{\max} - {FS}_{\min}},$ (5) where ${Fit}_{j}$ represents the probability value associated with position of jth cat, ${FS}_{j}$ is the fitness of jth cat, ${FS}_{\max}$ is the maximum fitness value and ${FS}_{\min}$ represents the minimum fitness value.

The steps involved in this mode are given as:

Make “i” copies of ${cat}_{j}$ , where “i” equals to the seeking memory pool (SMP) of ${cat}_{j}$ , if “i” is one of the candidate solution then $“ i ” = SMP - 1$ else $“ i ” = SMP$ .

Determine the shifting value for each of “i” copies using SRD ∗ position of ${cat}_{j}$ .

Determine the number of copies undergo for mutation (randomly add or subtract the shifting value to “i” copies).

Evaluate the fitness of all copies.

Pick the best candidate from “i” copies and place it on the position of the jth cat.

3.2. Tracing mode

The tracing mode of CSO algorithm describes the acting state of cats. This mode describes the movement of cats towards the targets i.e. ability of the cats to trace the targets. The movement of cats are directly proportional to their velocities in each dimension and these velocities are in turn used to update their positions. The position $X_{j}$ and velocity $V_{j}$ of a ${cat}_{j}$ in the D-dimensional space can be described as $X_{j} = (X_{j, 1}, X_{j, 2}, X_{j, 3}, \dots, X_{j, N})$ and $V_{j} = (V_{j, 1}, V_{j, 2}, V_{j, 3}, \dots, V_{j, N})$ where D ( $1 ⩽ d ⩾ N$ ) and $j = 1, 2, 3, \dots, M$ . This mode of CSO algorithm performs local search to compute the solution set for an optimization problem. The velocities and positions of cats are updated using Eqs (6) and (7), $\begin{array}{rclr} V_{j d, new} = V_{j d} + c \times r \times (X_{g d} - X_{j d}), & (6) \\ X_{j d, new} = X_{j d} + V_{j d} . & (7) \end{array}$ Here $V_{j d}$ is the velocity of the jth cat in dth dimension, r is a random number in the range of $[0, 1]$ , c is a constant value, $X_{g d}$ is the global best position of cat in dth dimension and $X_{j d}$ is the position of the jth cat in the dth dimension.

The variable mixture ratio (MR) is used to combine the seeking and tracing mode of the CSO algorithm and it also determines the number of cats in seeking mode and tracing mode.

The steps of the CSO algorithm are as follows.

Initialize the population of Cats.

Define the parameters and specify the numbers of cats for seeking mode as well as tracing mode according to MR.

Evaluate the fitness function for each cat to determine the position and memorize the best position.

According to the flag:

If Cat is in seeking mode, apply the seeking mode process.

Otherwise, apply tracing mode process.

Again set the number of cats in tracing and seeking mode according the value of MR.

Repeat steps 3–5 until the termination condition is satisfied.

4. Proposed improved CSOKHM algorithm

From the literature, it has been observed that the efficiency of optimization algorithms mainly depends on the balance between the local and global search. These searches can be perceived as exploitation and exploration of solutions in solution space. Therefore, all the swarm based algorithms follow some mechanism to ensure the balance between the exploration and exploitation. CSO algorithm has good exploration ability, but sometimes suffers with poor exploitation ability. Therefore, there is a need to enhance the exploitation ability or local search of the CSO algorithm by using some other strategy to obtain the best results. In order to improve the searching ability and the convergence rate of CSO algorithm, few modifications are made in the original CSO algorithm. These modifications are:

A selection mechanism is adopted in tracing mode of the CSO algorithm to improve the exploitation ability.

Adaptive inertia weight concept is used to enhance the diversity of CSO algorithm.

Boundary level constraints are handled in a more efficient way to overcome the premature convergence.

4.1. Proposed modifications in CSO algorithm

This subsection illustrates the proposed modifications in the CSO algorithm.

4.1.1. Selection mechanism

To enhance the exploitation ability, a selection mechanism is introduced in the tracing mode of the CSO algorithm. The purpose of the selection mechanism is to find more promising position of cats in the solution space. This mechanism not only improves the local search of CSO (as the small region of solution space is searched carefully for solution component) but also increases the population diversity. The steps of selection mechanism are described in Algorithm 1.

Algorithm 1.

Selection mechanism for tracing mode of CSO

The aim of the selection mechanism is to pick a cat with better probability value ( $p_{j}$ ) which represents the position of cat in solution space and also gives the possible solution. Probability $p_{j}$ is calculated on the basis of the fitness function of the CSO algorithm which can be described as follows, $p_{j} = \frac{{SSE}_{j}}{\sum_{j = 1}^{K} {SSE}_{j}},$ (8) where $p_{j}$ denotes a probability value associated to the jth position of a cat and ${SSE}_{j}$ represents the fitness function (sum of squared error) of the jth cat.

4.1.2. Adaptive inertia weight

Second modification in CSO algorithm is the idea of inertia weight. Shy et al. [29] proposed the concept of inertia weight to overcome the diversity problem in optimization techniques and also claimed that the large value of inertia weight initiates the global search, while the small value initiates the local search. Inertia weight techniques are classified into three classes i.e. constant or random inertia weight, time varying inertia weight and adaptive inertia weight [18]. In this paper, an adaptive inertia weight technique is used to compute weight “w”. It analyzes the search space direction of an optimization algorithm and calculates the weight based on one or more feedback parameters. The inertia weight “w” is introduced into Eq. (9), resulting in the improvement in diversity of CSO algorithm. Now the Eq. (9) can be rewritten as $V_{j d, new} = w \times V_{j d} + c \times r \times (X_{g d} - X_{j d}) .$ (9) The value of inertia weight “w” can be computed as follows, $w_{j} = w_{initial} - \frac{SSE (X_{Gbest})}{SSE (X_{Tbest}) + SSE (X_{Sbest})},$ (10) where the value of inertia weight “w” dynamically changes throughout the execution and it varies in between $0.5 < w_{j} < 1$ . The variable $w_{initial}$ represents the initial value of inertia weight and it is equal to 1.1. $SSE (X_{Gbest})$ is the sum of squared error of the global best position of a Cat. $SSE (X_{Tbest})$ is the sum of squared error of best position of Cat in tracing mode. $SSE (X_{Sbest})$ is the sum of squared error of best position of Cat in seeking mode.

4.1.3. Boundary level constraints

Third issue related to CSO is the boundary level constraints. To handle this, two modifications are proposed in existing CSO algorithm: firstly, for seeking mode and secondly, in tracing mode.

Seeking mode modification.

In seeking mode, cats are continuously changing their positions without tracing any target and find an appropriate position. This behavior of cats is implemented using SMP parameter. SMP parameter represents the number of possible position (movement) of a cat in the search space. The movement (position) of cats in seeking mode is obtained by randomly adding or subtracting the shifting values from the present cluster centers, which in turn results in (SMP ∗ K) positions. Addition or subtraction of shifting values from the cluster centers may lead the data vectors to cross the boundary of the dataset. Thus, a mechanism is adopted to deal with such data vectors. The proposed mechanism can be described as follows. If the position of cat

X_{j d} < X_{\min}^{d}

then Eq. (11) is used to determine the position of cats in the range of

X_{\min}^{d} < X_{j d} > X_{\max}^{d}

where

d = 1, 2, 3, \dots, N

\begin{array}{rclr} X_{(j d, new)} & = & X_{\min}^{d} + rand (0, 1) \\ \times (X_{\max}^{d} - X_{\min}^{d}) + a, & (11) \end{array}

where “a” is a variable which is used to prevent the data vectors from being stuck in local optima near the boundary of data set. It is calculated using the following equation,

a = (1 + \frac{current iteration}{maximum iteration}) .

(12) If the position of cat

X_{j d} < X_{\min}^{d}

, then Eq. (13) is used to determine the position of cats in the range of

X_{\min}^{d} < X_{j d} > X_{\max}^{d}

where

d = 1, 2, 3, \dots, N

\begin{array}{rclr} X_{(j d, new)} & = & X_{\max}^{d} - rand (0, 1) \\ \times (X_{\max}^{d} - X_{\min}^{d}) + a, & (13) \end{array}

where “a” is a variable which is used to prevent the data vectors from being stuck in local optima near the boundary of data set. It is calculated using the following equation,

a = (1 - \frac{current iteration}{maximum iteration}) .

(14)

Tracing mode modification.

Tracing mode of the CSO algorithm is considered as local search for obtaining the solution set. In tracing mode, a cat traces its target with high speed. Mathematically, it can be achieved by defining the position and velocity of cat in D-dimensional search space. The position of a cat is obtained by Eq. (9). Hence, there is also a possibility that data vectors may go beyond the boundary limits. To deal with such data vectors, the following mechanism is outlined as follows.

When the position of cat $X_{j d} < X_{\min}^{d}$ , then the new position of cat is adjusted according to Eq. (16), $\begin{array}{rclr} V_{j d, new} = rand (0, 1) \times V_{j d}, & (15) \\ X_{j d, new} = X_{\min}^{d} + V_{j d, new} . & (16) \end{array}$

When the position of cat $X_{j d} > X_{\min}^{d}$ , then the new position of cat is adjusted according to Eq. (18), $\begin{array}{rclr} V_{j d, new} = - rand (0, 1) \times V_{j d}, & (17) \\ X_{j d, new} = X_{\min}^{d} + V_{j d, new} . & (18) \end{array}$

The above mentioned three modifications (selection algorithm, inertia weight, boundary level constraints handling) constitute the Improved CSO algorithm. It may be mentioned that CDC and MR parameters are not used in Improved CSO. The justification of MR and CDC parameters are as follow.

Table 1

An example dataset with $M = 10$ and $d = 4$

Dataset (M)	Attribute (d)

	Att1	Att2	Att3	Att4
1	5.5	3.5	1.3	0.2
2	4.9	3.1	1.5	0.1
3	4.4	3	1.3	0.2
4	5.1	3.4	1.5	0.2
5	5	3.5	1.3	0.3
6	4.5	2.3	1.3	0.3
7	4.4	3.2	1.3	0.2
8	5	3.5	1.6	0.6
9	5.1	3.8	1.9	0.4
10	4.8	3	1.4	0.3

CDC parameter is used to determine the dimensions of a cat’s position to be mutated [25] and the shifting value is computed only for mutated dimensions. The updated position of a cat is obtained using the mutated dimensions. As a result, only mutated dimensions are updated and rest of dimensions remain unchanged. Thus, the position of cat is partially updated. To update the position of cat in each dimension, CDC parameter either should be equal to the SMP, or it can be removed. To illustrate this, consider an example dataset given in Table 1. Present position of the cat is assumed to be $(4.4, 3, 1.3, 0.2)$ i.e. 3rd data instance of Table 1. SMP parameter is 5, then the five copies of present position are replicated as $(4.4, 3, 1.3, 0.2)$ , $(4.4, 3, 1.3, 0.2)$ , $(4.4, 3, 1.3, 0.2)$ , $(4.4, 3, 1.3, 0.2)$ and $(4.4, 3, 1.3, 0.2)$ . CDC parameter is set to 50%. Out of four dimensions, two dimensions of present position of cat to be mutated and the mutated dimensions are $(4.4, 1.3)$ , $(4.4, 0.2)$ , $(3, 1.3)$ , $(4.4, 1.3)$ and $(3, 0.2)$ . According to CDC parameter, the shifting value is computed for each mutated dimension which can be calculated using SRD ∗ mutative dimension. The shifting values are $(3.5, 1.05)$ , $(3.9, 0.18)$ , $(0.38, 0.16)$ , $(4.01, 1.18)$ and $(1.8, 0.12)$ . These values are randomly added or subtracted to each replicated copy of present position of cat to get new position and replace the old ones. The new position of cat for ith copy is $(0.9, 3, 0.25, 0.2)$ , $(8.3, 3, 1.3, 0.02)$ , $(4.4, 3.38, 1.68, 0.2)$ , $(8.41, 3, 0.12, 0.2)$ and $(4.4, 1.2, 1.3, 0.08)$ . From this, it is noticed that only mutated dimensions are updated while the rest of dimensions remain same. Due to the mentioned reason, the CDC parameter is removed so that all copies of the present position will be considered for mutation. The value outside the boundary of dataset is handled using Eqs (11) and (13).

Mixture ratio (MR) is a control parameter that can limit the number of cats to be moved into seeking mode and tracing mode [25]. The number of possible clusters in a dataset are described in terms of the number of cats and the final position of cats are used to obtain the optimal cluster centers. However, to determine the clusters accurately, the number of cats to be moved in seeking mode and tracing mode should be equal. If the number of cats in seeking mode are different from the number of cats in tracing mode, then it is not possible to compute the correct number of cluster centers. For example, Table 1 consists of three clusters. So, three cats are used to initialize the possible clusters in random order and the positions of cats correspond to the cluster centers. The objective function is computed using the position of cats, and the data is arranged into three clusters using objective function values. MR parameter is used to determine the number of cats to move into tracing mode according to its value. But the number of cats in the tracing mode should be equal to number of cats initialized, otherwise it will be conceptually wrong. In the above discussion, three cats are initialized to cluster the data. If the number of cats in tracing mode is 2 as per MR parameter then data is clustered in two clusters rather than three in tracing mode which is hypothetically wrong. Therefore, in this paper, the MR parameter is removed so that the number of cats in seeking mode and tracing mode remains same.

4.1.4. Improved CSO (ICSO) pseudo-code

This section summarizes the pseudo-code of the seeking and tracing modes of the Improved CSO (ICSO) algorithm. Algorithms 2 and 3 provide the pseudo-code of seeking and tracing modes of the ICSO algorithm.

Algorithm 2.

Seeking mode pseudo-code of ICSO algorithm

Algorithm 3.

Tracing mode pseudo-code of ICSO algorithm

4.2. Proposed ICSOKHM algorithm

This section describes the proposed ICSOKHM algorithm. The proposed algorithm is the combination of Improved CSO (discussed above) and KHM algorithms. KHM is a fast and efficient algorithm which requires less number of function evaluations but suffers from local optima problem. The proposed hybrid algorithm includes the qualities of both the algorithms and it is given the name Improved CSOKHM (ICSOKHM). The objective of the proposed algorithm is to prevent the trapping of KHM algorithm in the local optima and also to improve the convergence speed of the CSO algorithm. In short, we start with Improved CSO algorithm and feed the output of Improved CSO algorithm to KHM algorithm to obtain final cluster centers. The main steps of the proposed ICSOKHM algorithm are summarized in Algorithm 4 and corresponding flowchart is mentioned in Fig. 1.

Algorithm 4.

Pseudo-code of ICSOKHM algorithm

Fig. 1.

Flowchart of proposed CSOKHM.

5. Experimental results

The efficiency of the proposed algorithm is investigated on seven datasets. Out of seven datasets, two of are synthetic which are generated in Matlab, and rest of are real that are downloaded from the UCI repository. The characteristics of these datasets are summarized in Table 2. Matlab 2010a environment is used to implement the proposed algorithm using Windows 7 operating system with Core i5 processor and 4 GB RAM. The result of the proposed algorithm is obtained by average over 10 simulations. The “p” values of objective function (in KHM), which denotes the power, also play a vital role in obtaining the results [31]. The proposed algorithm is also tested on different “p” values i.e. 3 and 3.5. The parameters of the proposed algorithm are shown in Table 3. The values of

V_{\max}

and

V_{\min}

are in the range of maximum and minimum values of each dimension of dataset and number of iteration is set to 100. Experimental results of the proposed algorithm are compared with existing algorithms like KHM, PSO, PSOKHM, GSAKHM, ACA, ACAKHM and CSO algorithms [8,32,33].

Table 2
Characteristics of datasets

Dataset	Class	Attribute	Total data	Data in each class
Synthetic1	3	2	300	$(100, 100, 100)$
Synthetic2	3	3	300	$(100, 100, 100)$
Iris	3	4	150	$(50, 50, 50)$
Glass	6	9	214	$(70, 17, 76, 13, 9, 29)$
Cancer	2	9	683	$(444, 239)$
CMC	3	9	1473	$(629, 334, 510)$
Wine	3	13	178	$(59, 71, 48)$

Table 3

Characteristics of datasets

Parameter	Value
SRD (mutative ratio)	Random number in $[0, 1]$
SMP	5
Population size	$K * d$
$r_{1}$	Random number in $[0, 1]$
$c_{1}$	2

Fig. 2.

Represents the cluster centers in Synthetic2 dataset. (Colors are visible in the online version of the article; https://dx-doi-org.web.bisu.edu.cn/10.3233/AIC-150677.)

5.1. Datasets

1. Synthetic1 ( $Total instances = 300$ , $attributes = 2$ , $classes = 3$ ): This dataset is generated in Matlab using independent bivariate normal distribution. Data are distributed in the classes using μ and Σ where μ is the mean vector and Σ is the covariance matrix. Figure 2 shows the Synthetic1 dataset and the description of generated dataset is given below, $\begin{array}{rcl} (μ) = [\begin{matrix} μ_{(i 1)} \\ μ_{(i 2)} \end{matrix}], \\ Σ = [\begin{matrix} 0.4 & 0.04 \\ 0.04 & 0.04 \end{matrix}], i = 1, 2, 3, \dots, n, \\ μ_{(11)} = μ_{(12)} = - 2, μ_{(21)} = μ_{(22)} = 2, \\ μ_{(31)} = μ_{(32)} = 6 . \end{array}$

2. Synthetic2 ( $Total instances = 300$ , $attributes = 3$ , $classes = 3$ ): This dataset includes 300 instances with three attributes and three classes. Figure 3 describes the Synthetic2 dataset and the attributes are scattered using uniform distribution, $\begin{array}{rcl} Class 1 \sim Uniform (10, 25), \\ Class 2 \sim Uniform (25, 40), \\ Class 3 \sim Uniform (40, 55) . \end{array}$

Fig. 3.

Represents the cluster centers in Synthetic2 dataset. (Colors are visible in the online version of the article; https://dx-doi-org.web.bisu.edu.cn/10.3233/AIC-150677.)

3. Iris dataset ( $Total instances = 150$ , $attributes = 4$ , $classes = 3$ ): Iris dataset contains the three species of the iris flower: Iris Setosa, Iris Versicolour and Iris Virginica. The attributes of the dataset are sepal length, sepal width, petal length and petal width.

4. Wine dataset ( $Total instance = 178$ , $attribute = 13$ , $classes = 3$ ): It contains the chemical analysis of wine in the same region of Italy but three different cultivators. The attributes of the dataset are alcohol, malic acid, ash, alkalinity of ash, magnesium, total phenols, flavanoids, non-flavanoid phenols, proanthocyanins, color intensity, hue, OD280/OD315 of diluted wines and proline.

5. Glass ( $Total instances = 214$ , $attributes = 9$ , $classes = 6$ ): It contains the information about six different types of glass. The attributes are Id number, refractive index, sodium, magnesium, aluminium, silicon, potassium, calcium, barium and iron.

6. Cancer ( $Total instances = 683$ , $attributes = 9$ , $classes = 2$ ): This dataset contains information of cell nuclei present in the image of breast mass. The attributes in this dataset are clump thickness, cell size uniformity, cell shape uniformity, marginal adhesion, single epithelial cell size, bare nuclei, bland chromatin, normal nucleoli and mitoses. Malignant class contains 444 instances and benign class contains 239 instances.

7. Contraceptive algorithm choice (CMC) ( $Total instances = 1473$ , $attributes = 9$ , $classes = 3$ ): This dataset contains information about the married women who were either pregnant (but did not know about pregnancy) or not. Classes of CMC dataset are classified into no use, long term algorithm and short term algorithm classes. Each class contains 629, 334 and 510 instances. The attributes of dataset are Age, Wife’s education, Husband’s education, Number of children ever born, Wife’s religion, Wife now working, Husband’s occupation, Standard-of-living index and Media exposure.

5.2. Performance parameters

1. K-harmonic mean ( $KHM (X, C)$ ): This parameter is used to measure the quality of clusters. It is proportional to the distance function; smaller the sum of distances higher the quality of the cluster and vice versa. Equation (1) is used to compute the harmonic mean.

2. F-measure: It is weighted harmonic mean of recall and precision from an information retrieval system [3,5]. The value of F-measure ( $F (i, j)$ ) can be described as $F (i, j) = \frac{2 * (Recall * Precision)}{(Recall + Precision)} .$ (20) The final value of F-measure parameter can be computed using Eq. (21): $F (i, j) = \sum_{i = 1}^{n} \frac{n_{i}}{n} * max_{i} F (i, j) .$ (21)

5.3. Result and discussion

This subsection illustrates the results of our study. The effectiveness of the proposed algorithm is tested on the seven datasets and compared with the other existing algorithms like KHM, PSO, PSOKHM, ACA, ACAKHM, GSA, GSAKHM and CSO. Tables 4–10 describe the results of our proposed algorithm and other clustering algorithms using

KHM (X, C)

, F-measure and runtime parameters.

Table 4
Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for Synthetic1 and Synthetic2 datasets

Algorithm	Dataset (p value = 3)

	Synthetic1			Synthetic2

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	742.116(0.004)	1	0.001	278,758(0)	1	0.22
PSO	741.682(0.076)	1	1.796	8,675,172(6,625,756)	0.682	5.42
GSA	741.688(0.012)	1	1.716	8,675,812(6,625,896)	0.681	3.542
CSO	741.663(0.013)	1	1.746	8,674,735(6,625,865)	0.687	3.686
PSOKHM	741.458(0.002)	1	1.921	278,541(33)	1	2.844
ACAKHM	741.467(0.017)	1	1.929	278,545(11)	1	4.18
GSAKHM	741.453(0.000)	1	1.789	278,541(11)	1	2.524
CSOKHM	741.441	1	1.834	278,537(16)	1	2.654
ICSOKHM	739.23(0.028)	1	1.816	276,356(23)	1	2.536

Table 5

Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for iris and glass datasets

Algorithm	Dataset (p value = 3)

	Iris			Glass

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	126.517(0.000)	0.744	0.19	1535.198(0.000)	0.422	4.042
PSO	147.217(22.896)	0.74	3.096	18,191.700(1870.044)	0.378	43.594
ACA	147.378(26)	0.746	4.27	18,298.62(1540)	0.371	44.67
GSA	147.209(20.634)	0.743	2.998	18,246.0119(956.238)	0.378	45.385
CSO	146.92(21.453)	0.742	3.213	16,167.562(1570.782)	0.373	47.246
PSOKHM	125.951(0.052)	0.744	2.796	1442.847(35.871)	0.427	17.609
ACAKHM	126.216(0.078)	0.746	3.7	1448.366(86)	0.422	16.28
GSAKHM	125.951(0)	0.751	1.65	1400.950(0.630)	0.442	15.958
CSOKHM	125.736(0)	0.756	2.865	1389.278(0.563)	0.448	23.083
ICSOKHM	124.46(0.97)	0.769	2.923	1319.21(5.43)	0.454	19.64

Table 6

Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for cancer and CMC datasets

Algorithm	Dataset (p value = 3)

	Cancer			CMC

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	119,458(0)	0.834	2.27	187,525(0)	0.303	8.627
PSO	119,333(3770)	0.817	16.15	205,548(60,798)	0.25	54.895
ACA	120,104(3580)	0.807	13.926	208,278(55,768)	0.256	54.242
GSA	118,412(1236)	0.826	15.638	204,986(61,369)	0.267	56.559
CSO	118,936(378)	0.829	16.135	203,474(55,989)	0.291	57.835
PSOKHM	117,418(237)	0.834	9.594	186,722(111)	0.303	39.785
ACAKHM	117,468(196)	0.836	12.53	186,856(42)	0.296	39.576
GSAKHM	117,418(55)	0.847	7.91	186,722(94)	0.472	32.107
CSOKHM	117,418(46)	0.821	8.091	186,713(85)	0.493	36.141
ICSOKHM	117,424(32)	0.843	9.826	184,817(48)	0.502	37.946

Table 7

Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for Synthetic1 and Synthetic2 datasets

Algorithm	Dataset (p value = 3.5)

	Synthetic1			Synthetic2

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	807.548(0.016)	1	0.106	697,215(0.00)	1	0.22
PSO	806.811(0.079)	1	1.628	80,729,943(33,400,802)	0.66	3.601
ACA	807.742(0.08)	1	5.39	8,073,042,333(350,517)	0.645	6.513
GSA	806.779(0.027)	1	1.616	80,726,839(33,401,426)	0.662	3.624
CSO	806.708(0.037)	1	2.725	80,727,883(33,378,953)	0.679	4.896
PSOKHM	806.619(0.014)	1	1.921	696,349(78)	1	2.842
ACAKHM	807.514(0.06)	1	3.53	697,105(0.00)	1	4.19
GSAKHM	806.613(0.012)	1	1.766	696,281(34)	1	2.471
CSOKHM	806.532(0.072)	1	1.856	696,226(26)	1	2.814
ICSOKHM	798.68(0.26)	1	1.802	690,596(24.58)	1	3.136

Table 8

Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for iris and glass datasets

Algorithm	Dataset (p value = 3.5)

	Iris			Glass

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	113.413(0.085)	0.77	0.194	1871.812(0.000)	0.396	4.0568
PSO	255.763(117.388)	0.66	3.078	32,933.349(1398.602)	0.373	43.35
ACA	209.38(1619)	0.643	4.27	76,125(1415)	0.27	41.067
GSA	242.566(112)	0.672	3.107	32,933.349(1398.602)	0.386	42.218
CSO	228.534(87.216)	0.693	3.458	31,786.789(264.534)	0.382	41.108
PSOKHM	110.004(0.260)	0.762	1.873	1857.152(4.937)	0.396	17.651
ACAKHM	112.466(1.2)	0.801	3.7	1871.81161(0.00)	0.402	16.28
GSAKHM	109.94(0.002)	0.766	1.587	1857.152(0.035)	0.421	15.799
CSOKHM	110.004(0.026)	0.767	1.793	1857.152(0.456)	0.416	17.867
ICSOKHM	104.72(0.018)	0.776	1.649	1691.24(0.213)	0.426	17.108

Table 9

Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for cancer and CMC datasets

Algorithm	Dataset (p value = 3.5)

	Cancer			CMC

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	243,440(0)	0.832	2.072	381,444(0)	0.332	8.528
PSO	240,634(8842)	0.82	42.097	426,562(43,932)	0.298	49.881
ACA	241,682(6327)	0.824	45.26	424,744(36,214)	0.311	46.236
GSA	240,484(6032)	0.823	41.513	423,096(39,973)	0.301	49.035
CSO	240,118(1040)	0.826	41.679	422,846(40,565)	0.309	49.485
PSOKHM	235,441(696)	0.835	39.859	379,678(247)	0.332	32.707
ACAKHM	236,341(125.78)	0.876	36.53	380,462(578.98)	0.514	36.268
GSAKHM	236,125(15)	0.862	31.521	380,183(16)	0.506	31.521
CSOKHM	235,965(45)	0.889	36.521	380,069(47)	0.521	38.456
ICSOKHM	232,917(32)	0.891	35.438	378,947(41)	0.528	35.276

Table 10

Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for wine dataset

Algorithm	Dataset

	Wine (p value = 3)			Wine (p value = 3.5)

	$KHM (X, C)$	F-measure	Runtime (s)	$KHM (X, C)$	F-measure	Runtime (s)
KHM	298,230,848(24,270,951)	0.538	2.084	8,568,319,639(2075)	0.502	2.04
PSO	276,508,278(23,807,035)	0.519	35.284	363,757,595(202,759,448)	0.53	35.072
ACA	276,506,876(21,670,255)	0.519	35.46	7,285,431,684(2,784,324)	0.519	35.846
GSA	276,506,778(237,867,414)	0.521	34.756	363,756,341(202,747,531)	0.53	35.512
CSO	273,473,397(19,405,216)	0.526	38.972	363,756,474(202,754,594)	0.531	35.902
PSOKHM	252,522,504(766)	0.553	6.598	3,546,930,579(1,214,985)	0.535	6.508
ACAKHM	252,526,114(274)	0.551	6.16	3,549,156,713(208.78)	0.534	7.57
GSAKHM	252,522,000(0)	0.553	5.71	3,540,920,000(232)	0.536	5.536
CSOKHM	252,492,116(324)	0.554	7.857	3,540,918,728(476)	0.542	6.236
ICSOKHM	251,649,384(89)	0.561	7.128	3,539,527,392(393)	0.547	5.836

Table 4 demonstrates the results of all the nine algorithms using Synthetic1 and 2 datasets when “p” is equal to 3. From results, it can be observed that performance of all the algorithms is almost equal (in terms of $KHM (X, C)$ and F-measure). But, significant differences lie between the aforementioned algorithms with respect to runtime parameter. KHM algorithm takes minimum runtime for both the datasets while ACA algorithm takes maximum time.

Results of the iris and glass datasets for all algorithms is described in Table 5 when “p” value is 3. Table 5 indicates that proposed ICSOKHM algorithm provides better results (in terms of $KHM (X, C)$ and F-measure) with both the datasets in comparison to other algorithms. ACA algorithm gives worst results amongst all the algorithms particularly with iris dataset in terms of $KHM (X, C)$ and runtime parameters. In case of iris dataset, PSO algorithm also exhibits poor performance in terms of F-measure. With glass dataset, ACA algorithm also exhibits poor performance in terms of $KHM (X, C)$ and F-measure while CSO algorithm takes maximum runtime.

Table 6 depicts the results of all algorithms for cancer and CMC datasets when “p” value is 3. From results, it can be concluded that GSAKHM algorithm gives better results in terms of $KHM (X, C)$ and F-measure parameters for cancer dataset. Again, ACA algorithm gives worst performance in terms of $KHM (X, C)$ and F-measure parameters amongst all the used algorithms. Performance of the proposed ICSOKHM algorithm is better in comparison to other algorithms in terms of $KHM (X, C)$ and F-measure parameters, except GSAKHM and PSOKHM. But, in case of CMC dataset, the proposed algorithm gives better results (in terms of $KHM (X, C)$ and F-measure parameters) in comparison to the all other algorithms. In addition to it, it can be seen that KHM algorithm takes less runtime for both datasets while CSO algorithm takes more runtime.

Table 7 shows the experimental results of all the nine algorithms for Synthetic1 and Synthetic2 datasets when “p” value is 3.5. From results, it can be stated that there is a slight variation between the performance (in terms of $KHM (X, C)$ and F-measure parameters) of all the algorithms except runtime parameter. Again, KHM algorithm takes minimum runtime for both datasets while ACA algorithm takes maximum runtime.

Results of all the nine algorithms for iris and glass datasets are summarized in Table 8 when “p” is equal to 3.5. Results indicate that the proposed ICSOKHM algorithm gives better results (in terms of $KHM (X, C)$ and F-measure parameters). But, it is also observed that proposed algorithm gives better runtime among all the algorithms except KHM and GSAKHM algorithms for iris dataset. In case of wine dataset, proposed algorithm also obtains better runtime except KHM, ACAKHM and GSAKHM algorithms. On the other hand, it is noticed that PSO provides worst performance (in terms of $KHM (X, C)$ and F-measure parameters) for iris dataset while ACA exhibits poor performance (in terms of $KHM (X, C)$ and F-measure parameters) for glass dataset. It is revealed that KHM algorithm again takes less runtime for both the dataset. While, ACA and PSO algorithms take more runtime for iris and glass datasets, respectively.

Table 9 illustrates the results for cancer and CMC datasets when “p” value is 3.5. It is noticed that proposed ICSOKHM provides better results (in terms of $KHM (X, C)$ and F-measure parameters) in comparison to the other algorithms. For cancer dataset, KHM algorithms exhibits poor performance in terms of $KHM (X, C)$ and F-measure. It is also observed that PSO algorithm gives worst performance (in terms of $KHM (X, C)$ , F-measure and runtime parameters) for CMC dataset. Table 10 demonstrates the results of wine dataset using all the algorithms when “p” values are 3 and 3.5. Results indicate that the proposed algorithm provides better results (in terms of $KHM (X, C)$ and F-measure parameters) among all the algorithms using both the “p” values. KHM algorithms exhibits poor performance (in terms of $KHM (X, C)$ and F-measure parameters) among all the algorithms for wine dataset for both the “p” values. It is also observed that the CSO algorithm takes more runtime amongst all.

Finally, from Tables 4–10, it can be concluded that average $KHM (X, C)$ and F-measure of proposed ICSOKHM are better than the other algorithms using real datasets. It is also noticed that ACA and KHM exhibit poor performance with most of the datasets. In case of synthetic datasets, performance (in terms of average $KHM (X, C)$ and F-measure) of all the algorithms are almost same except runtime parameter. It is noted that KHM algorithm requires minimum runtime than others but may be trapped in local optima. Moreover, it is also observed that the CSO algorithm takes more runtime with most of the datasets when “p” value is 3. While, ACA algorithm takes maximum runtime for synthetic dataset 2, cancer and wine datasets when “p” value is 3.5. PSO algorithm takes maximum runtime for glass and CMC datasets when “p” value is 3.5. From the performance comparison of PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM algorithms, it can be observed that proposed ICSOKHM algorithm provides better results (in terms of $KHM (X, C)$ and F-measure) except cancer dataset when “p” value is 3. From the above discussion, it can be predicted that GSAKHM algorithm requires less runtime for all datasets. Conversely, ACAKHM algorithm takes maximum runtime for most of the datasets using both the “p” values. Along this, it can be noticed that ICSOKHM takes more runtime only for wine and glass datasets when “p” value is 3. PSOKHM algorithm requires more runtime for cancer and glass datasets when “p” value 3.5. Overall conclusion can be summarized as:

Proposed algorithm gives better results for most datasets with large “p” values (3.5) except wine dataset.

Proposed ICSOKHM algorithm also provides better runtime results when “p” value is large (3.5).

Proposed algorithm gives significant results with linearly non-separable datasets such as iris and glass datasets.

6. Conclusion

In this paper, a new improved CSOKHM (ICSOKHM) algorithm is proposed by combining the Improved cat swarm optimization (CSO) and K-harmonic means (KHM) algorithms. The proposed algorithm contains the qualities of both the algorithms. In addition to it, a selection algorithm is also introduced in tracing mode of the CSO algorithm. Along this, some heuristic approaches are also used to make the CSO algorithm more robust and efficient. Finally, the performance of the proposed algorithm is evaluated on two synthetic and five benchmark datasets and compared with existing algorithms like KHM, PSO, PSOKHM, ACA, ACAKHM, GSA, GSAKHM CSO and CSOKHM. The experimental results prove that ICSOKHM is an effective and more competent algorithm than other existing algorithms being compared. In this paper, harmonic average is taken as objective function for proposed algorithm instead of Euclidean distance. With the same objective function, the CSO algorithm requires more time and higher convergence rate while KHM gets stuck in local optima. But, the proposed ICSOKHM algorithm not only improves the convergence speed of the CSO but also prevents the KHM from running into local optima.

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A hybrid data clustering approach based on improved cat swarm optimization and K -harmonic mean algorithm

Abstract

Keywords

1. Introduction

2. K-harmonic algorithm

3.1. Seeking mode

4. Proposed improved CSOKHM algorithm

4.1. Proposed modifications in CSO algorithm

4.1.1. Selection mechanism

Table 2 Characteristics of datasets

Table 4 Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for Synthetic1 and Synthetic2 datasets

References

Table 2
Characteristics of datasets

Table 4
Comparison of results obtained using KHM, PSO, ACA, GSA, CSO, PSOKHM, ACAKHM, GSAKHM, CSOKHM and ICSOKHM for Synthetic1 and Synthetic2 datasets