A Hybrid Data Clustering Algorithm Using Modified Krill Herd Algorithm and K-MEANS

Subject Areas : Pattern Analysis and Intelligent Systems

Jensi R ¹

1 - Dr.Sivanthi Aditanar College of Engineering Tiruchendur Tamilnadu India

Received: 2018-12-27 Accepted : 2019-03-23 Published : 2019-05-01

Keywords: Levy-flight distribution, Convergence rate, Data clustering, Krill Herd, K-means,

Abstract :

Data clustering is the process of partitioning a set of data objects into meaning clusters or groups. Due to the vast usage of clustering algorithms in many fields, a lot of research is still going on to find the best and efficient clustering algorithm. K-means is simple and easy to implement, but it suffers from initialization of cluster center and hence trapped in local optimum. In this paper, a new hybrid data clustering approach which combines the modified krill herd and K-means algorithms, named as K-MKH, is proposed. K-MKH algorithm utilizes the power of quick convergence behaviour of K-means and efficient global exploration of Krill Herd and random phenomenon of Levy flight method. The Krill-herd algorithm is modified by incorporating Levy flight in to it to improve the global exploration. The proposed algorithm is tested on artificial and real life datasets. The simulation results are compared with other methods such as K-means, Particle Swarm Optimization (PSO), Original Krill Herd (KH), hybrid K-means and KH. Also the proposed algorithm is compared with other evolutionary algorithms such as hybrid modified cohort intelligence and K-means (K-MCI), Simulated Annealing (SA), Ant Colony Optimization (ACO), Genetic Algorithm (GA), Tabu Search (TS), Honey Bee Mating Optimization (HBMO) and K-means++. The comparison shows that the proposed algorithm improves the clustering results and has high convergence speed.

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Full-Text:



A Hybrid Data Clustering Algorithm Using Modified Krill Herd Algorithm and K-Means

R.Jensi, AP/CSE, Dr.Sivanthi Aditanar College of Engineering, Tiruchendur

 

Abstract

Data clustering is the process of partitioning a set of data objects into meaning clusters or groups. Due to the vast usage of clustering algorithms in many fields, a lot of research is still going on to find the best and efficient clustering algorithm. K-means is simple and easy to implement, but it suffers from initialization of cluster center and hence trapped in local optimum. In this paper, a new hybrid data clustering approach which combines the modified krill herd and K-means algorithms, named as K-MKH, is proposed. K-MKH algorithm utilizes the power of quick convergence behaviour of K-means and efficient global exploration of Krill Herd and random phenomenon of Levy flight method. The Krill-herd algorithm is modified by incorporating Levy flight in to it to improve the global exploration. The proposed algorithm is tested on artificial and real life datasets. The simulation results are compared with other methods such as K-means, Particle Swarm Optimization (PSO), Original Krill Herd (KH), hybrid K-means and KH. Also the proposed algorithm is compared with other evolutionary algorithms such as hybrid modified cohort intelligence and K-means (K-MCI), Simulated Annealing (SA), Ant Colony Optimization (ACO), Genetic Algorithm (GA), Tabu Search (TS), Honey Bee Mating Optimization (HBMO)  and K-means++. The comparison shows that the proposed algorithm improves the clustering results and has high convergence speed.

Keywords: Data clustering, Krill Herd, Levy-flight distribution, K-means, Convergence rate

I. Introduction

Data clustering [5] is the method in which a group of data objects are divided into groups or clusters in such a way that the objects within the clusters are having high similarity while the data objects in different clusters are dissimilar. Data clustering is an unsupervised technique due to the unknown class labels. The similarity between data objects is measured by some distance metric. There are several distance measurements [2] such as Euclidean distance, Minkowski metric, Manhatten distance, Cosine similarity, Jaccard coefficient, Pearson correlation coefficient, and so on.

Clustering is widely used in many fields of science and engineering and it must often be solved as part of complicated tasks in pattern recognition, data mining, information retrieval and image analysis. The clustering algorithms are mainly classified into two [2]: hierarchical and partitional. Hierarchical clustering algorithms group data objects into tree-like structure and it is further classified into two types, agglomerative and divisive, based on how the hierarchical decomposition is formed. On the other hand, partitional clustering algorithm groups the data objects into a predefined number of clusters based on some optimization criterions. The most well known partitional clustering algorithm is K-means which is the center-based clustering algorithm. The advantage of K-means algorithm is simple and efficient. But K-means suffers from initial cluster seed selection since it is easily trapped in local minima. In order to overcome the shortcomings of K-means, several heuristic algorithms have been introduced in the literature.

Many nature-inspired algorithms, also known as Swarm Intelligence (SI) [3] have been introduced inspired by the clever behaviours of animal or insect groups, such as ant colonies, bird flocks or fish schools, bacterial swarms, bee colonies, cuckoos, fireflies and flower pollination. Swarm Intelligence is based on heuristic approach, so SI algorithms were used to solve the clustering problems.

Gandomi and Alavi(2012) [26] introduced Krill Herd (KH) optimization algorithm simulating the herding behaviour of krill individuals to solve the optimization problems. The implementation of KH is available from [37]. A novel variants of Krill Herd algorithm was presented in [27]-[30], [40-41]. Each algorithm was tested with several standard unimodal and multimodal functions. The main contribution of this study is to apply krill-herd algorithm with levy flight for data clustering. In order to speed up the convergence of the proposed algorithm, k-means algorithm is employed for generating initial cluster centers.

The remaining section of the paper is organized as follows. Section 2 lists out the various research works related to data clustering and Section 3 provides the clustering problem statement. Section 4 briefly explains the K-means algorithm, Original Krill Herd algorithm, Levy Flight method and Section 5 presents the proposed K-MKH approach. Section 6 provides the experimental results, and Section 7 concludes the paper.

 

II. Related Works

In this section the various research works related to data clustering proposed by the authors are given. Selim et.al [4] introduced a simulated annealing algorithm for solving the clustering problems. Maulik and Bandyopadhyay [5] proposed a clustering algorithm using genetic algorithm for improving global search capacity. Sung et.al. [6] presented a tabu search based clustering method to alleviate the local optima problem. An ant colony based clustering approach was proposed by Shelokar.et.al [7]. Liu et.al. [8] proposed a new tabu search based clustering approach to enhance the clustering solutions. In Kao et.al [9], a hybrid clustering technique K-NM-PSO based on K-means, Nedler-Mead simplex and PSO was presented by the authors. Fathian and Amiri [10] applied honey-bee mating technique for obtaining better cluster solutions. Dervis Karaboga [11] and Yan et.al [12] proposed novel clustering algorithms using Artificial Bee Colony (ABC). Miao Wan et.al [13] proposed data clustering using bacterial foraging optimization. Senthilnath et.al [14] performed a clustering study using firefly algorithm. Tunchan Cura [15] presented a particle swarm optimization approach to clustering for the known and unknown number of clusters. Data clustering using a binary search algorithm was done in Hatamlou [16] and also Hatamlou [17] presented a new heuristic algorithm for data clustering using black hole phenomenon. Hybrids of evolutionary and nature-inspired algorithms were developed for solving data clustering problem. These include ACO and SA [18], [21], PSO and SA [19], PSO, SA and K-means [20], PSO, ACO, K-means [22], PSO and K-means [23], modified imperialist competitive algorithm and K-means [24], Modified Cohert and K-means [25], FPA-K[38], MBA-LF [39]. In [42], Krill-herd algorithm is used for optimizing resource allocation in mobile cloud computing in an energy-efficient manner. Modified mutation operators and updated mechanisms are applied to Krill herd algorithm [43] to improve global optimization and the proposed algorithm is then applied to solve clustering problems. The authors of [44] used krill-herd strategy to investigate the energy saving opportunities in the single mixed refrigerant liquefaction process.  In [45], k-means algorithm is initialized by choosing random starting centers with specific probabilities.

III.The problem statement

 

Clustering is the process of partitioning the set of N data objects into K clusters or groups based on some distance (or similarity) metric. Let D = {d1, d2,…,dN} be a set of N data objects to be partitioned and each data object di ,i=1,2,… ,N is represented as di={di1,di2,….,dim} where dim represents mth dimension value of data object i.

The aim of clustering algorithm is to find a set of K partitions C = {C1, C2, …., Ck |  k : Ck  and l  : CkCl= in such a way that objects within the clusters are more similar and dissimilar to objects in different clusters. These similarities are measured by some optimization criterions, especially squared error function [30] and it has been calculated as follows:

 

                                                                                     

where cj represents a jth cluster center ; E is a distance measure between a data object di and a cluster center cj. This optimization criterion is used as the objective function value in this study. There are many distance metric used in literature. In this study Euclidean distance is used as distance metric which is defined as follows:

 

 

where, cj is cluster center for a cluster j and is calculated as follows:

 

 

where nj is the total number of objects in cluster j.

The main issue in data clustering is local optima problem and slow convergence speed. In order to achieve global optimal solution and speed up the convergence, a hybrid data clustering approach using modified krill herd and k-means is proposed.

IV.K-Means and Krill Herd Algorithm

 

4.1       K-Means algorithm

K-means [1] is the simplest partitional clustering algorithm and it is widely used due its simplicity and efficiency. Given a set of N data objects and the number of clusters k, the k-means algorithm proceeds as follows:

Step 1:         Randomly select ‘k’ cluster centers.

Step 2:         Step 2: Calculate the Euclidean distance between each data point and cluster centers.

Step 3:         Step 3: Assign the data point to the cluster center whose distance from the cluster center is minimum of all the cluster centers.

Step 4:         Update cluster center using Eq. (3).  

Step 5:         If no data point was reassigned then stop, otherwise repeat from step 2.

4.2       Krill Herd Algorithm

Krill Herd (KH) [26] is a new heuristic population based global optimization algorithm. The inspiration of KH algorithm is the herding behaviour of krill swarm when looking for food and communication with each other. The implementation of KH method is based on three movements such as

(i)                      Movement influenced by other krill individual

(ii)                      Foraging action

(iii)                      Physical diffusion

KH approach follows Lagrangian model for effective search and it is described as:

                                 (4)

where  is the movement induced by other krill individuals, is the foraging action and  is the random physical diffusion of the ith krill individuals.

The direction of motion induced , depends on the three components, namely local swarm density, a target swarm density and a repulsive swarm density. The movement of a krill individual Ni is defined as:

                         (5)

where

                                (6)

and  is the maximum induced speed,   is the inertia weight,  is the motion induced previously,  is the local effect offered by neighbours and  is the best krill individual’s target effect.

The second movement of KH approach foraging action  depends on two parameters, namely current food location and information about previous food location. The ith krill individual’s motion is described as:

                                 (7)

where

                                (8)

and  is the foraging speed,  is the inertia weight of the foraging action,  is the previous foraging motion,  is the food attractive and  is the best fitness found by the ith krill so far. The value for  is equal to 0.997 at the first iteration and decreases gradually to 0.1 at the end of the iteration.

The third movement of KH approach is random physical diffusion. The physical diffusion of the ith krill individual depends on two components, namely maximum diffusion speed and a random directional vector and it is defined as:

                                 (9)

where  is the maximum diffusion speed,  is the random vector in the range [-1, 1],  is the current generation and  is the maximum generation.

Based on the three movements defined above, the position of ith krill individual during the time interval is

                        (10)

It is clearly seen that  is an important parameter and its value determines the convergence speed.        For more details, refer [26].

4.3 Levy flight

Levy flight follows [31-34]; the generation of random numbers with levy flight consists of two steps: the choice of a random direction and the generation of steps which obey the chosen levy distribution. Random walks are drawn from Levy stable distribution. This distribution is a simple power-law formula L(s) ∼ |s|−1−β where 0 < β< 2 is an index.

     Definition 4.1 Mathematically, a simple version of Levy distribution can be defined as:

 

 

 

where  parameter is location or shift parameter,  > 0 parameter is scale (controls the scale of distribution) parameter.

  

Definition 4.2 In general, Levy distribution should be defined in terms of Fourier transform.

 

 

where  is a parameter within [−1, 1] interval and known as skewness or scale factor. An index of o stability  (0, 2) is also referred to as Levy index. The analytic form of the integral is not known for general  except for a few special cases.

For random walk, the step length S can be calculated by Mantegna’s algorithm as

 

 

 

where and  are drawn from normal distributions. That is

 

  

 

where 

  

 

Then the step size is calculated by

 

The new position of the krill individual is calculated as:

[0,1]        (15)

 

where ⊕ stands for entry-wise multiplication,  is the best position of the jth variable krill individual in the swarm.

V. The proposed hybrid K-MKH clustering approach

In [26], Gandomi and Alavi presented four different KH algorithms and they tested each algorithm and concluded that KH with crossover operator has the best performance in compare to those of other algorithms. Hence, in this study KH means it refers to KH with crossover operator. The shortcoming of KH algorithm is KH cannot escape from local optima due to the failure in global search capability. The search in KH is based on random physical activity and hence it cannot always produce the global optimal solution. In order to alleviate the shortcomings of KH, in this paper global search capability is included via levy-flight method.

5.1 Global search random walk using Levy Flight method

In order to explore search space globally, fine tuning of current position of ith krill individual is made with a chance of 0.5.  For random walk, a coin is flipped and if it is less than 0.5 then Levy walk is performed for making diversity of solutions as in Section 3.3, otherwise new position of the krill is created belonging to the search space.

The above mentioned modification of krill herd algorithm is combined with K-means to solve the clustering problem. The proposed novel hybrid data clustering approach is then referred to as Modified Krill Herd with K-means (K-MKH). In this hybrid algorithm, K-means is employed as the first step before entering into the generations for finding optimal solution. Instead of getting trapped in local minimum, this idea makes the proposed algorithm to converge quickly as well as attain best clustering quality. For solving the data clustering problem, K-MKH algorithms is neatly explained in detail in the following steps.

Step 1: Initialize algorithm parameters, define minimum and maximum bounds.

Step 2: Randomly generate krill individuals (solutions).  The population of solutions is represented as given below:

                                                     (16)

                                 (17)

                 (18)

where k is the number of clusters, m is the dimension of the data object, N is the number krill individuals. 

Thus the candidate solution is represented as a row vector of size k×m and it is shown in   Fig. 1.

Step 3: Run K-means with random cluster center generated in Step 2 as seed. Perform this step for each krill individual.

Step 4: Evaluate the objective function value f using Eq. (1) and find the worst and best fitness values.

Step 5: Store the pre-specified number of best krill.

Step 6: Calculate three movements.        

5.1 Movement influenced by other krill individual

5.2 Foraging action

5.3 Physical diffusion

Step 7: Implement crossover operator.

Step 8: Update krill position using Eq. (10).

Step 9: Generate a random integer between 0 and 1 and if it is less than 0.5, explore new krill individual position using Levy walk as in Section 3.3 using equation (15), otherwise krill individual new position is found using the below equation.

                   (19)

        

where UB and LB are maximum and minimum value of each feature of the data object respectively.

Step 10: Evaluate the objective function value f using Eq. (1) and update the krill individual if necessary.

Step 11: Repeat Step 6-10 for each krill individual.

Step 12: Replace the worst krill with the best krill stored before.

Step 13: Increment the iteration count and go to Step 5 if the maximum number of iterations is not reached.

The flowchart of the proposed algorithm for data clustering problem is shown in Fig. 2.

 

 

c11

c12

c13

…

c1m

c21

c22

c23

…

c2m

ck1

ck2

ck3

…

ckm

 

 

C1                                        C2                        ….                Ck

 

Fig.1. Representation of a candidate solution for k clusters and m features

 

 

Fig.2. Flowchart of the proposed algorithm

 

VI.                      Experimental results

The Kmeans, PSO, Krill Herd (KH) [26], Hybrid Kmeans and Krill Herd (K-KH) and proposed algorithm (K-MKH) are written in Matlab 8.3 and executed in a Windows 7 Professional OS environment using Intel i3, 2.30 GHz, 2 GB RAM. K-means, PSO, KH, K-KH and K-MKH are executed 20 times independently.  KH, K-KH and K-MKH algorithms are run with the parameters as [26]: Vf = 0.02, Dmax = 0.005, Nmax = 0.01, Max Generation=300, No. of Krills=25. The parameter setting for PSO algorithm is shown in Table 1

Table 1. PSO Parameter Settings

 

To evaluate the performance of proposed algorithm, ten datasets have been used.  Two artificial datasets, Art1 and Art2 are drawn from Kao. et.al (2008) [9].  The eight real datasets, namely, Iris, Wine, Glass, Wisconsin Breast Cancer (WBC), Contraceptive Method Choice (CMC), Crude Oil, Vowel and Liver Disorders (LD) are collected from Machine Learning Laboratory [35]. The datasets used in this study is summarized in Table 1.1. 

Table.1.1 Test Dataset Descriptions

Parameter	Value
Max Generation (Ngen)	300
Population Size(Popsize)	40
	0.9
	0.4
C1	2
C2	2

Dataset Name	# of features	# of classes	# of instances(size of each class)
Art1	2	4	600 (150,150,150,150)
Art2	3	5	250(50,50,50,50,50)
Iris	4	3	150(50,50,50)
Wine	13	3	178(59,71,48)
Glass	9	6	214 (70,17,76,13,9,29)
Wisconsin Breast Cancer (WBC)	9	2	683(444,239)
CMC	10	3	1473(629,333,511)
Crude Oil	5	3	56(7,11,38)
Vowel	3	6	871 (72, 89, 172, 151, 207, 180)
Liver Disorders (LD)	6	2	345(145,200)

 

In order to evaluate the performance and accuracy of the clustering result, three criteria are used. They are:

(i)                      Intra-cluster distances as defined in Eq. (1). The low value of the sum is, the higher the quality of the clustering is.

(ii)                      Number of fitness function evaluations (NFE). The smaller the NFE value is, the higher the convergence speed of the algorithm.

(iii)                      F-measure: This combines the precision and recall values used in information retrieval. The precision P(i,j) and recall R(i,j) for each class i of each cluster j are calculated as

                                            (20)

                                                             (21)

where,

γi : is the number of members of class i

γj : is the number of members of cluster j

γij: is the number of members of class i in cluster j

The corresponding F-measure F(i,j) is given in Eq. (22):

                                  (22)

Then the definition of F-measure of a class i is given as

                                             (23)

where, n is the total number of data objects in the collection. In general, the larger the F-measure gives the better clustering result.

Table 2-3 lists the best, worst, average and standard deviation of solutions for the five algorithms K-means, PSO, Original Krill Herd (KH), K-means Krill Herd (K-KH) and proposed algorithm (K-MKH) from 20 independent runs for artificial and real life datasets respectively and Table 4-5 lists the average and standard deviation of F-measures and mean time (in seconds) from 20 independent runs for artificial and real life datasets respectively.

 From the values given in Table 2., for the artificial datasets Art1 KH obtains better solution than the other algorithms whereas for the Art2 dataset K-KH and K-MKH algorithms achieve better results. As seen from Table 3, for the iris dataset KH, K-KH and K-MKH algorithms obtain the same result. K-KH performs well for the vowel dataset. For the datasets Wine, Glass, WBC, CMC, Crude Oil and Liver Disorders, the proposed algorithm obtains better optimal results compared to other algorithms. Thus the proposed approach reaches the optimal values in almost all the 20 independent runs.

The convergence behaviour of K-means, PSO, KH, K-KH and K-MKH algorithms for the artificial and real datasets are shown in Fig. 3-12. On seeing the graph, K-means algorithm converges quickly and at the same time it gets stuck in local optima. The proposed algorithm converges quickly compared to PSO, KH, and K-KH and also achieves better optimal solutions than those algorithms.

Also the performance of the proposed algorithm is compared with several heuristic methods in the literature such as K-means++ [36], SA [4], GA [5], ACO [7], TS [8], HBMO [10], K-MCI [25] whose results are directly taken from [25] and its values is given in Table 6. Table 6 lists the best, worst, average and standard deviation of solutions from 20 independent runs and also includes the number of fitness function evaluations (NFE) required to attain the best solution.

The experimental results given in Table 6 show that proposed algorithm obtains near optimal solutions and converge quickly in compare to those of other methods. The proposed algorithm achieves much better results for almost all datasets with small standard deviation. The number of function evaluations required for obtaining best solution over 20 independent runs is much smaller than all other methods. For iris dataset, K-MCI converges to 96.6554 for each run, but K-MKH obtains 96.6555 for each run. While comparing the number of function evaluations required to achieve the optimal result is 3145 for proposed algorithm whereas 3200 for K-MCI which indicates that proposed algorithm obtains near optimal value in small amount of time.

For wine dataset, K-MKH obtains better solution for best, worst, mean and standard deviation than K-means ++, GA, SA, TS, ACO, HBMO, K-MCI, KH. As for glass data set, K-MKH achieves best global optimum value of 210.45 which needs only 5511 fitness function evaluations while the mean values for glass dataset for the algorithms K-means ++, GA, SA, TS, ACO, HBMO,K-MCI and K-MKH are 217.56, 278.37, 282.19, 279.87, 269.72, 245.73,212.47 and 212.387 respectively. And also the NFE for glass dataset is very high for all methods when compared to K-MKH algorithm.

 

The best solutions for Cancer dataset obtained by K-MKH algorithm is 2964.38 in 3015 function evaluations, while K-MCI achieves 2964.38 in 5000 function evaluations. For CMC dataset, K-MKH achieves best, worst and mean solutions of 5693.727, 5693.731 and 5693.729 with a standard deviation of 0.002328, while K-means ++, GA, SA, TS, ACO, and HBMO cannot reach the global solution in all runs whereas K-MCI obtains the best solution with a standard deviation of 0.014. As proposed K-MKH algorithm reached the solution in almost all runs with a small standard deviation values indicates that K-MKH has high convergence speed as well as reaches optimal solution. For vowel dataset, K-MCI performs well than all other algorithms including K-MKH. Nevertheless, K-MKH achieves best solutions of 148967.24 in 5667 function evaluations. As a conclusion, the proposed scheme reaches near global optimal solution with a small standard deviation and smaller number of fitness function evaluations. As well as when compared to the other algorithms, K-MKH obtains first rank.

 

 

Table.2. Intra-cluster distances for different algorithms for artificial datasets

Data set	Criteria	K-means	PSO	KH	K-KH	K-MKH
Art1	Average (Std)	531.5287 (0)	531.0678 (0.44678)	530.8739 (2.73E-05)	531.5287 (0)	531.5287 (0)
	Best	531.5287	530.8742	530.8739	531.5287	531.5287
	Worst	531.5287	532.7425	530.874	531.5287	531.5287
Art2	Average (Std)	2173.115 (375.0322)	1761.966 (25.06701)	1828.604 (249.3881)	1727.153 (0.000166)	1727.153 (0.00061)
	Best	1728.798	1736.722	1727.153	1727.153	1727.153
	Worst	2516.083	1813.282	2504.093	1727.154	1727.154

 

Table.3. Intra-cluster distances for different algorithms for real life datasets

Data set	Criteria	K-means	PSO	KH	K-KH	K-MKH
Iris	Average (Std)	98.58575 (5.576794)	97.39148 (0.633658)	96.65549 (4.15E-06)	96.65549 (4.32E-06)	96.65549 (6.39E-06)
	Best	97.32592	96.68199	96.65548	96.65549	96.65548
	Worst	122.2789	98.77545	96.6555	96.6555	96.6555
Wine	Average (Std)	16707.38 (467.1996)	16316.56 (8.289929)	16943.49 (472.6777)	16483.52 (81.87778)	16292.59 (0.196126)
	Best	16555.68	16304.9	16331.95	16292.29	16292.21
	Worst	18123.03	16331.63	18277.05	16555.68	16292.7
Glass	Average (Std)	223.7238 (11.16914)	229.0883 (5.301659)	216.3489 (2.67677)	212.7181 (1.239082)	212.504 (1.243256)
	Best	215.6775	218.1326	212.2917	210.5065	210.4447
	Worst	254.5833	237.9754	220.8886	215.1111	213.355
Wisconsin Breast Cancer(WBC)	Average (Std)	3099.078 (498.2934)	2991.599 (19.21758)	2964.388 (0.000543)	2969.012 (8.249199)	2964.387 (0.000255)
	Best	2986.961	2972.092	2964.387	2964.387	2964.387
	Worst	5216.089	3040.519	2964.389	2984.637	2964.388
CMC	Average (Std)	5704.184 (0.882939)	5834.046 (66.04917)	5771.027 (240.3774)	5697.486 (4.654867)	5693.727 (0.002554)
	Best	5703.438	5737.539	5693.725	5693.724	5693.725
	Worst	5705.275	5995.226	6594.83	5703.438	5693.734
Crude Oil	Average (Std)	279.6207 (0.182929)	278.3461 (0.665132)	282.5664 (10.20102)	277.492 (0.615446)	277.2588 (0.044633)
	Best	279.271	277.5075	277.2107	277.2107	277.2107
	Worst	279.7432	279.9806	315.0369	279.271	277.3018
Vowel	Average (Std)	151569 (2570.489)	155672.9 (3718.922)	149474.5 (1022.676)	149027.7 (45.45572)	149038.9 (92.98758)
	Best	149400.6	151161.6	148967.3	148967.3	148967.2
	Worst	157814.6	164610.6	153127	149072.4	149383.1
Liver Disorders (LD)	Average (Std)	10213.49 (4.223201)	9881.303 (20.63139)	9851.812 (0.158778)	9905.162 (130.113)	9851.776 (0.131)
	Best	10212.55	9856.047	9851.722	9851.722	9851.721
	Worst	10231.44	9930.972	9852.081	10212.55	9852.081

 

Table.4. F-measure and Computational Time values for different algorithms for artificial datasets

Data set	Criteria	K-means	PSO	KH	K-KH	K-MKH
Art1	Average F-Measure	0.996667	0.996667	0.996667	0.99667	0.996667
	Std F-Measure	3.42E-16	3.42E-16	3.42E-16	5.7E-17	3.42E-16
	Time	0.005221	3.619262	4.009741	19.45976	19.17784
Art2	Average F-Measure	0.876562	1	0.969524	1	1
	Std F-Measure	0.104329	0	0.074721	0	0
	Time	0.004638	3.255233	3.79901	4.257772	4.365127

 

Table.5. F-measure and Computational Time values for different algorithms for real life datasets

Data set	Criteria	K-means	PSO	KH	K-KH	K-MKH
Iris	Average F-Measure	0.876254	0.900314	0.898775	0.898775	0.898775
	Std F-Measure	0.051157	0.010492	1.14E-16	1.14E-16	1.14E-16
	Time	0.018827	3.102192	4.158826	4.292506	4.352969
Wine	Average F-Measure	0.709064	0.722999	0.717422	0.71326	0.725405
	Std F-Measure	0.017896	0.00459	0.010765	0.010398	0.001972
	Time	0.004443	3.179471	4.145985	4.50042	4.77045
Glass	Average F-Measure	0.537838	0.523456	0.52628	0.558264	0.55857
	Std F-Measure	0.021938	0.02433	0.025186	0.004022	0.004598
	Time	0.007306	3.453983	4.777963	4.652005	5.4064
Wisconsin Breast Cancer(WBC)	Average F-Measure	0.9477	0.963706	0.964755	0.964385	0.964755
	Std F-Measure	0.059807	0.002181	2.28E-16	0.001064	2.28E-16
	Time	0.002859	3.443544	5.035167	4.489766	5.093428
CMC	Average F-Measure	0.402879	0.402819	0.403912	0.412995	0.412968
	Std F-Measure	0.002171	0.002938	0.008196	0.016432	4.21E-05
	Time	0.009258	5.186614	6.072309	5.727927	6.108674
Crude Oil	Average F-Measure	0.671124	0.69824	0.680466	0.709203	0.710883
	Std F-Measure	0.022764	0.020161	0.033381	0.015954	0.01116
	Time	0.004041	2.990571	3.778886	4.119627	3.979876
Vowel	Average F-Measure	0.5233	0.533555	0.531858	0.533769	0.531142
	Std F-Measure	0.036181	0.026008	0.019983	0.00623	0.005701
	Time	0.015468	4.587014	5.036707	5.111952	5.210973
Liver Disorders (LD)	Average F-Measure	0.626148	0.62266	0.622763	0.627467	0.628642
	Std F-Measure	0.000213	0.002002	0.000929	0.012294	0.005325
	Time	0.003944	3.273448	5.21026	5.194463	4.901854

 

 

 

Fig.3. Convergence behavior of Art1 dataset

 

Fig.4. Convergence behavior of Art2 dataset

 

Fig.5. Convergence behavior of  Iris dataset

                   Fig.6. Convergence behavior of  Wine dataset

 

Fig.7. Convergence behavior of  Glass dataset

 

Fig.8. Convergence behavior of WBC dataset

 

Fig.9. Convergence behavior of  CMC dataset

 

Fig.10. Convergence behavior of  Crude Oil dataset

 

Fig.11. Convergence behavior of Liver Disorder dataset

 

Fig.12. Convergence behavior of Vowel dataset

 

 

Table 6.  Comparison of objective function values for different datasets with other methods

Dataset	Criteria	K-means++	GA	SA	TS	ACO	HBMO	K-MCI	K-MKH
Iris	Best	97.3259	113.98650	97.4573	97.36597	97.10077	96.752047	96.6554	96.6555
	Worst	122.279	139.77827	102.0100	98.569485	97.808466	97.757625	96.6554	96.6555
	Mean	98.5817	125.1970	99.9570	97.868008	97.171546	96.95316	96.6554	96.6555
	Std	5.578	14.563	2.01	0.53	0.367	0.531	0	6.6E-06
	NFE	71	38128	5314	20201	10998	11214	3500	3145
	Rank	6	8	7	5	4	3	1	2
Wine	Best	16555.9	16,530.53	16473.4	16,666.227	16,530.53	16,357.284	16292.44	16292.19
	Worst	18294.9	16,530.533	18083.25	16,837.536	16,530.53	16,357.284	16292.88	16292.71
	Mean	16816.5	16,530.533	17521.09	16,785.459	16,530.533	16,357.28	16292.70	16292.57
	Std	637.140	0	753.084	52.073	0	0	0.130	0.216862
	NFE	261	33551	17264	22716	15473	7238	6250	6837
	Rank	7	5	8	6	4	3	2	1
Glass	Best	215.36	282.32	275.16	283.79	273.46	247.71	212.34	210.45
	Worst	223.71	286.77	287.18	286.47	280.08	249.54	212.80	215.197
	Mean	217.56	278.37	282.19	279.87	269.72	245.73	212.57	212.387
	Std	2.455	4.138712	4.238	4.192734	3.584829	2.438120	0.135	1.474966
	NFE	510	199892	199438	199574	196581	195439	25000	5511
	Rank	3	6	8	7	5	4	2	1
Cancer	Best	2986.96	3,249.46	2993.45	3,251.37	3,046.06	3,112.42	2964.38	2964.38
	Worst	2988.43	3,427.43	3421.95	3,434.16	3,242.01	3,210.78	2964.38	2964.39
	Mean	2987.99	2,999.32	3239.17	2,982.84	2,970.49	2,989.94	2964.38	2964.38
	Std	0.689	229.734	230.192	232.217	90.50028	103.471	0	0.0006
	NFE	112	20221	17387	18981	15983	19982	5000	3015
	Rank	5	7	8	4	3	6	1	1
CMC	Best	5703.20	5,756.5984	5849.03	5,993.5942	5,819.1347	5,713.9800	5693.73	5693.727
	Worst	5705.37	5,812.6480	5966.94	5,999.8053	5,912.4300	5,725.3500	5693.80	5693.731
	Mean	5704.19	5,705.6301	5893.48	5,885.0621	5,701.9230	5,699.2670	5693.75	5693.727
	Std	0.955	50.3694	50.867	40.84568	45.63470	12.690000	0.014	0.002328
	NFE	163	29483	26829	28945	20436	19496	15000	4783
	Rank	5	6	8	7	4	3	2	1
Vowel	Best	149394.6	159,153.49	149370.47	162,108.54	159,458.14	161,431.04	148967.2	148967.2
	Worst	161845.5	165,991.65	165986.42	165,996.43	165,939.83	165,804.67	149048.6	149083.9
	Mean	151445.29	149,513.73	161566.28	149,468.27	149,395.60	149,201.63	148987.6	149030.4
	Std	3119.751	3,105.5445	2847.085	2,846.2351	3,485.3816	2,746.0416	36.086	45.88285
	NFE	129	10548	9423	9528	8046	8436	7500	5667
	Rank	7	6	8	5	4	3	1	2
Mean Rank		5.6	6.3333333	7.8333333	5.6666667	4	3.6666667	1.5	1.333333
Final Rank		5	7	8	6	4	3	2	1

 

 

 

VII. Conclusion

Krill Herd (KH) is an optimization method for solving many complex global optimization problems. In this paper, a new hybrid data clustering based on modified krill algorithm and K-means is proposed. The original krill herd saturated quickly and hence trapped in local minimum. To alleviate the shortcomings of krill herd, modified krill herd was proposed by using levy walk random global search capability. Using these modifications, K-MKH algorithm converges to optimal solutions quickly. The simulation results show that the proposed method is fast and efficient for data clustering problem. In future, the proposed method can be applied to cluster text documents.

 

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