A Modified Grey Wolf Optimizer Based Data Clustering Algorithm

Figures & data

Figure 1. Possible next positions of a gray wolf

Figure 2. Flowchart of gray wolf optimizer

Figure 3. Example of solution encoding

Table 1. Definition of standard test functions

Function	D	R	${\mathrm{f}}_{\mathrm{m}\mathrm{i}\mathrm{n}}$
$f_1(x)=\sum _{i=1}^n{x}_i^2$	30	[−100,100]	0
$f_2(x)=\sum _{i=1}^n|x_i|+\prod _{i=1}^n|x_i|$	30	[−10,10]	0
$f_3(x)=\sum _{i=1}^n{(\sum _{j-1}^i{x}_j)}^2$	30	[−100,100]	0
$f_4(x)=ma{x}_i\{|x_i|,1\le i\le n\}$	30	[−100,100]	0
$f_5(x)=\sum _{i=n}^{n-1}\left[100{\left(x_{i+1}-x_i^2\right)}^2+{\left(x_i-1\right)}^2\right]$	30	[−30,30]	0
$f_6(x)=\sum _{i=1}^n{([x_i+0.5])}^2$	30	[−100,100]	0
$f_7(x)=\sum _{i=1}^{n}i{x}_i^4+random[0,1)$	30	[−1.28,1.28]	0
$f_8(x)=\sum _{i=1}^n-x_{i}sin(\sqrt{|x_i|})$	30	[−500,500]	−418.9829 × 5
$f_9(x)=\sum _{i=1}^n\left[x_i^2-10cos\left(2\pi x_i\right)+10\right]$	30	[−5.12,5.12]	0
$f_{10}(x)=-20exp(-0.2\sqrt{\frac{1}{n}\sum _{i=1}^n{x}_i^2})-exp(\frac{1}{n}\sum _{i=1}^{n}cos(2\pi x_i))+20+e$	30	[−32,32]	0
$f_{11}(x)=\frac{1}{4000}\sum _{i=1}^n{x}_i^2-\prod _{i=1}^{n}cos(\frac{x_i}{\sqrt{i}})+1$	30	[−600,600]	0
$\begin{array}{rl}& f_{12}(x)=\frac{\pi }{n}\{10sin(\pi y_i)+\sum _{i=1}^{n-1}{(y_i-1)}^2[1+10{sin}^2(\pi y_{i+1})]+(y_n-1{)}^2\}\\ & +\sum _{i=1}^{n}u(x_i,10,100,4)\\ & y_i=1+\frac{x_i+1}{4}\\ & u(x_i,a,k,m)=\left\{\begin{array}{cc}k{(x_i-a)}^m& x_i>a\\ 0& -a<x_i<a\\ k{(-x_i-a)}^m& x_i<-a\end{array}\right.\end{array}$	30	[−50,50]	0
$\begin{array}{rl}& f_{13}(x)=0.1\{si{n}^2(3\pi x_1)+\sum _{i=1}^n{(x_i-1)}^2[1+si{n}^2(3\pi x_i+1)]\\ & +(x_n-1{)}^2[1+si{n}^2(2\pi x_n)]\}+\sum _{i=1}^{n}u(x_i,5,100,4)\end{array}$	30	[−50,50]	0
$f_{14}(x)=(\frac{1}{500}+\sum _{j=1}^{25}\frac{1}{j+\sum _{i=1}^2{(x_i-a_{ij})}^6}{)}^{-1}$	2	[−65,65]	1
$f_{15}(x)=\sum _{i=1}^{11}{[a_i-\frac{x_i(b_i^2+b_i{x}_2)}{b_i^2+b_i{x}_3+x_4}]}^2$	4	[−5,5]	0.00030
$f_{16}(x)=4x_1^2-2.1x_1^4+\frac{1}{3}{x}_1^6+x_1{x}_2-4x_2^2+4x_2^4$	2	[−5,5]	−1.0316
$f_{17}(x)=(x_2-\frac{5.1}{4{\pi }^2}{x}_1^2+\frac{5}{\pi }{x}_1-6{)}^2+10(1-\frac{1}{8\pi })cos{x}_i+10$	2	[−5,5]	0.398
$\begin{array}{rl}& f_{18}(x)=[1+(x_1+x_2+1{)}^2(19-14x_1+3x_1^2-14x_2+6x_1{x}_2+3x_2^2)]\\ & \times [30+(2x_1-3x_2{)}^2\times (18-32x_1+12x_1^2+48x_2-36x_1{x}_2+27x_2^2)]\end{array}$	2	[−2,2]	3
$f_{19}(x)=-\sum _{i=1}^4{c}_{i}exp(-\sum _{j=1}^3{a}_{ij}(x_j-{p_{ij}}^2)$	3	[1,3]	−3.86
$f_{20}(x)=-\sum _{i=1}^4{c}_{i}exp(-\sum _{j=1}^6{a}_{ij}(x_j-{p_{ij}}^2)$	6	[0,1]	−3.32
$f_{21}(x)=-\sum _{i=1}^5{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	4	[0,10]	−10.1532
$f_{22}(x)=-\sum _{i=1}^7{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	4	[0,10]	−10.4028
$f_{23}(x)=-\sum _{i=1}^{10}{[(X-a_i){(X-a_i)}^T+c_i]}^{-1}$	4	[0,10]	−10.5363

Table 2. Results of test problems $F_1$ to $F_{23}$

	MGWO		GWO		PSO		GSA		WOA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	5.7849E-36	1.3675E-35	1.7929E-27	3.1886E-27	0.000136	0000202	2.53E-16	9.67E-17	1.41E-30	4.91E-30
F2	9.4232E-22	7.4645E-22	1.2656E-16	1.7063E-16	0.042144	0.045421	0.055655	0.194074	1.06E-21	2.39E-21
F3	6.4878E-07	2.6693E-06	4.3995E-05	0.00014684	70.12562	22.11924	896.5347	318.9559	5.39E-07	2.93E-06
F4	1.2982E-09	2.0487E-09	6.1031E-07	5.5882E-07	1.086481	0.317039	7.35487	1.741452	0.072581	0.39747
F5	26.8472	0.55809	27.1034	0.77555	96.71832	60.11559	67.54309	62.22534	27.86558	0.763626
F6	0.66062	0.38643	0.6712	0.3577	0.000102	8.28E-05	2.5E-16	1.74E-16	3.116266	0.532429
F7	0.0016112	0.00091939	0.0020548	0.0012676	0.122854	0.044957	0.089441	0.04339	0.001425	0.001149
F8	−5516.8023	1129.774	−5838.6035	815.6955	−4841.29	1152.814	−2821.07	493.0375	−5080.76	695.7968
F9	2.4632E-14	7.4165E-14	3.439	6.9524	46.70423	11.62938	25.96841	7.470068	0	0
F10	2.2797E-14	4.5786E-15	1.0581E-13	1.9453E-14	0.276015	0.50901	0.062087	0.23628	7.4043	9.897572
F11	0.0029237	0.0077608	0.0049926	0.0090249	0.009215	0.007724	27.70154	5.040343	0.000289	0.001586
F12	0.03737	0.018899	0.062244	0.097997	0.006917	0.026301	1.799617	0.95114	0.339676	0.214864
F13	0.45877	0.197	0.66336	0.26329	0.006675	0.008907	8.899084	7.126241	1.889015	0.266088
F14	4.556	4.2063	3.2234	3.3473	3.627168	2.560828	5.859838	3.831299	2.111973	2.498594
F15	0.00043824	0.008082	0.0030304	0.0069151	0.000577	0.000222	0.003673	0.001647	0.000572	0.000324
F16	−1.0316	2.0284E-08	−1.0316	3.0386E-08	−1.03163	6.25E-16	−1.03163	4.88E-16	−1.03163	4.2E-07
F17	0.398	4.9031E-05	0.39789	2.8163E-06	0.397887	0	0.397887	0	0.397914	2.7E-05
F18	3	2.6543E-05	3	3.9183E-05	3	1.33E-15	3	4.17E-15	3	4.22E-15
F19	−3.8648	0.0025351	−3.8615	0.0021957	−3.86278	2.58E-15	−3.86278	2.29E-15	−3.85616	0.002706
F20	−3.2699	0.06636	−3.2693	0.066951	−3.26634	0.060516	−3.31778	0.023081	−2.98105	0.376653
F21	−9.8072	1.2858	−9.1408	2.0555	−6.8651	3.019644	−5.95512	3.737079	−7.04918	3.629551
F22	−10.3963	0.0039211	−10.1788	1.2191	−8.45653	3.087094	−9.68447	2.014088	−8.18178	3.829202
F23	−10.2731	1.3981	−9.9034	1.968	−9.95291	1.782786	−10.5364	2.6E-15	−9.34238	2.414737

Table 3. Characteristics of UCI data sets

Data set	Data objects	Classes	Attributes
Heart	303	2	35
E. Coli	327	5	7
Horse	364	3	58
Dermatology	366	6	34
Cancer	569	2	30
Balance	625	3	4
Credit	690	2	15
Cancer-Int	699	2	9
Diabetes	768	2	8

Table 4. Intra-cluster distances of each algorithm on data sets

Data set	MGWO	GWO	GSA	k-means	PSO	K-PSO	K-NM-PSO	FA
Heart	1683.51	2103.95	4350.15	10284.16	2298.83	2144.42	6693.52	9095.10
E.Coli	28.49	42.64	571.97	675.21	40.37	408.93	566.36	57.78
Horse	15.58	18.19	14.85	19.69	8.42	15.10	70.41	94.37
Dermatology	115.83	702.47	654.98	442.87	1862.78	742.20	555.49	1448.48
Cancer	19.53	118.96	34.89	90.25	169.88	29.62	195.20	126.67
Balance	12893.22	21870.25	131293.30	20137.76	61987.01	10966.71	17153.45	37640.44
Credit	1369.16	2403.58	1398.57	3349.21	5090.77	1973.18	4194.19	4036.46
Cancer-Int	96.72	137.45	260.61	148.82	230.30	272.94	150.33	117.52
Diabetes	134.60	3482.75	6747.03	156.52	3140.59	5148.62	9525.25	5488.42

Table 5. Mean error rate (%) of each algorithm on data sets

Data set	MGWO	GWO	GSA	k-means	PSO	K-PSO	K-NM-PSO	FA
Heart	9.18	17.98	25.01	37.81	16.82	16.26	38.20	19.46
E. Coli	3.92	12.52	7.15	15.51	38.82	21.71	26.31	8.71
Horse	14.22	15.69	7.22	12.31	24.56	12.91	4.06	16.90
Dermatology	11.39	26.73	34.29	40.47	14.39	27.51	5.13	25.37
Cancer	6.04	11.51	12.72	14.62	13.74	6.37	11.77	29.44
Balance	8.27	9.04	3.38	15.11	11.34	36.87	24.19	2.59
Credit	24.70	21.88	14.66	27.67	18.05	36.16	33.33	24.09
Cancer-Int	18.76	16.95	10.09	26.06	8.41	5.76	25.86	7.92
Diabetes	4.53	12.14	7.59	38.30	12.47	2.62	30.76	31.54
Average	11.22	16.04	13.56	25.31	17.62	18.46	22.17	18.44
Rank	1	3	2	8	4	6	7	5