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Table 3 Validations of 6 internal and 2 external measures on the clustering results of 5 synthetic datasets with 2 sample sizes (A = 5000; B = 10,000) and 6 different subsets of random samples listed in column es

From: An ensemble method for estimating the number of clusters in a big data set using multiple random samples

Dataset

n

es

SSE

Silhouette

DB index

Homogeneity

Completeness

V-measure

AMI

ARI

DS1

A

5%

7.398 ± 3.013

0.849 ± 0.016

0.231 ± 0.037

0.971 ± 0.025

1.000 ± 0.000

0.985 ± 0.013

0.971 ± 0.025

0.967 ± 0.028

10%

5.716 ± 3.128

0.859 ± 0.016

0.209 ± 0.038

0.986 ± 0.025

1.000 ± 0.000

0.993 ± 0.013

0.986 ± 0.025

0.984 ± 0.028

20%

3.993 ± 0.116

0.867 ± 0.001

0.190 ± 0.003

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

30%

3.939 ± 0.053

0.868 ± 0.001

0.188 ± 0.002

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

40%

3.910 ± 0.024

0.868 ± 0.000

0.188 ± 0.001

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

50%

3.894 ± 0.033

0.868 ± 0.000

0.188 ± 0.001

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

B

5%

22.386 ± 15.800

0.830 ± 0.036

0.270 ± 0.080

0.954 ± 0.035

0.999 ± 0.002

0.976 ± 0.019

0.954 ± 0.035

0.948 ± 0.041

10%

14.165 ± 5.580

0.850 ± 0.015

0.227 ± 0.035

0.974 ± 0.024

1.000 ± 0.000

0.987 ± 0.012

0.974 ± 0.024

0.971 ± 0.026

20%

10.542 ± 5.089

0.860 ± 0.014

0.206 ± 0.031

0.989 ± 0.022

1.000 ± 0.000

0.995 ± 0.011

0.989 ± 0.022

0.988 ± 0.024

30%

7.900 ± 0.214

0.867 ± 0.001

0.189 ± 0.004

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

40%

7.635 ± 0.229

0.868 ± 0.002

0.188 ± 0.004

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

50%

7.514 ± 0.129

0.869 ± 0.001

0.186 ± 0.002

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

DS2

A

5%

10.928 ± 5.194

0.855 ± 0.031

0.309 ± 0.036

0.939 ± 0.029

1.000 ± 0.000

0.968 ± 0.016

0.938 ± 0.029

0.896 ± 0.052

10%

3.754 ± 3.128

0.908 ± 0.027

0.182 ± 0.093

0.981 ± 0.018

1.000 ± 0.000

0.991 ± 0.009

0.981 ± 0.019

0.971 ± 0.028

20%

1.667 ± 1.788

0.926 ± 0.015

0.120 ±  0.045

0.993 ± 0.012

1.000 ± 0.000

0.997 ± 0.006

0.993 ± 0.012

0.989 ± 0.019

30%

0.638 ± 0.010

0.935 ± 0.001

0.093 ± 0.001

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

40%

0.635 ± 0.011

0.935 ± 0.000

0.093 ± 0.001

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

50%

0.632 ± 0.006

0.935 ± 0.001

0.093 ± 0.001

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

B

5%

18.394 ± 10.015

0.865 ± 0.037

0.287 ± 0.051

0.948 ± 0.032

1.000 ± 0.000

0.974 ± 0.017

0.948 ± 0.032

0.911 ± 0.061

10%

11.054 ± 8.737

0.894 ± 0.036

0.226 ± 0.096

0.971 ± 0.028

1.000 ± 0.000

0.985 ± 0.015

0.971 ± 0.028

0.951 ± 0.051

20%

7.470 ± 4.989

0.908 ± 0.022

0.191 ± 0.078

0.982 ± 0.015

1.000 ± 0.000

0.991 ± 0.007

0.982 ± 0.015

0.972 ± 0.022

30%

7.518 ± 4.322

0.907 ± 0.019

0.197 ± 0.069

0.982 ± 0.013

1.000 ± 0.000

0.991 ± 0.006

0.982 ± 0.013

0.972 ± 0.019

40%

6.027 ± 3.173

0.914 ± 0.014

0.176 ± 0.060

0.987 ± 0.009

0.987 ± 0.009

0.993 ± 0.005

0.987 ± 0.009

0.979 ± 0.015

50%

1.264 ± 0.008

0.935 ± 0.000

0.093 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

1.000 ± 0.000

DS3

A

5%

14.204 ± 8.207

0.761 ± 0.075

0.411 ± 0.096

0.907 ± 0.050

1.000 ± 0.000

0.951 ± 0.027

0.905 ± 0.050

0.846 ± 0.083

10%

9.462 ± 3.442

0.803 ± 0.034

0.342 ± 0.028

0.933 ± 0.023

1.000 ± 0.000

0.965 ± 0.012

0.932 ± 0.023

0.885 ± 0.043

20%

4.252 ± 1.560

0.861 ± 0.021

0.263 ± 0.044

0.968 ± 0.012

1.000 ± 0.000

0.984 ± 0.007

0.968 ± 0.013

0.947 ± 0.022

30%

3.357 ± 2.198

0.873 ± 0.029

0.229 ± 0.077

0.976 ± 0.019

1.000 ± 0.000

0.988 ± 0.010

0.976 ± 0.019

0.961 ± 0.032

40%

2.531 ± 1.776

0.884 ± 0.025

0.201 ± 0.066

0.983 ± 0.017

1.000 ± 0.000

0.991 ± 0.009

0.982 ± 0.017

0.971 ± 0.030

50%

2.095 ± 1.392

0.889 ± 0.019

0.196 ± 0.047

0.986 ± 0.013

1.000 ± 0.000

0.993 ± 0.007

0.986 ± 0.013

0.979 ± 0.024

B

5%

39.314 ± 10.666

0.715 ± 0.038

0.478 ± 0.077

0.875 ± 0.026

1.000 ± 0.000

0.933 ± 0.015

0.874 ± 0.026

0.796 ± 0.040

10%

29.491 ± 10.190

0.752 ± 0.041

0.410 ± 0.053

0.900 ± 0.029

1.000 ± 0.000

0.947 ± 0.016

0.899 ± 0.029

0.832 ± 0.047

20%

17.259 ± 8.164

0.810 ± 0.045

0.327 ± 0.070

0.936 ± 0.027

1.000 ± 0.000

0.967 ± 0.014

0.936 ± 0.028

0.890 ± 0.046

30%

15.913 ± 7.190

0.818 ± 0.039

0.323 ± 0.058

0.941 ± 0.024

1.000 ± 0.000

0.970 ± 0.013

0.940 ± 0.025

0.898 ± 0.041

40%

11.339 ± 6.358

0.843 ± 0.039

0.285 ± 0.059

0.958 ± 0.023

1.000 ± 0.000

0.978 ±  0.012

0.957  ±  0.023

0.927 ± 0.042

50%

6.694 ± 3.438

0.873 ± 0.022

0.238 ± 0.046

0.975 ± 0.015

1.000 ± 0.000

0.988 ± 0.008

0.975 ± 0.015

0.960 ± 0.028

DS4

A

5%

17.860 ± 5.959

0.687 ± 0.051

0.543 ±  0.075

0.855 ± 0.040

0.999 ± 0.001

0.921 ± 0.023

0.852 ± 0.040

0.785 ± 0.070

10%

9.654 ± 2.088

0.767 ± 0.027

0.423 ± 0.034

0.913 ± 0.017

0.999 ± 0.001

0.954 ± 0.009

0.910 ± 0.017

0.877 ± 0.025

20%

8.015 ± 2.516

0.792 ± 0.037

0.384 ± 0.059

0.928 ± 0.022

0.999 ± 0.001

0.962 ± 0.012

0.925 ± 0.022

0.899 ± 0.033

30%

4.050 ± 2.283

0.847 ± 0.033

0.295 ± 0.064

0.963 ± 0.022

1.000 ± 0.001

0.981 ± 0.012

0.962 ± 0.023

0.955 ± 0.037

40%

3.149 ± 1.398

0.861 ± 0.017

0.267 ± 0.027

0.972 ± 0.013

1.000 ± 0.000

0.986 ± 0.006

0.971 ± 0.014

0.970 ± 0.022

50%

1.984 ± 0.834

0.875 ± 0.012

0.233 ± 0.036

0.984 ± 0.009

1.000 ± 0.000

0.992 ± 0.004

0.983 ± 0.009

0.985 ± 0.010

B

5%

45.941 ± 3.544

0.650 ± 0.016

0.557 ± 0.045

0.819 ± 0.009

0.996 ± 0.005

0.900 ±  0.005

0.817 ± 0.009

0.722 ±  0.006

10%

36.157 ± 9.622

0.688 ± 0.040

0.509 ± 0.050

0.852 ± 0.034

0.997 ± 0.004

0.919 ± 0.020

0.850 ± 0.035

0.776 ± 0.060

20%

22.751 ± 7.440

0.748 ± 0.039

0.452 ± 0.054

0.902 ± 0.028

1.000 ± 0.001

0.948 ± 0.016

0.900 ± 0.029

0.860 ± 0.043

30%

14.541 ± 7.407

0.798 ± 0.050

0.373 ± 0.077

0.934 ± 0.030

1.000 ± 0.000

0.966 ± 0.016

0.933 ±  0.030

0.909 ±  0.048

40%

8.740 ± 4.905

0.841 ± 0.035

0.306 ± 0.063

0.959 ± 0.022

1.000 ± 0.000

0.979 ± 0.011

0.958 ± 0.022

0.948 ± 0.034

50%

5.240 ± 1.944

0.866 ± 0.016

0.257 ± 0.036

0.976 ± 0.011

1.000 ± 0.000

0.988 ± 0.006

0.975 ± 0.012

0.974 ± 0.017

DS5

A

5%

10.964 ± 1.646

0.694 ± 0.026

0.499 ± 0.046

0.879 ± 0.015

1.000 ± 0.000

0.936 ± 0.008

0.873 ± 0.015

0.748 ± 0.022

10%

8.272 ± 2.108

0.738 ± 0.035

0.444 ± 0.037

0.906 ± 0.023

1.000 ± 0.000

0.951 ± 0.013

0.902 ± 0.024

0.800 ± 0.048

20%

6.011 ± 0.872

0.778 ± 0.018

0.401 ± 0.024

0.929 ± 0.008

1.000 ± 0.000

0.963 ± 0.004

0.925 ±  0.008

0.847 ±  0.016

30%

5.904 ± 0.743

0.779 ± 0.015

0.398 ± 0.020

0.930 ± 0.006

1.000 ± 0.000

0.964 ± 0.003

0.926 ± 0.006

0.848 ± 0.013

40%

5.086 ± 0.483

0.796 ± 0.012

0.373 ± 0.017

0.936 ± 0.006

1.000 ± 0.000

0.967 ± 0.003

0.932 ± 0.006

0.860 ± 0.011

50%

4.118 ± 0.516

0.819 ± 0.011

0.342 ± 0.014

0.949 ± 0.006

1.000 ± 0.000

0.974 ± 0.004

0.946 ± 0.007

0.883 ± 0.012

B

5%

33.551 ± 4.208

0.623 ± 0.019

0.583 ± 0.058

0.828 ± 0.012

1.000 ± 0.000

0.906 ± 0.007

0.825 ± 0.012

0.654 ± 0.020

10%

25.541 ± 3.411

0.665 ± 0.021

0.505 ± 0.011

0.857 ± 0.019

0.999 ± 0.002

0.922 ± 0.011

0.854 ± 0.020

0.699 ± 0.038

20%

18.505 ± 2.550

0.720 ± 0.014

0.469 ± 0.023

0.897 ± 0.009

1.000 ± 0.001

0.946 ± 0.005

0.895 ± 0.009

0.786 ± 0.017

30%

17.424 ± 1.054

0.725 ± 0.009

0.458 ± 0.019

0.900 ± 0.005

1.000 ± 0.000

0.948 ± 0.003

0.898 ± 0.005

0.789 ± 0.013

40%

15.016 ± 1.346

0.751 ± 0.016

0.425 ± 0.010

0.914 ± 0.009

1.000 ± 0.000

0.955 ± 0.005

0.912 ± 0.009

0.814 ± 0.022

50%

12.226 ± 1.438

0.778 ± 0.014

0.400 ± 0.024

0.929 ± 0.007

1.000 ± 0.000

0.963 ± 0.004

0.927 ± 0.007

0.850 ± 0.019