Advertisement

General correlation coefficient based agglomerative clustering

  • 146 Accesses

Abstract

Correlation analysis is an important concept for studying patterns in data and making predictions. There have been many interesting revelations by applying this concept as patterns emerg out of seemingly unrelated data. In this paper, the focus is on exploring the role of correlation analysis in data clustering. We propose an algorithm, that defines an intuitive and accurate correlation coefficient metric known as the general correlation coefficient (G). We then define a framework for agglomerative clustering, based on this metric, called G based agglomerative clustering (GBAC). This framework is validated by performing experiments using synthetic as well as real datasets. The real world dataset is taken from http://databank.worldbank.org, a high dimensional dataset on human development indicators. The objective of these evaluations is to compare the performance of the proposed framework on different types of datasets. Comparative studies are performed in order to validate the proposed metric and the clustering framework. Our approach is found to be better than the existing agglomerative clustering techniques and correlation coefficient based clusterings. It is found to be effective for small, large, as well as high dimensional data. Finally, the clusters generated using this framework are validated against the existing validation measures. It is found that GBAC generates clean, more cohesive clusters. This framework combines the predictive power of correlation coefficients with the ability of finding patterns in data obtained from agglomerative hierarchical clustering. GBAC can be applied on a wide range of clustering based applications such as social network analysis, customer segmentation, collaborative filtering, construction of biological models, etc.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

References

  1. 1.

    Abdi, H.: The Kendall rank correlation coefficient. In: Encyclopedia of Measurement and Statistics. Sage, Thousand Oaks, pp. 508–510 (2007)

  2. 2.

    Atkins, J., Sharma, D.P.: Visualization of babble-speech interactions using Andrews curves. Circuits Syst. Signal Process. 35(4), 1313–1331 (2016)

  3. 3.

    Campbell, J., Chang, V., Hosseinian-Far, A.: Philosophising data: a critical reflection on the hidden issues. Int. J. Organ. Collect. Intell. 5(1), 1–12 (2015)

  4. 4.

    Chaudhuri, B., Bhattacharya, A.: On correlation between two fuzzy sets. Fuzzy Sets Syst. 118(3), 447–456 (2001)

  5. 5.

    Chen, H., Chiang, R.H., Storey, V.C.: Business intelligence and analytics: from big data to big impact. MIS Q. 36(4), 1165–1188 (2012)

  6. 6.

    Chen, N., Xu, Z., Xia, M.: Correlation coefficients of hesitant fuzzy sets and their applications to clustering analysis. Appl. Math. Model. 37(4), 2197–2211 (2013)

  7. 7.

    Collobert, R., Bengio, S.: SVMTorch: support vector machines for large-scale regression problems. J. Mach. Learn. Res. 1, 143–160 (2001)

  8. 8.

    Deufemia, V., Giordano, M., Polese, G., Tortora, G.: A visual language-based system for extraction-transformation-loading development. Softw. Pract. Exp. 44(12), 1417–1440 (2014)

  9. 9.

    Didelez, V., Pigeot, I.: Judea Pearl: causality: models, reasoning, and inference. Polit. Vierteljahresschr. 42(2), 313–315 (2001)

  10. 10.

    Dwork, C., Hardt, M., Pitassi, T., Reingold, O., Zemel, R.: Fairness through awareness. In: Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 214–226. ACM (2012)

  11. 11.

    Galton, F.: Co-relations and their measurement, chiefly from anthropometric data. Proc. R. Soc. Lond. 45(273–279), 135–145 (1888)

  12. 12.

    Ginsberg, J., Mohebbi, M.H., Patel, R.S., Brammer, L., Smolinski, M.S., Brilliant, L.: Detecting influenza epidemics using search engine query data. Nature 45(7232), 1012–1014 (2009)

  13. 13.

    Granville, V.: Developing Analytic Talent: Becoming a Data Scientist. Wiley, New York (2014)

  14. 14.

    Gratton, G., Kolotilin, A.: Euclidean fairness and efficiency. Econ. Inq. 53(3), 1689–1690 (2015)

  15. 15.

    Halkidi, M., Batistakis, Y., Vazirgiannis, M.: On clustering validation techniques. J. Intell. Inf. Syst. 17(2–3), 107–145 (2001)

  16. 16.

    Härdle, W., Simar, L.: Applied Multivariate Statistical Analysis, vol. 22007. Springer, Berlin (2007)

  17. 17.

    Hong, D.H.: Fuzzy measures for a correlation coefficient of fuzzy numbers under T W (the weakest t-norm)-based fuzzy arithmetic operations. Inf. Sci. 176(2), 150–160 (2006)

  18. 18.

    Hong, Y.: On computing the distribution function for the Poisson binomial distribution. Comput. Stat. Data Anal. 59, 41–51 (2013)

  19. 19.

    Hung, W.L.: Using statistical viewpoint in developing correlation of intuitionistic fuzzy sets. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 9(04), 509–516 (2001)

  20. 20.

    Huo, X., Székely, G.J.: Fast computing for distance covariance. Technometrics 58(4), 435–447 (2016)

  21. 21.

    Hurley, C.B., Oldford, R.: Pairwise display of high-dimensional information via Eulerian tours and Hamiltonian decompositions. J. Comput. Graph. Stat. 19, 861–886 (2012)

  22. 22.

    Inglehart, R., Klingemann, H.D.: Genes, culture, democracy, and happiness. In: Culture and Subjective Well-Being, pp. 165–183. MIT Press, Cambridge (2000)

  23. 23.

    Jain, A.K., Dubes, R.C.: Algorithms for Clustering Data. Prentice-Hall, Inc., Upper Saddle River (1988)

  24. 24.

    James, G., Witten, D., Hastie, T., Tibshirani, R.: Unsupervised learning. In: An Introduction to Statistical Learning, pp. 373–418. Springer, New York (2013)

  25. 25.

    Kauffman, L., Rousseeuw, P.: Finding Groups in Data. An Introduction to Cluster Analysis. Wiley, New York (1990)

  26. 26.

    Kitano, H.: Systems biology: a brief overview. Science 295(5560), 1662–1664 (2002)

  27. 27.

    Kong, J., Klein, B.E., Klein, R., Lee, K.E., Wahba, G.: Using distance correlation and SS-ANOVA to assess associations of familial relationships, lifestyle factors, diseases, and mortality. Proc. Natl Acad. Sci. USA 109(50), 20352–20357 (2012)

  28. 28.

    Lerman, I.C.: Foundations and Methods in Combinatorial and Statistical Data Analysis and Clustering. Springer, London (2016)

  29. 29.

    Li, R., Zhong, W., Zhu, L.: Feature screening via distance correlation learning. J. Am. Stat. Assoc. 107(499), 1129–1139 (2012)

  30. 30.

    Liao, H., Xu, Z., Zeng, X.J.: Novel correlation coefficients between hesitant fuzzy sets and their application in decision making. Knowl. Based Syst. 82, 115–127 (2015a)

  31. 31.

    Liao, H., Xu, Z., Zeng, X.J., Merigó, J.M.: Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets. Knowl. Based Syst. 76, 127–138 (2015b)

  32. 32.

    Linden, G., Smith, B., York, J.: Amazon.com recommendations: item-to-item collaborative filtering. IEEE Internet Comput. 7(1), 76–80 (2003)

  33. 33.

    Liu, S.T., Kao, C.: Fuzzy measures for correlation coefficient of fuzzy numbers. Fuzzy Sets Syst. 128(2), 267–275 (2002)

  34. 34.

    Lord, N.: Binomial averages when the mean is an integer. Math. Gaz. 94(530), 331–332 (2010)

  35. 35.

    Lyons, R.: Distance covariance in metric spaces. Ann. Probab. 41(5), 3284–3305 (2013)

  36. 36.

    Marsaglia, G.: Evaluating the normal distribution. J. Stat. Softw. 11(4), 1–7 (2004)

  37. 37.

    McGregor, C.: Big data in neonatal intensive care. Computer 46(6), 54–59 (2013)

  38. 38.

    Meilă, M.: Comparing clusterings? An information based distance. J. Multivar. Anal. 98(5), 873–895 (2007)

  39. 39.

    Mitchell, H.: A correlation coefficient for intuitionistic fuzzy sets. Int. J. Intell. Syst. 19(5), 483–490 (2004)

  40. 40.

    Mouret, G.: No-mean clustering algorithm. PhD Thesis, École Polytechnique de Montréal (2014)

  41. 41.

    Murphy, K.P.: Machine Learning: A Probabilistic Perspective. MIT Press, Cambridge (2012)

  42. 42.

    Murtagh, F., Contreras, P.: Algorithms for hierarchical clustering: an overview. Wiley Interdiscip. Rev. Data Min. Knowl. Discov. 2(1), 86–97 (2012)

  43. 43.

    Murthy, C., Pal, S., Majumder, D.D.: Correlation between two fuzzy membership functions. Fuzzy Sets Syst. 17(1), 23–38 (1985)

  44. 44.

    Park, S.Y., Bera, A.K.: Maximum entropy autoregressive conditional heteroskedasticity model. J. Econom. 150(2), 219–230 (2009)

  45. 45.

    Reshef, D.N., Reshef, Y.A., Finucane, H.K., Grossman, S.R., McVean, G., Turnbaugh, P.J., Lander, E.S., Mitzenmacher, M., Sabeti, P.C.: Detecting novel associations in large data sets. Science 334(6062), 1518–1524 (2011)

  46. 46.

    Ritala, P., Golnam, A., Wegmann, A.: Coopetition-based business models: the case of Amazon.com. Ind. Mark. Manag. 43(2), 236–249 (2014)

  47. 47.

    Sarkar, D.: Labels and legends. In: Lattice: Multivariate Data Visualization with R, pp. 151–163. Springer, New York (2008)

  48. 48.

    Scism, L., Maremont, M.: Insurers test data profiles to identify risky clients. Wall Str. J. 19, 1–6 (2010)

  49. 49.

    Sen, A., Dacin, P.A., Pattichis, C.: Current trends in web data analysis. Commun. ACM 49(11), 85–91 (2006)

  50. 50.

    Székely, G.J., Rizzo, M.L.: On the uniqueness of distance covariance. Stat. Probab. Lett. 82(12), 2278–2282 (2012)

  51. 51.

    Tan, P.N., Steinbach, M., Kumar, V.: Data mining cluster analysis: basic concepts and algorithms. In: Introduction to Data Mining (2013)

  52. 52.

    Volpone, S.D., Tonidandel, S., Avery, D.R., Castel, S.: Exploring the use of credit scores in selection processes: beware of adverse impact. J. Bus. Psychol. 30(2), 357–372 (2015)

  53. 53.

    Waller, M.A., Fawcett, S.E.: Data science, predictive analytics, and big data: a revolution that will transform supply chain design and management. J. Bus. Logist. 34(2), 77–84 (2013)

  54. 54.

    Weiss, N.A., Weiss, C.A.: Introductory Statistics. Pearson Education, London (2012)

  55. 55.

    Xiao, C., Ye, J., Esteves, R.M., Rong, C.: Using Spearman’s correlation coefficients for exploratory data analysis on big dataset. Concurr. Comput. Pract. Exp. (2015). https://doi.org/10.1002/cpe.3745

Download references

Author information

Correspondence to Divya Pandove.

Appendix 1: Value of D for a given value of n

Appendix 1: Value of D for a given value of n

n q r d n q r d n
1 0 1 − 1 126 31 2 6015 251
2 0 2 1 127 31 3 6140 252
3 0 3 2 128 32 0 6144 253
4 1 0 6 129 32 1 6271 254
5 1 1 9 130 32 2 6401 255
6 1 2 15 131 32 3 6530 256
7 1 3 20 132 33 0 6534 257
8 2 0 24 133 33 1 6665 258
9 2 1 31 134 33 2 6799 259
10 2 2 41 135 33 3 6932 260
11 2 3 50 136 34 0 6936 261
12 3 0 54 137 34 1 7071 262
13 3 1 65 138 34 2 7209 263
14 3 2 79 139 34 3 7346 264
15 3 3 92 140 35 0 7350 265
16 4 0 96 141 35 1 7489 266
17 4 1 111 142 35 2 7631 267
18 4 2 129 143 35 3 7772 268
19 4 3 146 144 36 0 7776 269
20 5 0 150 145 36 1 7919 270
21 5 1 169 146 36 2 8065 271
22 5 2 191 147 36 3 8210 272
23 5 3 212 148 37 0 8214 273
24 6 0 216 149 37 1 8361 274
25 6 1 239 150 37 2 8511 275
26 6 2 265 151 37 3 8660 276
27 6 3 290 152 38 0 8664 277
28 7 0 294 153 38 1 8815 278
29 7 1 321 154 38 2 8969 279
30 7 2 351 155 38 3 9122 280
31 7 3 380 156 39 0 9126 281
32 8 0 384 157 39 1 9281 282
33 8 1 415 158 39 2 9439 283
34 8 2 449 159 39 3 9596 284
35 8 3 482 160 40 0 9600 285
36 9 0 486 161 40 1 9759 286
37 9 1 521 162 40 2 9921 287
38 9 2 559 163 40 3 10,082 288
39 9 3 596 164 41 0 10,086 289
40 10 0 600 165 41 1 10,249 290
41 10 1 639 166 41 2 10,415 291
42 10 2 681 167 41 3 10,580 292
43 10 3 722 168 42 0 10,584 293
44 11 0 726 169 42 1 10,751 294
45 11 1 769 170 42 2 10,921 295
46 11 2 815 171 42 3 11,090 296
47 11 3 860 172 43 0 11,094 297
48 12 0 864 173 43 1 11,265 298
49 12 1 911 174 43 2 11,439 299
50 12 2 961 175 43 3 11,612 300
51 12 3 1010 176 44 0 11,616 301
52 13 0 1014 177 44 1 11,791 302
53 13 1 1065 178 44 2 11,969 303
54 13 2 1119 179 44 3 12,146 304
55 13 3 1172 180 45 0 12,150 305
56 14 0 1176 181 45 1 12,329 306
57 14 1 1231 182 45 2 12,511 307
58 14 2 1289 183 45 3 12,692 308
59 14 3 1346 184 46 0 12,696 309
60 15 0 1350 185 46 1 12,879 310
61 15 1 1409 186 46 2 13,065 311
62 15 2 1471 187 46 3 13,250 312
63 15 3 1532 188 47 0 13,254 313
64 16 0 1536 189 47 1 13,441 314
65 16 1 1599 190 47 2 13,631 315
66 16 2 1665 191 47 3 13,820 316
67 16 3 1730 192 48 0 13,824 317
68 17 0 1734 193 48 1 14,015 318
69 17 1 1801 194 48 2 14,209 319
70 17 2 1871 195 48 3 14,402 320
71 17 3 1940 196 49 0 14,406 321
72 18 0 1944 197 49 1 14,601 322
73 18 1 2015 198 49 2 14,799 323
74 18 2 2089 199 49 3 14,996 324
75 18 3 2162 200 50 0 15,000 325
76 19 0 2166 201 50 1 15,199 326
77 19 1 2241 202 50 2 15,401 327
78 19 2 2319 203 50 3 15,602 328
79 19 3 2396 204 51 0 15,606 329
80 20 0 2400 205 51 1 15,809 330
81 20 1 2479 206 51 2 16,015 331
82 20 2 2561 207 51 3 16,220 332
83 20 3 2642 208 52 0 16,224 333
84 21 0 2646 209 52 1 16,431 334
85 21 1 2729 210 52 2 16,641 335
86 21 2 2815 211 52 3 16,850 336
87 21 3 2900 212 53 0 16,854 337
88 22 0 2904 213 53 1 17,065 338
89 22 1 2991 214 53 2 17,279 339
90 22 2 3081 215 53 3 17,492 340
91 22 3 3170 216 54 0 17,496 341
92 23 0 3174 217 54 1 17,711 342
93 23 1 3265 218 54 2 17,929 343
94 23 2 3359 219 54 3 18,146 344
95 23 3 3452 220 55 0 18,150 345
96 24 0 3456 221 55 1 18,369 346
97 24 1 3551 222 55 2 18,591 347
98 24 2 3649 223 55 3 18,812 348
99 24 3 3746 224 56 0 18,816 349
100 25 0 3750 225 56 1 19,039 350
101 25 1 3849 226 56 2 19,265 351
102 25 2 3951 227 56 3 19,490 352
103 25 3 4052 228 57 0 19,494 353
104 26 0 4056 229 57 1 19,721 354
105 26 1 4159 230 57 2 19,951 355
106 26 2 4265 231 57 3 20,180 356
107 26 3 4370 232 58 0 20,184 357
108 27 0 4374 233 58 1 20,415 358
109 27 1 4481 234 58 2 20,649 359
110 27 2 4591 235 58 3 20,882 360
111 27 3 4700 236 59 0 20,886 361
112 28 0 4704 237 59 1 21,121 362
113 28 1 4815 238 59 2 21,359 363
114 28 2 4929 239 59 3 21,596 364
115 28 3 5042 240 60 0 21,600 365
116 29 0 5046 241 60 1 21,839 366
117 29 1 5161 242 60 2 22,081 367
118 29 2 5279 243 60 3 22,322 368
119 29 3 5396 244 61 0 22,326 369
120 30 0 5400 245 61 1 22,569 370
121 30 1 5519 246 61 2 22,815 371
122 30 2 5641 247 61 3 23,060 372
123 30 3 5762 248 62 0 23,064 373
124 31 0 5766 249 62 1 23,311 374
125 31 1 5889 250 62 2 23,561 375
501 125 1 94,249 626 156 2 147,265 751
502 125 2 94,751 627 156 3 147,890 752
503 125 3 95,252 628 157 0 147,894 753
504 126 0 95,256 629 157 1 148,521 754
505 126 1 95,759 630 157 2 149,151 755
506 126 2 96,265 631 157 3 149,780 756
507 126 3 96,770 632 158 0 149,784 757
508 127 0 96,774 633 158 1 150,415 758
509 127 1 97,281 634 158 2 151,049 759
510 127 2 97,791 635 158 3 151,682 760
511 127 3 98,300 636 159 0 151,686 761
512 128 0 98,304 637 159 1 152,321 762
513 128 1 98,815 638 159 2 152,959 763
514 128 2 99,329 639 159 3 153,596 764
515 128 3 99,842 640 160 0 153,600 765
516 129 0 99,846 641 160 1 154,239 766
517 129 1 100,361 642 160 2 154,881 767
518 129 2 100,879 643 160 3 155,522 768
519 129 3 101,396 644 161 0 155,526 769
520 130 0 101,400 645 161 1 156,169 770
521 130 1 101,919 646 161 2 156,815 771
522 130 2 102,441 647 161 3 157,460 772
523 130 3 102,962 648 162 0 157,464 773
524 131 0 102,966 649 162 1 158,111 774
525 131 1 103,489 650 162 2 158,761 775
526 131 2 104,015 651 162 3 159,410 776
527 131 3 104,540 652 163 0 159,414 777
528 132 0 104,544 653 163 1 160,065 778
529 132 1 105,071 654 163 2 160,719 779
530 132 2 105,601 655 163 3 161,372 780
531 132 3 106,130 656 164 0 161,376 781
532 133 0 106,134 657 164 1 162,031 782
533 133 1 106,665 658 164 2 162,689 783
534 133 2 107,199 659 164 3 163,346 784
535 133 3 107,732 660 165 0 163,350 785
536 134 0 107,736 661 165 1 164,009 786
537 134 1 108,271 662 165 2 164,671 787
538 134 2 108,809 663 165 3 165,332 788
539 134 3 109,346 664 166 0 165,336 789
540 135 0 109,350 665 166 1 165,999 790
541 135 1 109,889 666 166 2 166,665 791
542 135 2 110,431 667 166 3 167,330 792
543 135 3 110,972 668 167 0 167,334 793
544 136 0 110,976 669 167 1 168,001 794
545 136 1 111,519 670 167 2 168,671 795
546 136 2 112,065 671 167 3 169,340 796
547 136 3 112,610 672 168 0 169,344 797
548 137 0 112,614 673 168 1 170,015 798
549 137 1 113,161 674 168 2 170,689 799
550 137 2 113,711 675 168 3 171,362 800
551 137 3 114,260 676 169 0 171,366 801
552 138 0 114,264 677 169 1 172,041 802
553 138 1 114,815 678 169 2 172,719 803
554 138 2 115,369 679 169 3 173,396 804
555 138 3 115,922 680 170 0 173,400 805
556 139 0 115,926 681 170 1 174,079 806
557 139 1 116,481 682 170 2 174,761 807
558 139 2 117,039 683 170 3 175,442 808
559 139 3 117,596 684 171 0 175,446 809
560 140 0 117,600 685 171 1 176,129 810
561 140 1 118,159 686 171 2 176,815 811
562 140 2 118,721 687 171 3 177,500 812
563 140 3 119,282 688 172 0 177,504 813
564 141 0 119,286 689 172 1 178,191 814
565 141 1 119,849 690 172 2 178,881 815
566 141 2 120,415 691 172 3 179,570 816
567 141 3 120,980 692 173 0 179,574 817
568 142 0 120,984 693 173 1 180,265 818
569 142 1 121,551 694 173 2 180,959 819
570 142 2 122,121 695 173 3 181,652 820
571 142 3 122,690 696 174 0 181,656 821
572 143 0 122,694 697 174 1 182,351 822
573 143 1 123,265 698 174 2 183,049 823
574 143 2 123,839 699 174 3 183,746 824
575 143 3 124,412 700 175 0 183,750 825
576 144 0 124,416 701 175 1 184,449 826
577 144 1 124,991 702 175 2 185,151 827
578 144 2 125,569 703 175 3 185,852 828
579 144 3 126,146 704 176 0 185,856 829
580 145 0 126,150 705 176 1 186,559 830
581 145 1 126,729 706 176 2 187,265 831
582 145 2 127,311 707 176 3 187,970 832
583 145 3 127,892 708 177 0 187,974 833
584 146 0 127,896 709 177 1 188,681 834
585 146 1 128,479 710 177 2 189,391 835
586 146 2 129,065 711 177 3 190,100 836
587 146 3 129,650 712 178 0 190,104 837
588 147 0 129,654 713 178 1 190,815 838
589 147 1 130,241 714 178 2 191,529 839
590 147 2 130,831 715 178 3 192,242 840
591 147 3 131,420 716 179 0 192,246 841
592 148 0 131,424 717 179 1 192,961 842
593 148 1 132,015 718 179 2 193,679 843
594 148 2 132,609 719 179 3 194,396 844
595 148 3 133,202 720 180 0 194,400 845
596 149 0 133,206 721 180 1 195,119 846
597 149 1 133,801 722 180 2 195,841 847
598 149 2 134,399 723 180 3 196,562 848
599 149 3 134,996 724 181 0 196,566 849
600 150 0 135000 725 181 1 197289 850
601 150 1 135,599 726 181 2 198,015 851
602 150 2 136,201 727 181 3 198,740 852
603 150 3 136,802 728 182 0 198,744 853
604 151 0 136,806 729 182 1 199,471 854
605 151 1 137,409 730 182 2 200,201 855
606 151 2 138,015 731 182 3 200,930 856
607 151 3 138,620 732 183 0 200,934 857
608 152 0 138,624 733 183 1 201,665 858
609 152 1 139,231 734 183 2 202,399 859
610 152 2 139,841 735 183 3 203,132 860
611 152 3 140,450 736 184 0 203,136 861
612 153 0 140,454 737 184 1 203,871 862
613 153 1 141,065 738 184 2 204,609 863
614 153 2 141,679 739 184 3 205,346 864
615 153 3 142,292 740 185 0 205,350 865
616 154 0 142,296 741 185 1 206,089 866
617 154 1 142,911 742 185 2 206,831 867
618 154 2 143,529 743 185 3 207,572 868
619 154 3 144,146 744 186 0 207,576 869
620 155 0 144,150 745 186 1 208,319 870
621 155 1 144,769 746 186 2 209,065 871
622 155 2 145,391 747 186 3 209,810 872
623 155 3 146,012 748 187 0 209,814 873
624 156 0 146,016 749 187 1 210,561 874
625 156 1 146,639 750 187 2 211,311 875
n q r d n q r d
1 62 3 23,810 376 94 0 53,016
2 63 0 23,814 377 94 1 53,391
3 63 1 24,065 378 94 2 53,769
4 63 2 24,319 379 94 3 54,146
5 63 3 24,572 380 95 0 54,150
6 64 0 24,576 381 95 1 54,529
7 64 1 24,831 382 95 2 54,911
8 64 2 25,089 383 95 3 55,292
9 64 3 25,346 384 96 0 55,296
10 65 0 25,350 385 96 1 55,679
11 65 1 25,609 386 96 2 56,065
12 65 2 25,871 387 96 3 56,450
13 65 3 26,132 388 97 0 56,454
14 66 0 26,136 389 97 1 56,841
15 66 1 26,399 390 97 2 57,231
16 66 2 26,665 391 97 3 57,620
17 66 3 26,930 392 98 0 57,624
18 67 0 26,934 393 98 1 58,015
19 67 1 27,201 394 98 2 58,409
20 67 2 27,471 395 98 3 58,802
21 67 3 27,740 396 99 0 58,806
22 68 0 27,744 397 99 1 59,201
23 68 1 28,015 398 99 2 59,599
24 68 2 28,289 399 99 3 59,996
25 68 3 28,562 400 100 0 60,000
26 69 0 28,566 401 100 1 60,399
27 69 1 28,841 402 100 2 60,801
28 69 2 29,119 403 100 3 61,202
29 69 3 29,396 404 101 0 61,206
30 70 0 29,400 405 101 1 61,609
31 70 1 29,679 406 101 2 62,015
32 70 2 29,961 407 101 3 62,420
33 70 3 30,242 408 102 0 62,424
34 71 0 30,246 409 102 1 62,831
35 71 1 30,529 410 102 2 63,241
36 71 2 30,815 411 102 3 63,650
37 71 3 31,100 412 103 0 63,654
38 72 0 31,104 413 103 1 64,065
39 72 1 31,391 414 103 2 64,479
40 72 2 31,681 415 103 3 64,892
41 72 3 31,970 416 104 0 64,896
42 73 0 31,974 417 104 1 65,311
43 73 1 32,265 418 104 2 65,729
44 73 2 32,559 419 104 3 66,146
45 73 3 32,852 420 105 0 66,150
46 74 0 32,856 421 105 1 66,569
47 74 1 33,151 422 105 2 66,991
48 74 2 33,449 423 105 3 67,412
49 74 3 33,746 424 106 0 67,416
50 75 0 33,750 425 106 1 67,839
51 75 1 34,049 426 106 2 68,265
52 75 2 34,351 427 106 3 68,690
53 75 3 34,652 428 107 0 68,694
54 76 0 34,656 429 107 1 69,121
55 76 1 34,959 430 107 2 69,551
56 76 2 35,265 431 107 3 69,980
57 76 3 35,570 432 108 0 69,984
58 77 0 35,574 433 108 1 70,415
59 77 1 35,881 434 108 2 70,849
60 77 2 36,191 435 108 3 71,282
61 77 3 36,500 436 109 0 71,286
62 78 0 36,504 437 109 1 71,721
63 78 1 36,815 438 109 2 72,159
64 78 2 37,129 439 109 3 72,596
65 78 3 37,442 440 110 0 72,600
66 79 0 37,446 441 110 1 73,039
67 79 1 37,761 442 110 2 73,481
68 79 2 38,079 443 110 3 73,922
69 79 3 38,396 444 111 0 73,926
70 80 0 38,400 445 111 1 74,369
71 80 1 38,719 446 111 2 74,815
72 80 2 39,041 447 111 3 75,260
73 80 3 39,362 448 112 0 75,264
74 81 0 39,366 449 112 1 75,711
75 81 1 39,689 450 112 2 76,161
76 81 2 40,015 451 112 3 76,610
77 81 3 40,340 452 113 0 76,614
78 82 0 40,344 453 113 1 77,065
79 82 1 40,671 454 113 2 77,519
80 82 2 41,001 455 113 3 77,972
81 82 3 41,330 456 114 0 77,976
82 83 0 41,334 457 114 1 78,431
83 83 1 41,665 458 114 2 78,889
84 83 2 41,999 459 114 3 79,346
85 83 3 42,332 460 115 0 79,350
86 84 0 42,336 461 115 1 79,809
87 84 1 42,671 462 115 2 80,271
88 84 2 43,009 463 115 3 80,732
89 84 3 43,346 464 116 0 80,736
90 85 0 43,350 465 116 1 81,199
91 85 1 43,689 466 116 2 81,665
92 85 2 44,031 467 116 3 82,130
93 85 3 44,372 468 117 0 82,134
94 86 0 44,376 469 117 1 82,601
95 86 1 44,719 470 117 2 83,071
96 86 2 45,065 471 117 3 83,540
97 86 3 45,410 472 118 0 83,544
98 87 0 45414 473 118 1 84,015
99 87 1 45,761 474 118 2 84,489
100 87 2 46,111 475 118 3 84,962
101 87 3 46,460 476 119 0 84,966
102 88 0 46,464 477 119 1 85,441
103 88 1 46,815 478 119 2 85,919
104 88 2 47,169 479 119 3 86,396
105 88 3 47,522 480 120 0 86,400
106 89 0 47,526 481 120 1 86,879
107 89 1 47,881 482 120 2 87,361
108 89 2 48,239 483 120 3 87,842
109 89 3 48,596 484 121 0 87,846
110 90 0 48,600 485 121 1 88,329
111 90 1 48,959 486 121 2 88,815
112 90 2 49,321 487 121 3 89,300
113 90 3 49,682 488 122 0 89,304
114 91 0 49,686 489 122 1 89,791
115 91 1 50,049 490 122 2 90,281
116 91 2 50,415 491 122 3 90,770
117 91 3 50,780 492 123 0 90,774
118 92 0 50,784 493 123 1 91,265
119 92 1 51,151 494 123 2 91,759
120 92 2 51,521 495 123 3 92,252
121 92 3 51,890 496 124 0 92,256
122 93 0 51,894 497 124 1 92,751
123 93 1 52,265 498 124 2 93,249
124 93 2 52,639 499 124 3 93,746
125 93 3 53,012 500 125 0 93,750
501 187 3 212,060 876 219 0 287,766
502 188 0 212,064 877 219 1 288,641
503 188 1 212,815 878 219 2 289,519
504 188 2 213,569 879 219 3 290,396
505 188 3 214,322 880 220 0 290,400
506 189 0 214,326 881 220 1 291,279
507 189 1 215,081 882 220 2 292,161
508 189 2 215,839 883 220 3 293,042
509 189 3 216,596 884 221 0 293,046
510 190 0 216,600 885 221 1 293,929
511 190 1 217,359 886 221 2 294,815
512 190 2 218,121 887 221 3 295,700
513 190 3 218,882 888 222 0 295,704
514 191 0 218,886 889 222 1 296,591
515 191 1 219,649 890 222 2 297,481
516 191 2 220,415 891 222 3 298,370
517 191 3 221,180 892 223 0 298,374
518 192 0 221,184 893 223 1 299,265
519 192 1 221,951 894 223 2 300,159
520 192 2 222,721 895 223 3 301,052
521 192 3 223,490 896 224 0 301,056
522 193 0 223,494 897 224 1 301,951
523 193 1 224,265 898 224 2 302,849
524 193 2 225,039 899 224 3 303,746
525 193 3 225,812 900 225 0 303,750
526 194 0 225,816 901 225 1 304,649
527 194 1 226,591 902 225 2 305,551
528 194 2 227,369 903 225 3 306,452
529 194 3 228,146 904 226 0 306,456
530 195 0 228,150 905 226 1 307,359
531 195 1 228,929 906 226 2 308,265
532 195 2 229,711 907 226 3 309,170
533 195 3 230,492 908 227 0 309,174
534 196 0 230,496 909 227 1 310,081
535 196 1 231,279 910 227 2 310,991
536 196 2 232,065 911 227 3 311,900
537 196 3 232,850 912 228 0 311,904
538 197 0 232,854 913 228 1 312,815
539 197 1 233,641 914 228 2 313,729
540 197 2 234,431 915 228 3 314,642
541 197 3 235,220 916 229 0 314,646
542 198 0 235,224 917 229 1 315,561
543 198 1 236,015 918 229 2 316,479
544 198 2 236,809 919 229 3 317,396
545 198 3 237,602 920 230 0 317,400
546 199 0 237,606 921 230 1 318,319
547 199 1 238,401 922 230 2 319,241
548 199 2 239,199 923 230 3 320,162
549 199 3 239,996 924 231 0 320,166
550 200 0 240,000 925 231 1 321,089
551 200 1 240,799 926 231 2 322,015
552 200 2 241,601 927 231 3 322,940
553 200 3 242,402 928 232 0 322,944
554 201 0 242,406 929 232 1 323,871
555 201 1 243,209 930 232 2 324,801
556 201 2 244,015 931 232 3 325,730
557 201 3 244,820 932 233 0 325,734
558 202 0 244,824 933 233 1 326,665
559 202 1 245,631 934 233 2 327,599
560 202 2 246,441 935 233 3 328,532
561 202 3 247,250 936 234 0 328,536
562 203 0 247,254 937 234 1 329,471
563 203 1 248,065 938 234 2 330,409
564 203 2 248,879 939 234 3 331,346
565 203 3 249,692 940 235 0 331,350
566 204 0 249,696 941 235 1 332,289
567 204 1 250,511 942 235 2 333,231
568 204 2 251,329 943 235 3 334,172
569 204 3 252,146 944 236 0 334,176
570 205 0 252,150 945 236 1 335,119
571 205 1 252,969 946 236 2 336,065
572 205 2 253,791 947 236 3 337,010
573 205 3 254,612 948 237 0 337,014
574 206 0 254,616 949 237 1 337,961
575 206 1 255,439 950 237 2 338,911
576 206 2 256,265 951 237 3 339,860
577 206 3 257,090 952 238 0 339,864
578 207 0 257,094 953 238 1 340,815
579 207 1 257,921 954 238 2 341,769
580 207 2 258,751 955 238 3 342,722
581 207 3 259,580 956 239 0 342,726
582 208 0 259,584 957 239 1 343,681
583 208 1 260,415 958 239 2 344,639
584 208 2 261,249 959 239 3 345,596
585 208 3 262,082 960 240 0 345,600
586 209 0 262,086 961 240 1 346,559
587 209 1 262,921 962 240 2 347,521
588 209 2 263,759 963 240 3 348,482
589 209 3 264,596 964 241 0 348,486
590 210 0 264,600 965 241 1 349,449
591 210 1 265,439 966 241 2 350,415
592 210 2 266,281 967 241 3 351,380
593 210 3 267,122 968 242 0 351,384
594 211 0 267,126 969 242 1 352,351
595 211 1 267,969 970 242 2 353,321
596 211 2 268,815 971 242 3 354,290
597 211 3 269,660 972 243 0 354,294
598 212 0 269,664 973 243 1 355,265
599 212 1 270,511 974 243 2 356,239
600 212 2 271361 975 243 3 357212
601 212 3 272,210 976 244 0 357,216
602 213 0 272,214 977 244 1 358,191
603 213 1 273,065 978 244 2 359,169
604 213 2 273,919 979 244 3 360,146
605 213 3 274,772 980 245 0 360,150
606 214 0 274,776 981 245 1 361,129
607 214 1 275,631 982 245 2 362,111
608 214 2 276,489 983 245 3 363,092
609 214 3 277,346 984 246 0 363,096
610 215 0 277,350 985 246 1 364,079
611 215 1 278,209 986 246 2 365,065
612 215 2 279,071 987 246 3 366,050
613 215 3 279,932 988 247 0 366,054
614 216 0 279,936 989 247 1 367,041
615 216 1 280,799 990 247 2 368,031
616 216 2 281,665 991 247 3 369,020
617 216 3 282,530 992 248 0 369,024
618 217 0 282,534 993 248 1 370,015
619 217 1 283,401 994 248 2 371,009
620 217 2 284,271 995 248 3 372,002
621 217 3 285,140 996 249 0 372,006
622 218 0 285,144 997 249 1 373,001
623 218 1 286,015 998 249 2 373,999
624 218 2 286,889 999 249 3 374,996
625 218 3 287,762 1000 250 0 375,000

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pandove, D., Goel, S. & Rani, R. General correlation coefficient based agglomerative clustering. Cluster Comput 22, 553–583 (2019). https://doi.org/10.1007/s10586-018-2863-y

Download citation

Keywords

  • General correlation coefficient
  • Cluster validation
  • Correlation analysis
  • Agglomerative clustering
  • Cluster visualization