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.
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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 |
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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
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DOI: https://doi.org/10.1007/s10586-018-2863-y