Identification Method of Optimal Copula Correlation Characteristic for Geological Parameters of Roof Structure
Abstract
:1. Introduction
2. A Multi-Dimensional Parameter Model Construction Method for Roofs Based on Copula Theory
2.1. Copula Theory
2.2. Coal Mine Roof Strength Parameters
2.3. Multidimensional Correlation Structure Characterization Methods
3. Identification of Multidimensional Roof Structures Based on Bayesian Approach
3.1. Bayesian Theory
3.2. Optimal Copula Function Identification Results
4. Comparison of Optimal Copula Recognition Methods
4.1. Determination of the Number of MCS
4.2. Analyses and Comparison of Identification Results
5. Factors Affecting the Recognition Accuracy of Bayesian
5.1. Changing the Number of Measured Samples
5.2. Changing the Correlation Coefficient
5.3. Changing the Prior Information
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Name of Coal Mine | Roof Geological Condition | Sample Data Size |
---|---|---|
Pingdingshan No. 11 coal mine [26] | Compound roof | 2 |
Shangwan mine [27] | Cracked roof under hallow mining conditions | 3 |
Jinchuan nickel mine [28] | Nickel mine roof | 5 |
Xinjulong coal mine [29] | Deep coal mine roadway roof located in fault zone | 5 |
Gaohe coal mine [30] | Paste false roof | 9 |
Great Wall #3 coal mine [31] | Thin roof | 2 |
He caogou coal mine [32] | Longwall coal mining | 5 |
Sanshandao gold mine [33] | Undersea roof | 3 |
No. 2130 coal mine [34] | Multi-layer gangue roof | 11 |
Boertai coal mine [35] | Thick and hard roof | 34 |
Sheilei coal mine [36] | Hard roof | 4 |
Yuwu coal mine [37] | Compound roof | 5 |
Gaokeng coal mine [38] | Compound roof | 4 |
Lishi coal mine [39] | Extra-thick compound mudstone roof | 8 |
Dongtan coal mine [40] | Thick and hard roof | 7 |
Sangshuping coal mine [41] | Hard roof | 7 |
Zhangshuanglou coal mine [42] | Hard roof | 5 |
Yungaishan coal mine [43] | Hard roof | 9 |
Zhuxianzhuang coal mine [44] | Hard thick roof | 7 |
a coal mine in Guizhou [45] | Compound roof | 12 |
Pingdingshan No. 2 coal mine [46] | Hard roof | 1 |
Linsheng coal mine [47] | Hard roof | 10 |
Sheilei coal mine [48] | Compound roof | 4 |
Dianping coal mine [49] | Hard and thick roof | 5 |
Distribution Type | F(x) | f(x) | Mean Value μ, Variance σ2 |
---|---|---|---|
Normal | µ = q σ2 = q2 | ||
Lognormal | µ = exp(p + 0.5q2) σ2 = [exp(q2) − 1] × exp(2p + q2) | ||
Extreme value type I | (1/q)exp[(x − p)/q] × exp[−exp(x − p)/q] | µ = p + 0.5772q σ2 = π2q2/6 | |
Gamma | µ = pq σ2 = pq2 |
Distribution Type | E | c | φ |
---|---|---|---|
Normal | 1681.4 | 1247.7 | 1142.4 |
Lognormal | −623.449 | −528.2 | 7068.8 |
Extreme value type I | 1878 | 1774.1 | 1178.8 |
Gamma | 16,160 | 3668 | 10,232 |
Copula | C(u1,u2; θ) | D(u1,u2; θ) | Range of θ |
---|---|---|---|
Gaussian | [−1, 1] | ||
Plackett | |||
Frank | |||
Clayton | |||
No.16 |
Parameter | Gaussian | Plackett | Frank | Clayton | No.16 |
---|---|---|---|---|---|
(E, c) | 90.21% | 0 | 9.79% | 0 | - |
(c, φ) | 46.61% | 22.72% | 30.66% | - | 0.01% |
Real Copula | n | τ = 0.2 | τ = 0.3 | τ = 0.4 | τ = 0.5 | τ = 0.6 | τ = 0.7 |
---|---|---|---|---|---|---|---|
Gaussian | 20 | 370 | 460 | 488 | 539 | 574 | 599 |
50 | 463 | 533 | 579 | 618 | 663 | 684 | |
100 | 561 | 658 | 664 | 694 | 730 | 765 | |
150 | 606 | 728 | 752 | 746 | 780 | 852 | |
200 | 664 | 761 | 766 | 764 | 826 | 894 | |
Plackett | 20 | 241 | 402 | 553 | 599 | 658 | 735 |
50 | 300 | 493 | 682 | 778 | 807 | 894 | |
100 | 346 | 583 | 835 | 891 | 914 | 967 | |
150 | 404 | 637 | 885 | 951 | 968 | 984 | |
200 | 442 | 682 | 925 | 974 | 979 | 996 | |
Frank | 20 | 326 | 342 | 378 | 402 | 419 | 455 |
50 | 397 | 408 | 436 | 544 | 628 | 736 | |
100 | 452 | 496 | 568 | 711 | 776 | 871 | |
150 | 489 | 563 | 614 | 765 | 807 | 916 | |
200 | 526 | 619 | 703 | 821 | 879 | 945 | |
Clayton | 20 | 578 | 593 | 649 | 658 | 672 | 679 |
50 | 669 | 756 | 798 | 805 | 828 | 831 | |
100 | 790 | 877 | 899 | 920 | 910 | 897 | |
150 | 833 | 925 | 946 | 949 | 955 | 947 | |
200 | 899 | 951 | 970 | 976 | 970 | 968 |
Real Copula | n | τ = 0.2 | τ = 0.3 | τ = 0.4 | τ = 0.5 | τ = 0.6 | τ = 0.7 |
---|---|---|---|---|---|---|---|
Gaussian | 20 | 342 | 451 | 486 | 512 | 556 | 572 |
50 | 434 | 550 | 559 | 628 | 635 | 736 | |
100 | 565 | 642 | 672 | 688 | 726 | 781 | |
150 | 597 | 700 | 744 | 754 | 802 | 797 | |
200 | 671 | 755 | 782 | 788 | 808 | 872 | |
Plackett | 20 | 239 | 362 | 548 | 638 | 632 | 705 |
50 | 294 | 491 | 674 | 756 | 782 | 884 | |
100 | 344 | 570 | 794 | 902 | 910 | 949 | |
150 | 390 | 619 | 857 | 947 | 953 | 982 | |
200 | 413 | 677 | 905 | 972 | 980 | 996 | |
Frank | 20 | 180 | 192 | 199 | 207 | 301 | 346 |
50 | 281 | 283 | 377 | 462 | 562 | 716 | |
100 | 402 | 394 | 559 | 697 | 611 | 836 | |
150 | 464 | 557 | 606 | 762 | 766 | 893 | |
200 | 505 | 605 | 698 | 808 | 856 | 915 | |
Clayton | 20 | 507 | 571 | 632 | 639 | 661 | 707 |
50 | 663 | 735 | 791 | 803 | 811 | 807 | |
100 | 773 | 876 | 892 | 910 | 904 | 909 | |
150 | 830 | 925 | 942 | 945 | 939 | 933 | |
200 | 891 | 955 | 965 | 968 | 969 | 959 |
Real Copula | n | τ = −0.2 | τ = −0.3 | τ = −0.4 | τ = −0.5 | τ = −0.6 | τ = −0.7 |
---|---|---|---|---|---|---|---|
Gaussian | 20 | 424 | 512 | 557 | 581 | 595 | 600 |
50 | 546 | 607 | 598 | 633 | 649 | 695 | |
100 | 659 | 659 | 666 | 681 | 733 | 786 | |
150 | 676 | 689 | 738 | 723 | 786 | 839 | |
200 | 731 | 758 | 778 | 799 | 829 | 879 | |
Plackett | 20 | 269 | 281 | 307 | 366 | 389 | 512 |
50 | 365 | 371 | 382 | 466 | 479 | 630 | |
100 | 388 | 436 | 439 | 554 | 561 | 757 | |
150 | 419 | 455 | 481 | 621 | 630 | 805 | |
200 | 445 | 471 | 509 | 659 | 687 | 859 | |
Frank | 20 | 312 | 335 | 361 | 392 | 412 | 445 |
50 | 385 | 392 | 415 | 535 | 623 | 721 | |
100 | 422 | 462 | 496 | 702 | 786 | 873 | |
150 | 475 | 536 | 587 | 762 | 819 | 902 | |
200 | 512 | 601 | 698 | 823 | 883 | 951 | |
No.16 | 20 | 805 | 759 | 689 | 608 | 558 | 550 |
50 | 927 | 913 | 819 | 691 | 630 | 581 | |
100 | 978 | 965 | 857 | 778 | 720 | 671 | |
150 | 993 | 989 | 905 | 823 | 765 | 712 | |
200 | 991 | 990 | 945 | 873 | 810 | 782 |
Real Copula | n | τ = −0.2 | τ = −0.3 | τ = −0.4 | τ = −0.5 | τ = −0.6 | τ = −0.7 |
---|---|---|---|---|---|---|---|
Gaussian | 20 | 502 | 529 | 545 | 564 | 573 | 593 |
50 | 567 | 587 | 630 | 644 | 658 | 684 | |
100 | 643 | 667 | 671 | 680 | 737 | 794 | |
150 | 683 | 714 | 720 | 733 | 768 | 838 | |
200 | 710 | 757 | 765 | 772 | 827 | 851 | |
Plackett | 20 | 265 | 271 | 275 | 342 | 379 | 491 |
50 | 336 | 349 | 360 | 461 | 479 | 606 | |
100 | 377 | 419 | 449 | 535 | 549 | 749 | |
150 | 414 | 453 | 480 | 610 | 602 | 803 | |
200 | 443 | 458 | 491 | 634 | 659 | 856 | |
Frank | 20 | 162 | 178 | 193 | 213 | 295 | 352 |
50 | 272 | 312 | 362 | 415 | 509 | 679 | |
100 | 362 | 452 | 476 | 612 | 698 | 827 | |
150 | 392 | 527 | 561 | 703 | 782 | 873 | |
200 | 424 | 576 | 669 | 812 | 832 | 928 | |
No.16 | 20 | 790 | 742 | 671 | 580 | 450 | 228 |
50 | 910 | 882 | 802 | 620 | 590 | 310 | |
100 | 958 | 929 | 831 | 750 | 686 | 660 | |
150 | 962 | 958 | 881 | 790 | 735 | 701 | |
200 | 981 | 979 | 940 | 809 | 792 | 762 |
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Cao, J.; Wang, T.; Zhu, C.; Yu, J.; Chen, X.; Zhang, X. Identification Method of Optimal Copula Correlation Characteristic for Geological Parameters of Roof Structure. Sustainability 2023, 15, 14932. https://doi.org/10.3390/su152014932
Cao J, Wang T, Zhu C, Yu J, Chen X, Zhang X. Identification Method of Optimal Copula Correlation Characteristic for Geological Parameters of Roof Structure. Sustainability. 2023; 15(20):14932. https://doi.org/10.3390/su152014932
Chicago/Turabian StyleCao, Jiazeng, Tao Wang, Chuanqi Zhu, Jianxin Yu, Xu Chen, and Xin Zhang. 2023. "Identification Method of Optimal Copula Correlation Characteristic for Geological Parameters of Roof Structure" Sustainability 15, no. 20: 14932. https://doi.org/10.3390/su152014932