Application of PCA and Bayesian MCMC to discriminate between water sources in seabed gold mines
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摘要: 海底金礦礦山水害對礦山生產、人員施工及礦山設備等產生較大威脅，是礦山開采中的自然災害之一，快速有效的判別出礦山水害水源對于事故的防治有重要意義。三山島金礦的巷道圍巖裂隙普遍并長期存在涌水現象，礦區開采中礦井水害的水源主要有海水、第四系水、基巖裂隙水、地下水等，為了準確快速的判別礦井水水源，有效預防礦井水突水及水害威脅，本研究結合監測點水樣的水文地質條件與不同監測點水樣的水化學成分分析，選取Mg2+、Na++K+、Ca2+、SO42?、Cl?和HCO3 ?共6項指標作為判別因子，通過主成分分析得出不同水樣的礦化程度。在貝葉斯算法分析原理的基礎上，將馬爾可夫鏈蒙特卡洛(Markov Chain Monte Carlo, MCMC)引入到貝葉斯方法中，運用統計軟件SPSS統計，構建貝葉斯判別分析模型，得出基于水樣樣本信息的算法估計的后驗分布，得出礦山水害水源的分析方法。運用三山島金礦水害取水點的水樣分析數據進行詳細的分析驗證，建立礦井突水水源模型，進行不同水樣的信息分析，得出貝葉斯統計函數并進行水源判別結果分析，驗證了貝葉斯礦山水害水源判別模型的準確性和實用性，對現場工作的開展和水害防治有一定的指導意義。Abstract: Water hazards in submarine gold mines pose a great threat to mine production, construction personnel, and mining equipment, and represent one of the natural disasters that occur in mining. To prevent and control accidents, it is critical to quickly and effectively identify water sources. Cracks in the rocks surrounding the roadway in the Sanshandao Gold Mine are a widespread and long-term water gushing phenomenon. The main sources of mine water hazards in mining areas are seawater, Quaternary water, bedrock fissure water, and groundwater. To accurately and quickly identify mine water sources and effectively prevent inrushes of mine water and water-hazard threats, the hydrogeological conditions and chemical composition of water samples from different monitoring points were analyzed and six indicators, i.e., Mg2+, Na++K+, Ca2+, SO4 2?, Cl?, and HCO3 ?, were selected as discriminant factors. Based on the analysis principle of the Bayesian algorithm, the Markov chain Monte Carlo (MCMC) approach was introduced into the Bayesian method. A Bayesian discriminant analysis model was then constructed using SPSS Statistics and the MCMC Bayesian method. The posterior distribution estimated by the algorithm is based on water-sample information, which enables the analysis of the mine water source. Based on the water-sample data from a water intake point at the Sanshandao Gold Mine, detailed analysis and verification were performed, and a water-source model for the inrush of mine water was established. An analysis of different water samples was then performed. Through the selection of variables, variables with a strong discriminant ability and high degree of correlation were introduced into the discriminant function to obtain the Bayesian statistical function, thus enabling a discriminatory analysis of the water sources. The accuracy and practicability of the proposed Bayesian mine-water-source identification model were verified. This model has certain significance for guiding future field work and water-hazard prevention and control efforts.
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圖 1 水樣主成分分析結果
Figure 1. Principal component analysis of water samples
PC1(Principal Component1) = −0.004[18O] -0.034[2H] + 0.208[TDS] + 0.209[Cl] + 0.141[SO4] + 0.047[HCO3] + 0.193[Na] + 0.140[Ca] + 0.205[Mg]PC2(Principal Component2) = 0.306[18O] + 0.034[2H] + 0.015[TDS] + 0.034[Cl] + 0.155[SO4] + 0.239[HCO3] + 0.035[Na] + 0.251[Ca] + 0.042[Mg]
圖 3 水樣中的離子比分布圖. （a）HCO3 ?+SO4 2?與Ca2++Mg2+的對比；（b）HCO3 ?與Ca2+的對比；（c）Cl?與Na+的對比；（d）SO4 2?與Ca2+的對比
Figure 3. Distribution of ion ratios in water samples: (a) comparison of HCO3 ?+SO4 2? and Ca2++Mg2+; (b) comparison of HCO3 ? and Ca2+; (c) comparison of Cl? and Na+; (d) comparison of SO4 2? and Ca2+
圖 4 先驗信息和后驗信息分布. （a）先驗信息的直方圖；（b）先驗信息的常規殘差；（c）后驗信息的直方圖； （d）后驗信息的常規殘差
Figure 4. Distribution of prior and posterior information: (a) histogram of prior information; (b) conventional residuals of prior information; (c) histogram of posterior information; (d) conventional residuals of posterior information
序號 ρ(HCO3 ?)/(mg·L?1) ρ(Cl?)/(mg·L?1) ρ(SO4 2?)/(mg·L?1) ρ(K++Na+)/(mg·L?1) ρ(Ca2+)/(mg·L?1) ρ(Mg2+)/(mg·L?1) 水樣水源結果 貝葉斯法判別結果 1 268.4 20585.5 2713.7 11707.5 716.4 1175.5 Ⅰ Ⅰ 2 295.3 23158.4 3097.9 13025.5 651.3 1427.6 Ⅰ Ⅰ 3 244.0 22644.0 3001.9 13025.5 676.4 1412.4 Ⅰ Ⅰ 4 237.9 24702.3 2977.9 13442.5 876.8 1576.5 Ⅰ Ⅰ 5 262.3 21408.6 2929.8 11957.5 651.3 1366.9 Ⅰ Ⅰ 6 274.5 23981.9 3122.0 13783 751.5 1549.1 Ⅱ Ⅱ 7 244.0 39112.3 4418.8 19692.5 2204.4 3049.7 Ⅲ Ⅲ 8 268.4 21099.8 2881.8 11957.5 576.2 1351.7 Ⅰ Ⅰ 9 274.5 21614.6 2833.8 12395.5 626.3 1369.9 Ⅰ Ⅰ 10 286.7 24393.5 3314.1 13783 776.6 1582.5 Ⅱ Ⅱ 11 268.4 27790.3 3362.1 16362.5 891.8 1767.8 Ⅱ Ⅱ 12 317.3 32936.6 4130.6 18505 1052.1 2126.3 Ⅲ Ⅲ 13 323.4 36024.3 4514.8 19448 1262.5 2423.9 Ⅲ Ⅲ 14 329.5 38082.9 4682.9 21295 1067.1 2530.2 Ⅲ Ⅲ 15 335.6 33965.7 4156.0 19365 926.9 2098.9 Ⅲ Ⅲ 16 268.4 46317.2 5283.3 25574 1703.4 3462.8 Ⅲ Ⅲ 17 280.6 29848.9 3001.9 15760 1392.8 1889.3 Ⅱ Ⅱ 18 207.4 39626.7 4755.0 21170 1853.7 3007.1 Ⅲ Ⅲ 19 280.6 31392.7 3962.5 17490 1112.2 2272.1 Ⅲ Ⅲ 20 219.6 33348.2 3266.0 16582 2148.3 2196.7 Ⅲ Ⅲ 21 286.7 33183.7 3746.3 17362 1242.5 2308.5 Ⅲ Ⅲ 22 250.1 28654.9 3688.7 15928 1042.1 1905.1 Ⅰ Ⅰ 23 195.2 26184.4 3073.9 13518 1162.3 1715.6 Ⅰ Ⅰ 24 250.1 23467.2 2881.8 12912 681.4 1458.0 Ⅰ Ⅰ 25 244.0 23467.2 2881.8 12692 825.6 1492.0 Ⅰ Ⅰ 26 250.1 23879.1 3035.5 12892 881.8 1569.8 Ⅰ Ⅰ 27 250.1 43641.1 4956.7 23226 1603.2 3134.7 Ⅲ Ⅲ 28 225.7 35406.8 4169.0 19250 1362.7 2478.6 Ⅲ Ⅲ 29 268.4 34171.7 4265.1 18910 1162.3 2269.6 Ⅲ Ⅲ 30 323.4 37053.8 4514.8 20530.8 1202.4 2527.2 Ⅲ Ⅲ 31 274.5 39112.3 4923.1 21270 1468.9 2694.3 Ⅲ Ⅲ 32 158.6 16468.3 2439.9 9412.5 380.8 1125.1 Ⅰ Ⅰ 33 152.5 16303.5 2267.1 9150 392.8 1122.7 Ⅰ Ⅰ 34 317.3 20585.5 2881.8 12847.5 551.1 1184.6 Ⅰ Ⅰ 35 225.7 19762.0 2473.5 11098.8 551.1 1233.2 Ⅰ Ⅰ 36 176.9 17456.3 2228.6 10112.5 320.6 1093.5 Ⅰ Ⅰ 37 140.3 89.0 76.8 61.2 100.2 14.6 Ⅳ Ⅳ 38 256.2 212.3 405.4 200 138.3 42.5 Ⅳ Ⅳ 39 209.9 21.3 55.7 32.2 81.0 4.4 Ⅳ Ⅳ 40 219.6 257.0 134.5 133.7 144.3 26.7 Ⅳ Ⅳ 
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表 2 主成分載荷解釋方差
Table 2. Interpretation variance of principal component load
主成分載荷 特征值 解釋方差/% 累計解釋方差/% PC1 4.952 72.694 75.627 PC2 1.338 17.354 80.735
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表 3 匯聚組內矩陣
Table 3. Convergence intra-group matrix
貝葉斯判別
主要離子協方差 相關性 HCO3? Cl? SO42? K++Na+ Ca2+ Mg2+ HCO3? Cl? SO42? K++Na+ Ca2+ Mg2+ HCO3? 1814.827 12616.899 5054.195 ?195.334 ?3807.587 ?1747.608 1.000 0.085 0.295 ?0.077 ?0.295 ?0.136 Cl? 12230598.974 1177325.895 ?77121.740 648781.708 985384.161 1.000 0.838 ?0.372 0.612 0.935 SO42? 161555.942 ?1547.831 28471.253 94643.546 1.000 ?0.065 0.234 0.781 K++Na+ 3520.493 ?11349.173 ?4738.708 1.000 ?0.631 ?0.265 Ca2+ 91887.254 60781.440 1.000 0.665 Mg2+ 90803.664 1.000 注：協方差矩陣的自由度為36.
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表 4 估算的分布參數(正態分布)
Table 4. Estimated distribution parameter
判別離子種類 特定位置質量濃度/(mg·L?1) 特定標度 HCO3? 254.0525 47.52158 Cl? 25785.2950 11531.13636 SO42? 3187.6700 1298.60022 K++Na+ 222.9925 70.41759 Ca2+ 930.3350 523.64955 Mg2+ 1725.9400 840.87415
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表 6 組統計
Table 6. Group statistics
貝葉斯判別指標 平均值 標準差 有效個案數（成列） 類別 因子 未加權 加權 Ⅰ HCO3? 239.4882 45.51605 17 17.000 Cl? 21849.5059 3281.77955 17 17.000 SO42? 2840.5529 349.32311 17 17.000 K++Na+ 232.9882 61.52192 17 17.000 Ca2+ 680.2529 225.75292 17 17.000 Mg2+ 1387.0647 223.60322 17 17.000 Ⅱ HCO3? 277.5500 7.87507 4 4.000 Cl? 26503.6500 2808.20950 4 4.000 SO42? 3200.0250 167.95121 4 4.000 K++Na+ 203.1250 46.15982 4 4.000 Ca2+ 953.1750 299.38190 4 4.000 Mg2+ 1697.1750 160.18371 4 4.000 Ⅲ HCO3? 276.9733 42.22783 15 15.000 Cl? 36891.7333 4177.20599 15 15.000 SO42? 4382.9933 514.12583 15 15.000 K++Na+ 247.9533 55.24753 15 15.000 Ca2+ 1424.8400 398.28648 15 15.000 Mg2+ 2572.0400 413.27733 15 15.000 Ⅳ HCO3? 206.5000 48.42761 4 4.000 Cl? 144.9000 108.79789 4 4.000 SO42? 168.1000 161.66808 4 4.000 K++Na+ 106.7750 75.39754 4 4.000 Ca2+ 115.9500 30.40181 4 4.000 Mg2+ 22.0500 16.39970 4 4.000 總計 HCO3? 254.0525 47.52158 40 40.000 Cl? 25785.2950 11531.13636 40 40.000 SO42? 3187.6700 1298.60022 40 40.000 K++Na+ 222.9925 70.41759 40 40.000 Ca2+ 930.3350 523.64955 40 40.000 Mg2+ 1725.9400 840.87415 40 40.000
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表 7 結構矩陣
Table 7. Structured matrix
變量 貝葉斯線性判別函數 1 2 3 Cl? 0.708 0.451 ?0.005 SO42? 0.698 0.241 ?0.215 Mg2+ 0.579 0.580 ?0.300 Ca2+ 0.304 0.502 ?0.002 HCO3? 0.116 0.162 0.607 K++Na+ 0.146 ?0.228 ?0.457
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表 8 特征值
Table 8. Eigenvalue
判別函數 特征值 方差百分數/% 累計方差百分數/% 典型相關性 1 21.018a 94.5 94.5 0.977 2 1.139a 5.1 99.6 0.730 3 0.091a 0.4 100.0 0.289 注：a表示在分析中使用了貝葉斯抽樣中涉及的前3個典則判別函數.
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表 9 Wilks’ lambda值統計
Table 9. Wilks’ lambda statistics
檢驗函數 Wilks’ Lambda值 卡方 自由度 顯著性 1, 2, 3 0.019 133.938 18 0 2, 3 0.428 28.815 10 0.001 3 0.917 2.961 4 0.564
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表 10 分類函數系數
Table 10. Coefficient of classification function
貝葉斯判別指標 系數 MCMC抽樣a 因子 類別 偏差 標準誤差 95%置信區間 下限 上限 HCO3? Ⅰ 0.141 0.115 0.235 ?0.108 0.819 Ⅱ 0.174 0.129b 0.255b ?0.083b 0.924b Ⅲ 0.167 0.140 0.303 ?0.186 1.022 Ⅳ 0.259 0.142c 0.202c 0.194c 0.957c Cl? Ⅰ 0.006 0.003 0.006 0.000 0.024 Ⅱ 0.007 0.004b 0.007b 0.000b 0.028b Ⅲ 0.008 0.004 0.008 0.000 0.032 Ⅳ ?0.002 0.000c 0.004c ?0.012c 0.006c SO42? Ⅰ 0.007 ?0.002 0.045 ?0.081 0.096 Ⅱ 0.003 ?0.004b 0.050b ?0.091b 0.101b Ⅲ 0.007 ?0.003 0.062 ?0.117 0.128 Ⅳ ?0.015 ?0.010c 0.022c ?0.083c 0.002c K++Na+ Ⅰ 0.274 0.171 0.243 0.145 1.105 Ⅱ 0.292 0.186b 0.266b 0.147b 1.200b Ⅲ 0.350 0.217 0.308 0.195 1.400 Ⅳ 0.114 0.088c 0.166c 0.035c 0.696c Ca2+ Ⅰ 0.052 0.030 0.061 ?0.007 0.239 Ⅱ 0.055 0.033b 0.068b ?0.007b 0.258b Ⅲ 0.065 0.035 0.081 ?0.027 0.314 Ⅳ 0.023 0.020c 0.041c 0.005c 0.151c Mg2+ Ⅰ ?0.073 ?0.034 0.085 ?0.321 0.012 Ⅱ ?0.075 ?0.036b 0.094b ?0.377b 0.025b Ⅲ ?0.088 ?0.038 0.119 ?0.422 0.048 Ⅳ 0.032 0.009c 0.049c ?0.040c 0.161c 總計 Ⅰ ?90.763 ?52.500 59.017 ?302.110 ?76.470 Ⅱ ?110.825 ?64.882b 71.692b ?376.573b ?92.475b Ⅲ ?164.246 ?94.746 102.085 ?534.464 ?141.437 Ⅳ ?34.523 ?21.562c 37.264c ?163.988c ?23.464c 注：除非另行說明，否則自助抽樣結果基于1000個自助抽樣樣本；b表示基于985個樣本；c表示基于993個樣本.
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