Application and Comparison of Weighted Weights of Evidence Models
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摘要: 证据权方法是目前最常用的信息综合方法之一，广泛应用于矿产资源定量预测与评价.然而，它要求变量间相互独立，地质上很难满足这一条件.如何削弱条件不独立对证据权预测结果的影响，已成为当前数学地球科学研究的热点.解决该问题的途径之一是对传统证据权模型进行校正，比如采取加权的方法对原证据权模型计算的证据权重进行修正，以便消除非条件独立性的影响.对近期提出的多种加权证据权模型进行了系统的对比研究，基于同样的应用实例和实验方案，对不同方法的应用效果进行了比较，结果表明，各种加权证据权模型均可不同程度地削弱证据图层条件不独立性的影响，其中，基于逻辑回归的加权证据权模型优于其他加权方法.Abstract: Weights of evidence (WofE) is one of the most widely used methods in prediction and evaluation of mineral resources. It requires the independence of geological variables. However, it is almost impossible to meet the requirement among geological variables. The study of reducing the influence of the independence of variables has become a hot topic. One way to solve the problem is to modify the traditional WofE model such as modifying the weights of each evidential layer by calculating correction factor. In this study, some weighted WofE models proposed in recent years are systematically studied, i.e., the same data of case and the same test program are used to compare these models. The results illustrate that weighted WofE models can reduce the influence of dependence among variables to some extent, and the weighted WofE model based on logistic regression performs better than others.
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表 1 实验数据(据Agterberg et al., 1993)
Table 1. Test data
编号 年龄 高程 接触距离 岩石类型 裂隙距离 通道数 面积 1 0 0 0 0 1 0 10 052 2 0 0 0 0 0 1 3 363 3 0 0 0 1 1 0 3 268 4 0 0 0 1 0 0 1 074 5 0 1 0 0 1 0 5 455 6 0 1 0 0 0 0 25 7 0 0 1 0 1 0 3 482 8 0 1 0 1 1 0 2 518 9 0 0 1 0 0 0 1 474 10 0 1 0 1 0 0 1 371 11 1 0 0 0 1 0 5 12 1 0 0 0 0 0 705 13 0 0 1 1 1 0 5 14 0 0 1 1 0 0 744 15 1 0 0 1 1 0 422 16 1 0 0 1 0 0 58 17 0 1 1 0 1 0 12 18 0 1 1 0 0 0 179 19 1 1 0 0 1 2 1 766 20 1 1 0 0 0 0 119 21 0 1 1 1 1 1 1 055 22 0 1 1 1 0 0 33 23 1 0 1 0 1 0 10 24 1 1 0 1 1 0 146 25 1 0 1 0 0 1 623 26 1 1 0 1 0 0 145 27 1 0 1 1 1 2 504 28 1 0 1 1 0 0 1 29 1 1 1 0 1 2 317 30 1 1 1 0 0 1 277 31 1 1 1 1 1 3 348 32 1 1 1 1 0 0 295 
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表 2 各加权证据权模型加权系数
Table 2. Weighted coefficients of different kinds of weighted WofE model
证据图层 S-K加权 秩相关系数加权 逻辑回归加权 *Deng加权W+ *Deng加权W- 年龄 0.44 0.76 0.84 0.82 0.80 高程 0.14 0.82 0.53 0.67 0.76 接触距离 0.27 0.94 0.84 0.90 0.93 岩石类型 0.21 0.91 0.00 0.71 0.45 裂隙距离 0.05 1.11 5.07 6.67 2.61 注：*Deng的加权方案下，W+和W-加权系数不同.
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表 3 各种证据权模型的权重
Table 3. Weights of different kinds of WofE model
图层 原始权重 S-K加权 秩相关系数加权 逻辑回归加权 Deng加权 W+ W- W+ W- W+ W- W+ W- W+ W- 年龄 1.77 -1.72 0.78 -0.76 1.34 -1.30 1.49 -1.44 1.45 -1.37 高程 0.67 -0.74 0.09 -0.10 0.55 -0.61 0.36 -0.39 0.45 -0.56 接触距离 1.19 -1.20 0.33 -0.33 1.11 -1.12 1.00 -1.01 1.07 -1.12 岩石类型 0.43 -0.26 0.09 -0.05 0.39 -0.24 0.00 0.00 0.30 -0.12 裂隙距离 0.04 -0.13 0.00 -0.01 0.05 -0.15 0.22 -0.67 0.29 -0.34
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表 4 各证据权模型预测后验概率
Table 4. Posterior probabilities using different kinds of WofE model
编号 普通证据权 S-K加权 秩相关系数加权 Deng加权 逻辑回归加权 1 7.00E-06 9.40E-05 1.31E-05 1.83E-05 2.36E-05 2 1.30E-05 1.09E-04 2.45E-05 2.79E-05 2.36E-05 3 1.10E-05 1.08E-04 2.02E-05 1.49E-05 9.76E-06 4 2.80E-05 1.15E-04 4.17E-05 5.04E-05 5.01E-05 5 2.30E-05 1.14E-04 3.44E-05 2.69E-05 2.07E-05 6 7.30E-05 1.82E-04 1.22E-04 1.63E-04 1.77E-04 7 5.60E-05 1.33E-04 7.80E-05 7.67E-05 5.01E-05 8 6.20E-05 1.80E-04 1.00E-04 8.68E-05 7.30E-05 9 4.70E-05 1.32E-04 6.43E-05 4.09E-05 2.07E-05 10 2.21E-04 4.38E-04 1.82E-04 3.08E-04 4.40E-04 11 1.86E-04 4.35E-04 1.51E-04 1.64E-04 1.82E-04 12 1.46E-04 2.10E-04 2.28E-04 2.48E-04 1.77E-04 13 1.23E-04 2.08E-04 1.88E-04 1.32E-04 7.29E-05 14 4.41E-04 5.06E-04 3.42E-04 4.68E-04 4.39E-04 15 3.70E-04 5.02E-04 2.82E-04 2.49E-04 1.82E-04 16 3.03E-04 2.21E-04 3.89E-04 4.48E-04 3.74E-04 17 2.55E-04 2.19E-04 3.20E-04 2.39E-04 1.55E-04 18 7.67E-04 5.29E-04 4.80E-04 4.51E-04 3.85E-04 19 5.07E-04 2.53E-04 6.00E-04 3.63E-04 1.55E-04 20 2.40E-03 8.44E-04 1.70E-03 2.73E-03 3.28E-03 21 1.82E-03 6.15E-04 1.09E-03 1.29E-03 9.31E-04 22 1.53E-03 6.10E-04 8.99E-04 6.86E-04 3.85E-04 23 4.01E-03 9.66E-04 2.62E-03 2.21E-03 1.36E-03 24 1.64E-02 1.17E-03 8.32E-03 6.07E-03 2.87E-03 25 6.00E-06 9.40E-05 1.08E-05 9.76E-06 9.77E-06 26 6.04E-04 2.55E-04 7.27E-04 6.82E-04 3.74E-04 27 2.02E-03 8.37E-04 1.40E-03 1.46E-03 1.36E-03 28 8.27E-03 1.02E-03 4.46E-03 3.99E-03 2.87E-03 29 9.12E-04 5.33E-04 5.82E-04 8.46E-04 9.31E-04 30 4.77E-03 9.73E-04 3.18E-03 4.15E-03 3.28E-03 31 9.83E-03 1.03E-03 5.40E-03 7.47E-03 6.92E-03 32 1.94E-02 1.18E-03 1.01E-02 1.13E-02 6.92E-03
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表 5 各证据权模型预测后验概率排序
Table 5. The rank of posterior probabilities using different kinds of WofE model
编号 普通证据权 S-K加权 秩相关系数加权 Deng加权 逻辑回归加权 1 31 31 31 30 27 2 29 29 29 28 28 3 30 30 30 31 32 4 27 27 27 26 25 5 28 28 28 29 29 6 23 23 23 22 19 7 25 25 25 25 26 8 24 24 24 24 23 9 26 26 26 27 30 10 19 15 21 17 11 11 20 16 22 21 17 12 21 21 19 19 20 13 22 22 20 23 24 14 15 13 16 13 12 15 16 14 18 18 18 16 17 19 15 15 15 17 18 20 17 20 21 18 12 12 14 14 13 19 14 18 12 16 22 20 7 7 7 6 3 21 9 9 9 9 10 22 10 10 10 11 14 23 6 6 6 7 8 24 2 2 2 3 6 25 32 31 32 32 31 26 13 17 11 12 16 27 8 8 8 8 7 28 4 4 4 5 5 29 11 11 13 10 9 30 5 5 5 4 4 31 3 3 3 2 1 32 1 1 1 1 2
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表 6 各证据权模型之间秩相关系数
Table 6. Rank correlation coefficients between posterior probabilities obtained from different WofE models
普通证据权 S-K加权 秩相关系数加权 Deng加权 逻辑回归加权 不加权 1.00 S-K加权 0.99 1.00 秩相关系数加权 0.99 0.96 1.00 Deng加权 0.99 0.98 0.98 1.00 逻辑回归加权 0.94 0.96 0.92 0.97 1.00
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