Unified Framework for Probabilistic Prediction of Critical Mechanical Parameters of Loess by Machine Learning Methods
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摘要:
为实现黄土关键力学参数的准确预测，并合理刻画预测结果的不确定性，提出了基于机器学习方法的黄土关键力学参数概率预测统一框架体系，通过对训练集的预测偏差进行概率分布拟合，进而构建预测结果的95%置信区间，置信区间的大小反映了预测结果的合理与否.基于随机森林、决策树、极限梯度提升和自适应提升4种方法预测黄土黏聚力，对应的决定系数R2分别达到了0.84、0.75、0.81和0.79，4种方法所构建的95%置信区间包含真正的试验结果的比例均在95%左右.表明通过训练集的预测偏差得到的95%置信区间是相对可靠的，可对预测结果的不确定性进行合理量化.此外，基于上述4种方法可实现黄土黏聚力的相对准确的预测.
Abstract:In order to predict the criticalmechanical parameters of loess accurately and quantify the uncertainty corresponding to the prediction results reasonably, anunified framework for probabilistic prediction of critical mechanical parameters of loess by machine learning methods is proposed. By fitting probability density function to the bias of the training dataset, a 95% confidence interval for the prediction results is constructed, and the size of the confidence interval reflects the rationality of the prediction results.(Result) Predicting cohesion of loess based on four machine learning methods, namely, random forest, decision tree, extreme gradient boosting and adaptive boosting, the corresponding coefficients of determination R2 reached 0.84, 0.75, 0.81 and 0.79, respectively. The proportion of measurement data included in the 95% confidence interval constructed by the four methods is around 95%. It is shown that the 95% confidence interval obtained from the bias based on the training dataset is relatively reliable and can quantify the uncertainty of the prediction results reasonably. In addition, the cohesion of loess can be predicted accurately using the four machine learning methods.
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Key words:
- loess /
- cohesion /
- data-driven /
- probabilistic prediction /
- bias /
- uncertainty
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图 1 黄土物理力学参数分布直方图及相关关系图
Fig. 1. The histograms of the physical and mechanical parameters of loess and correlations among them
图 2 基于框架体系二的预测偏差与黄土黏聚力预测值分布散点图
Fig. 2. Scatter plots between bias obtained from unified framework 2 and predicted cohesion of loess
a. RF; b. DT; c. XGBoost; d. AdaBoost
图 3 基于框架体系二的不同机器学习方法预测结果及其不确定性
Fig. 3. Results of cohesion prediction and uncertainty quantification for different machine learning methodsbased on unified framework 2
a. RF; b. DT; c. XGBoost; d. AdaBoost
图 4 基于框架体系二的100次随机试验黄土黏聚力预测性能与不确定性量化结果统计特征分析
a. RF方法：R2；b. RF方法：MAPE；c. RF方法：CP95；d. DT方法：R2；e. DT方法：MAPE；f. DT方法：CP95；g. XGBoost方法：R2；h. XGBoost方法：MAPE；i. XGBoost方法：CP95；j. AdaBoost方法：R2；k. AdaBoost方法：MAPE；l. AdaBoost方法：CP95
Fig. 4. Statistical characteristic results of predicted cohesion of loess and uncertainty quantification for 100 experiments based on unified framework 2
图 5 基于框架体系一和二的100次随机试验黄土黏聚力概率预测结果统计特征分析
Fig. 5. Statistical characteristic results of probabilisticprediction of cohesion of loess for 100 experiments based on unified framework 1 and 2
a. RF; b. DT; c. XGBoost; d. AdaBoost
图 6 基于预测残差的100次随机试验黄土黏聚力概率预测结果统计特征分析
Fig. 6. Statistical characteristic results of probabilisticprediction of cohesion of loess for 100 experiments based on residual
a. RF; b. DT; c. XGBoost; d. AdaBoost
表 1 黄土物理力学参数的基本统计特征
Table 1. Statistical characteristics of the physical and mechanical parameters of loess
黄土参数 单位 最大值 最小值 平均值 标准差 偏度 峰度 黏聚力 kPa 54.60 11.70 28.20 8.03 0.37 2.99 埋深 m 44.00 1.00 17.13 9.89 0.34 2.41 含水率 % 25.30 8.20 16.63 3.72 0.00 2.30 孔隙比 1 1.15 0.55 0.74 0.11 0.80 3.39 液限 % 33.70 21.90 24.86 2.02 1.44 5.32 塑限 % 20.30 15.20 16.82 0.79 1.21 4.86 液性指数 1 1.11 -1.17 -0.06 0.42 -0.30 2.56 塑性指数 1 13.40 6.20 8.04 1.29 1.44 5.25 饱和度 % 100.00 24.00 62.55 17.48 -0.17 1.89 干密度 g/cm3 1.75 1.26 1.56 0.09 -0.51 2.80 
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表 2 基于框架体系二的预测偏差与黄土黏聚力预测值及输入特征的斯皮尔曼相关系数及P值检验结果
Table 2. Spearman's correlation coefficientbetween bias obtained from unified framework 2 and predicted cohesion/ input variables and corresponding results of P-value test
黄土参数 RF DT XGBoost AdaBoost rs P值 rs P值 rs P值 rs P值 y=黏聚力 0.089 0.138 -0.156 0.009 -0.076 0.203 0.163 0.006 X1=埋深 0.140 0.019 0.213 0.000 0.052 0.389 0.122 0.041 X2=含水率 0.068 0.256 0.049 0.413 0.027 0.652 0.105 0.078 X3=孔隙比 -0.154 0.010 -0.091 0.131 -0.062 0.298 -0.244 0.000 X4=液限 0.122 0.041 0.089 0.137 0.064 0.287 0.225 0.000 X5=塑限 0.085 0.154 0.060 0.315 0.052 0.391 0.175 0.003 X6=液性指数 0.060 0.319 0.044 0.463 0.021 0.732 0.074 0.218 X7=塑性指数 0.129 0.031 0.097 0.104 0.062 0.300 0.240 0.000 X8=饱和度 0.131 0.028 0.083 0.168 0.048 0.423 0.204 0.001 X9=干密度 0.161 0.007 0.095 0.115 0.068 0.257 0.253 0.000
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表 3 基于框架体系一的预测偏差与黄土黏聚力预测值及输入特征的斯皮尔曼相关系数及P值检验结果
Table 3. Spearman's correlation coefficient between bias obtained from unified framework 1 and predicted cohesion/ input variables and corresponding results of P-value test
黄土参数 RF DT XGBoost AdaBoost rs P值 rs P值 rs P值 rs P值 y=黏聚力 0.418 0.000 0.024 0.693 0.196 0.001 0.305 0.000 X1=埋深 0.116 0.053 0.134 0.025 0.026 0.667 0.175 0.003 X2=含水率 0.083 0.167 0.019 0.749 0.001 0.990 0.091 0.131 X3=孔隙比 -0.144 0.016 -0.119 0.047 0.017 0.781 -0.279 0.000 X4=液限 0.148 0.013 0.062 0.304 0.012 0.847 0.222 0.000 X5=塑限 0.111 0.063 0.027 0.659 0.027 0.649 0.186 0.002 X6=液性指数 0.074 0.216 0.005 0.930 -0.008 0.888 0.054 0.365 X7=塑性指数 0.154 0.010 0.077 0.201 -0.002 0.971 0.224 0.000 X8 =饱和度 0.134 0.025 0.073 0.224 -0.017 0.775 0.204 0.001 X9=干密度 0.152 0.011 0.123 0.040 0.008 0.894 0.286 0.000
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表 4 预测残差与黄土黏聚力预测值及输入特征的斯皮尔曼相关系数及P值检验结果
Table 4. Spearman's correlation coefficient between residual and predicted cohesion/ input variables and corresponding results of P-value test
黄土参数 RF DT XGBoost AdaBoost rs P值 rs P值 rs P值 rs P值 y=黏聚力 0.098 0.101 -0.169 0.004 -0.075 0.211 0.164 0.006 X1=埋深 0.151 0.011 0.224 0.000 0.057 0.341 0.129 0.032 X2=含水率 0.078 0.193 0.036 0.547 0.022 0.718 0.109 0.068 X3=孔隙比 -0.160 0.007 -0.079 0.185 -0.060 0.320 -0.247 0.000 X4=液限 0.132 0.027 0.079 0.186 0.067 0.266 0.231 0.000 X5=塑限 0.094 0.115 0.052 0.382 0.056 0.352 0.183 0.002 X6=液性指数 0.067 0.262 0.032 0.597 0.012 0.842 0.074 0.216 X7=塑性指数 0.139 0.020 0.087 0.149 0.065 0.279 0.245 0.000 X8 =饱和度 0.143 0.016 0.070 0.242 0.044 0.461 0.212 0.000 X9=干密度 0.168 0.005 0.082 0.169 0.066 0.275 0.256 0.000
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