Evaluation mthods for performance of post-construction settlement prediction models in thick loess filled ground
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摘要: 工后沉降预测结果是黄土高填方场地变形稳定性评价和建筑物规划布局的重要参考依据。为遴选适合黄土高填方场地的工后沉降预测模型，基于某典型黄土高填方工程的实测沉降数据，分析了工后沉降曲线的变化规律和发展趋势，建立了17种回归参数模型，提出了模型预测效果的评价指标和方法。结果表明，该工程填方区工后沉降历时曲线呈“缓变型”变化，土方填筑完工初期无陡增段，随时间增加沉降速率逐步降低，尚未出现沉降趋于稳定的水平段；将外推预测误差、内拟合误差和后验误差比最小化作为综合控制目标，可遴选出理想的回归参数模型；MMF模型（Ⅱ型）和双曲线模型具有较高的预测精度、较好的稳定性和较强的适应性，在17种模型中的预测效果最佳；沉降数据的变化越平稳，模型预测效果越好；增大建模数据的时间跨度，会提升预测精度，但增大至一定值后，预测精度提升效果不再显著。Abstract: The prediction of post-construction settlement is an important reference for the evaluation of deformation stability evaluation and building layout planning in thick loess filled ground. To choose suitable models for predicting post-construction settlement in thick loess filled grounds, the characteristics of post-construction settlement curves are analyzed based on the measured settlement of a thick loess fill ground project. Seventeen regression parameter models are established, and some evaluation indexes and methods for models are proposed. The best prediction models for post-construction settlement prediction are optimized. The results indicate that the post-construction settlement curves of the filling area change slowly, with no steep increase in the initial stage of earthwork filling. The settlement rate gradually decreases with time, and there is no horizontal section where the settlement tends to be stable. The optimal regression parameter model can be selected by minimizing the extrapolation prediction error, the internal fitting error, and the posteriori error ratio as the comprehensive control objective. The MMF model （TypeⅡ） and hyperbolic model show high prediction accuracy, good stability, and strong adaptability, with the prediction effect being the best among the 17 models. The more stable the settlement data changes, the better the model prediction effect. Increasing the time span of modeling data would improve the prediction accuracy, but the improvement effect on prediction accuracy would no longer be significant after reaching a certain value.
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Key words:
- loess /
- thick loess fill ground /
- post-construction settlement /
- prediction model /
- evaluation index
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图 1 试验场地内的代表性地表沉降监测点平面位置图
Figure 1. Distribution map of representative surface settlement monitoring points of post-construction settlement within the test site
图 2 试验场地内代表性监测点的工后沉降曲线
Figure 2. Post-construction settlement curve of representative monitoring points within the test site
图 3 各模型的内拟合及外推预测曲线
Figure 3. Internal-fitting and extrapolation prediction curves of each model
图 4 各模型对实测沉降数据的拟合及预测误差
Figure 4. Fitting and prediction errors of each model for measured settlement data
图 5 优选模型在建模数据不同时间跨度时的拟合及预测曲线
Figure 5. Curve Fitting and prediction results of the optimal model with different time span
表 1 沉降预测中常用的回归参数预测模型
Table 1. Summary of regression parameter prediction models for settlement prediction
类型 模型名称 数学表达式 沉降速率 备 注 第Ⅰ类模型 Logistic ${\hat s_t} = a/(1 + b{{\text{e}}^{ - ct}})$ ${\hat s'_t} = abc{{\text{e}}^{ - ct}}/{(1 + b{{\text{e}}^{ - ct}})^2}$ 收敛模型 Gompertz ${\hat s_t} = a{{\text{e}}^{ - {{\text{e}}^{b - ct}}}}$ ${\hat s'_t} = ac{{\text{e}}^{ - {{\text{e}}^{b - ct}}}}{{\text{e}}^{b - ct}}$ 收敛模型 Usher ${\hat s_t} = a/{(1 + b{{\text{e}}^{ - ct}})^d}$ ${\hat s'_t} = abcd{{\text{e}}^{ - ct}}{(1 + b{{\text{e}}^{ - ct}})^{ - 1 - d}}$ 收敛模型 Weibull ${\hat s_t} = a(1 - b{{\text{e}}^{ - c{t^d}}})$ ${\hat s'_t} = abcd{{\text{e}}^{ - c{t^d}}}{t^{ - 1 + d}}$ 收敛模型 MMF-Ⅰ ${\hat s_t} = (ab + c{t^d})/(b + {t^d})$ ${\hat s'_t} = cd{t^{d - 1}}/(b + {t^d}) - d{t^{d - 1}} \cdot (ab + c{t^d})/{(b + {t^d})^2}$ 收敛模型 MMF-Ⅱ ${\hat s_t} = a{t^b}/(c + {t^b})$ ${\hat s'_t} = ab{t^{b - 1}}/(c + {t^b}) - ab{t^{2b - 1}}/{(c + {t^b})^2}$ 收敛模型 Richards ${\hat s_t} = a{(1 - b{{\text{e}}^{ - ct}})^{1/(1 - d)}}$ ${\hat s'_t} = abc{{\text{e}}^{ - ct}}{(1 - b{{\text{e}}^{ - ct}})^{d/(1 - d)}}/(1 - d)$ 收敛模型 Knothe-Ⅰ ${\hat s_t} = a{(1 - {{\text{e}}^{ - b{t^c}}})^d}$ ${\hat s'_t} = abcd{t^{c - 1}}{{\text{e}}^{ - b{t^c}}} \cdot {(1 - {{\text{e}}^{ - b{t^c}}})^{d - 1}}$ 收敛模型 Knothe-Ⅱ ${\hat s_t} = a{(1 - {{\text{e}}^{ - bt}})^c}$ ${\hat s'_t} = abc{{\text{e}}^{ - bt}}{(1 - {{\text{e}}^{ - bt}})^{c - 1}}$ 收敛模型 Bertalanffy ${\hat s_t} = {[{a^{1/3}} - {(a - b)^{1/3}} \cdot {{\text{e}}^{ - ct}}]^3}$ ${\hat s'_t} = 3{(a - b)^{1/3}}c{{\text{e}}^{ - ct}} \cdot {[{a^{1/3}} - {(a - b)^{1/3}} \cdot {{\text{e}}^{ - ct}}]^2}$ 收敛模型 邓英尔 ${\hat s_t} = a/(1 + b{{\text{e}}^{ - c{t^d}}})$ ${\hat s'_t} = abcd{{\text{e}}^{ - c{t^d}}}{t^{d - 1}}/{(1 + b{{\text{e}}^{ - c{t^d}}})^2}$ 收敛模型 第Ⅱ类模型 Spillman ${\hat s_t} = a - (a - b){{\text{e}}^{ - ct}}$ ${\hat s'_t} = (a - b)c{{\text{e}}^{ - ct}}$ 收敛模型 指数曲线 ${\hat s_t} = a(1 - {{\text{e}}^{ - bt}})$ ${\hat s'_t} = ab{{\text{e}}^{ - bt}}$ 收敛模型 双曲线 ${\hat s_t} = t/(a + bt)$ ${\hat s'_t} = a/{(a + bt)^2}$ 收敛模型 幂函数 ${\hat s_t} = a{t^b}$ ${\hat s'_t} = ab{t^{b - 1}}$ 发散模型 平方根函数 ${\hat s_t} = a + b\sqrt t $ ${\hat s'_t} = b/2\sqrt t $ 发散模型 对数函数 ${\hat s_t} = a\ln t + b$ ${\hat s'_t} = a/t$ 发散模型 对数抛物线 ${\hat s_t} = a{(\lg t)^2} + b\lg t + c$ ${\hat s'_t} = \lg {\text{e}} \cdot (b + 2a\lg t) \cdot {t^{ - 1}}$ 发散模型 星野法 ${\hat s_t} = {s_0} + \dfrac{ {ab\sqrt {t - {t_0} } } }{ {\sqrt {1 + {b^2}\left( {t - {t_0} } \right)} } }$ ${\hat s'_t} = \dfrac{ {ab} }{ {2\sqrt {[1 + {b^2}(t - {t_0})](t - {t_0})} } } - \dfrac{ {a{b^3}\sqrt {t - {t_0} } } }{ {2{ {[1 + {b^2}(t - {t_0})]}^{3/2} } } }$ 收敛模型 
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表 2 各模型对典型监测点S6实测数据的内拟合及外推预测结果
Table 2. Internal-fitting and extrapolation prediction results of each model for the measured settlement data at a typical monitoring point, S6
类型 模型名称 模型参数 内拟合效果 外推预测效果 后验误差
比Cia b c d MAPE* /% SSE RMSE R2 MAPE /% MAD MSE MFE 第
Ⅰ
类
模
型Logistic 39.7540 7.6080 0.0252 / 9.88 8.06 0.95 0.9928 21.66 11.77 179.32 11.77 2.19 Gompertz 43.7854 0.9395 0.0150 / 6.10 3.09 0.59 0.9973 16.39 8.99 111.11 8.99 2.69 Usher 64.0873 −0.9931 0.0049 −1.0149 5.16 2.72 0.52 0.9981 5.03 2.74 10.97 2.31 0.98 Weibull 3.3377 −52.2248 14.7577 −0.4232 3.58 2.04 0.45 0.9986 3.09 1.65 3.87 1.21 0.86 MMF-Ⅰ 2.4888 869.7498 68.8715 1.3199 3.26 1.71 0.44 0.9985 7.18 3.93 21.54 3.62 2.20 MMF-Ⅱ 113.9160 0.9777 329.4676 / 5.84 3.04 0.55 0.9979 2.05 1.05 1.77 0.41 0.35 Richards 51.8874 0.8687 0.0085 0.3724 3.18 1.66 0.43 0.9985 9.98 5.49 42.78 5.25 3.13 Knothe-Ⅰ 41.0949 0.0000 4.0687 0.2098 4.49 2.17 0.49 0.9981 18.75 10.31 147.80 10.31 4.18 Knothe-Ⅱ 68.6817 240.2071 0.9562 / 5.65 2.87 0.56 0.9975 3.79 2.05 6.41 1.58 0.67 Bertalanffy 46.8972 36.1948 0.0116 / 4.45 2.03 0.47 0.9982 13.42 7.39 76.86 7.32 3.02 邓英尔 3.2623 −0.9558 28.7664 −1.2368 3.24 1.81 0.42 0.9987 6.32 3.45 16.62 3.11 1.95 第
Ⅱ
类
模
型Spillman 68.5727 0.7074 0.0043 / 4.50 2.47 0.52 0.9978 3.59 1.93 5.67 1.38 0.80 指数曲线 62.6226 0.0050 / / 5.95 3.01 0.58 0.9973 5.71 3.12 13.95 2.74 0.96 双曲线 3.1221 0.0095 / / 6.00 3.06 0.55 0.9979 2.75 1.46 3.16 0.93 0.46 幂函数 0.6425 0.7852 / / 4.34 5.38 0.73 0.9963 11.67 6.15 42.57 -6.15 2.69 平方根函数 −8.9505 3.3748 / / 13.18 14.00 1.25 0.9876 2.90 1.54 2.78 1.36 0.22 对数函数 11.6931 −28.2990 / / 26.75 113.31 3.55 0.8993 24.20 12.77 182.58 12.77 0.90 对数抛物线 21.4982 −43.5385 25.8604 / 4.12 2.83 0.56 0.9975 2.75 1.46 2.83 1.07 0.67 星野法 1367.0000 0.0018 / / 32.50 114.39 3.38 0.9205 16.81 8.80 84.05 8.80 0.52
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表 3 优选模型对不同监测点沉降数据的内拟合和外推预测结果
Table 3. Internal-fitting and extrapolation prediction of settlement data for different monitoring points using the optimal model
监测点 模型名称 模型参数 内拟合效果R2 外推预测效果 后验误差比Ci a b c MAPE /% MFE S2 双曲线 15.8265 0.0747 / 0.9805 8.33 −0.60 0.58 MMF-Ⅱ 7.4942 1.5515 829.1400 0.9880 7.42 0.56 0.55 S3 双曲线 4.9610 0.0145 / 0.9969 3.28 1.31 0.73 MMF-Ⅱ 64.7724 1.0199 344.1345 0.9969 4.11 1.60 0.88 S4 双曲线 3.4475 0.0099 / 0.9973 0.78 0.56 0.12 MMF-Ⅱ 90.2525 1.0341 350.0908 0.9974 2.21 1.28 0.32 S5 双曲线 3.7677 0.0095 / 0.9990 2.34 1.29 0.64 MMF-Ⅱ 86.2061 1.0584 397.9812 0.9991 4.82 2.51 1.14 S6 双曲线 3.1247 0.0095 / 0.9978 1.42 0.92 0.24 MMF-Ⅱ 112.9076 0.9804 329.8660 0.9979 0.57 0.46 0.10 S7 双曲线 4.4275 0.0108 / 0.9944 2.15 −0.75 0.22 MMF-Ⅱ 71.2408 1.0794 416.6811 0.9946 1.30 0.64 0.13 S8 双曲线 9.8039 0.1899 / 0.9419 31.96 2.24 3.26 MMF-Ⅱ −1.0396 0.1205 −2.3030 0.9814 0.55 −0.05 0.11 S9 双曲线 7.9448 0.0288 / 0.9816 8.11 1.79 0.54 MMF-Ⅱ 35.3201 0.9936 274.5082 0.9999 7.88 1.74 0.53 S10 双曲线 5.5294 0.0167 / 0.9908 4.16 1.37 0.40 MMF-Ⅱ 166.3711 0.8423 540.3470 0.9917 3.30 −0.98 0.36 S11 双曲线 4.0073 0.0100 / 0.9981 3.51 1.75 0.69 MMF-Ⅱ 143.1599 0.9305 451.8395 0.9982 0.26 0.22 0.06 S12 双曲线 3.0885 0.0101 / 0.9981 5.40 3.11 1.01 MMF-Ⅱ 139.2267 0.9179 325.7170 0.9984 1.82 1.14 0.40 S13 双曲线 3.6537 0.0104 / 0.9989 2.59 1.37 0.68 MMF-Ⅱ 145.9022 0.9138 397.9244 0.9992 1.51 −0.60 0.72 S14 双曲线 4.4223 0.0123 / 0.9986 3.52 1.52 0.73 MMF-Ⅱ 74.2716 1.0255 358.6842 0.9986 4.61 1.96 0.94 S15 双曲线 3.8173 0.0093 / 0.9982 1.93 1.12 0.38 MMF-Ⅱ 77.3464 1.1049 426.3157 0.9986 6.28 3.26 1.20 S16 双曲线 3.3989 0.0084 / 0.9967 1.54 1.02 0.22 MMF-Ⅱ 160.0072 0.9428 447.0479 0.9968 1.24 −0.51 0.20 S17 双曲线 3.2720 0.0110 / 0.9968 5.66 3.08 0.78 MMF-Ⅱ 100.1378 0.9706 296.3921 0.9968 4.46 2.46 0.63
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