Prediction of lateral wall deflections of excavations in water-rich sands by a modified multivariate-adaptive-regression-splines method

Abstract

Machine learning methods have advantages in predicting excavation-induced lateral wall displacements. Due to lack of sufficient field data, training data for prediction models were often derived from the results of numerical simulations, leading to poor prediction accuracy. Based on a specific quantity of data, a multivariate adaptive regression splines method (MARS) was introduced to predict lateral wall deflections caused by deep excavations in thick water-rich sands. Sensitivity of lateral wall deflections to affecting factors was analyzed. It is disclosed that dewatering mode has the most significant influence on lateral wall deflections, while the soil cohesion has the least influence. Using cross-validation analysis, weights were introduced to modify the MARS method to optimize the prediction model. Comparison of the predicted and measured deflections shows that the prediction based on the modified multivariate adaptive regression splines method (MMARS) is more accurate than that based on the traditional MARS method. The prediction model established in this paper can help engineers make predictions for wall displacement, and the proposed methodology can also serve as a reference for researchers to develop prediction models.

Graphical abstract

Keywords

lateral wall deflection / machine learning / multivariate adaptive regression splines method / excavation / database / water-rich sand

1 Introduction

With the increasing demand for underground space, a large number of excavations have been constructed in urban areas [1–18]. The lateral wall deflection of excavations is an important index for the construction-quality evaluation and risk assessment. Accurate prediction of excavation-caused lateral wall deflection has been a concern of scholars and engineers.

Analytical and numerical methods were often adopted to calculate the lateral wall deflections of excavations [19–28]. However, the values calculated by analytical and numerical methods often differed significantly from the field data due to the limitations of parameter settings, assumptions, soil constitutive models, and construction conditions. Using descriptive statistics, many scholars summarized the laws of lateral wall deflections by establishing an excavation database [7, 29–33]. Based on the measured data, the conclusions obtained by the descriptive statistics method are more reliable than those obtained by the numerical and analytical methods. It is worth noting that a descriptive statistics method cannot establish a functional relationship between the lateral wall deflection and affecting factors.

Some scholars have introduced neural network models in machine learning methods to analyze the response of ground and retaining walls during excavation as well as other geotechnical works [5,34–46]. The backpropagation neural network, which is currently the most used one in geotechnical engineering, uses a backpropagation algorithm to train the prediction model. Machine learning algorithms such as long short-term memory neural networks and deep neural networks have been introduced to consider more complex affecting factors. Notably, all of these methods use artificial neural networks as a black box for model training [34,35,38,45,46]. It makes the models poorly interpretable and significantly affected by hyperparameters. In addition, the model predictions are very susceptible to overfitting, which leads to poor prediction results. In contrast, the wall deflection prediction model established by regression methods in machine learning methods has good interpretability [40,44]. Moreover, the nonlinear regression method is more accurate in establishing multivariate complex models and thus has been introduced into geotechnical engineering for predictive analysis. Nevertheless, numerical methods were often used to obtain training data for the predictive model, which resulted in the poor predictive performance of the established regression model.

Based on the massive field data from 21 subway excavations of two metro lines in the city of Nantong, China, the multivariate adaptive regression splines (MARS) method was introduced to predict the lateral wall deflections caused by excavation in the thick water-rich sands. The effects of dewatering system, excavation size, wall penetration ratio, construction design, and soil parameters on the lateral wall deflections were considered in the prediction model. Based on the results of the cross-validation analysis, weights were introduced into the MARS method to optimize the prediction model. Comparison of the predicted values with field data shows that the prediction based on the modified multivariate adaptive regression splines (MMARS) method is more accurate than that based on the traditional MARS method.

2 Methodology

2.1 Overview

MARS is a data analysis method proposed by Friedman [47]. The method includes two steps: forward selection and backward deletion. The advantage of MARS is that it can deal with high-dimensional data. Moreover, the method is computationally fast and can give accurate predictions. MARS combines the advantages of the projection pursuit method and the recursive partitioning method. By introducing splines as the basis function, different regression slopes are used in different intervals of the independent variables to approximate the nonlinear relationship between the independent variables and the dependent variables. As shown in Fig.1(a), the MARS partitions the data by adaptively selecting nodes and generating the corresponding basis functions. Finally, the model is established by adding the basis functions with the following expression:

(1)f\left(x\right)={\alpha }_0+\sum _{i=1}^M{\alpha }_i{H}_i\left(x\right),

(2)H_i\left(x\right)=\{\begin{array}{c}\mathrm{m}\mathrm{a}\mathrm{x}(0,t-x),\\ \mathrm{m}\mathrm{a}\mathrm{x}(0,x-t),\end{array}

where f\left(x\right) is the predicted value of the dependent variable; {\alpha }_0 is the intercept; {\alpha }_i is the coefficient corresponding to the i\mathrm{t}\mathrm{h} basis function; H_i\left(x\right) is the i\mathrm{t}\mathrm{h} basis function; M is the total number of basis functions; E is the threshold of the input variable; x is the independent variable; t is a certain value of x.

2.2 Principles

The MARS method is an improved method based on the recursive segmentation regression algorithm. Each segmentation of the algorithm is conducted based on the completion of the previous segmentation, see Fig.1(b). The algorithm could fit any distribution model when the region was segmented enough. Whereas, recursive segmentation models are prone to overfitting when modeling continuous functions. Based on the above considerations, the MARS model uses a linear function in the forward selection procedure instead of the constant value in the recursive segmentation regression algorithm. Generalized cross-validation (GCV) is used in the backward deletion procedure.

(3)\epsilon =\frac{{\displaystyle \sum _{i=1}^N[y_i-f(x_i){]}^2}}{N{\left[1-{\displaystyle \frac{M+c\times {\displaystyle \frac{(M-1)}{2}}}{N}}\right]}^2},

where M is the total number of the basis function; c is the penalty coefficient; N is the number of training data sets; y_i is the measured value; f\left(x_i\right) is the predicted value. The process of establishing the MARS model is as follows:

1) Forward selection procedure

The mathematical model of MARS is:

(4)y=f\left(X_1,X_2,\dots ,X_{\mathrm{P}}\right)+e=f\left(X\right)+e,

where y is the output variable; X is the matrix of input variables,X=\left(X_1,X_2,\dots ,X_{\mathrm{P}}\right); e is the error.

The MARS uses basis functions to establish functional relationships between input and output variables. Linear functions are represented in the form of \mathrm{m}\mathrm{a}\mathrm{x}(0,x-t). The position of the node in the function is determined by t.

(5)

Expression of the MARS model is built from a linear combination of basis functions:

(6)f\left(X\right)={\beta }_0+\sum _{m=1}^M{\beta }_m{\lambda }_m\left(X\right),

where {\beta }_0 is a constant; M is the number of predetermined basis functions; {\beta }_m is the coefficient of the mth basis function obtained based on the least squares method; {\lambda }_m\left(X\right) is the m\mathrm{t}\mathrm{h} basis function.

MARS is a data modeling process. A forward selection procedure is performed on the training data set. Models are generated by using constant terms and basis functions. For a model containing M basis functions, the next basis function added to the model is as follows:

(7){\hat{\beta }}_{m+1}{\lambda }_m\left(X\right)\mathrm{m}\mathrm{a}\mathrm{x}\left(0,x_i-t\right)+{\hat{\beta }}_{m+2}{\lambda }_m\left(X\right)\mathrm{m}\mathrm{a}\mathrm{x}(0,t-x_i),

where \beta is obtained based on the least squares method. When a new basis function is added to the model, the interaction between the new basis function and existing basis functions is also considered. If the number of basis functions reached a predetermined maximum number, an overfitted model would be obtained.

2) Backward deletion procedure

To select an appropriate basis function, the GCV method is adopted to determine the final prediction model. The predictive performance of the model is evaluated using the parameter \epsilon. The smaller \epsilon is, the better the model predictive performance is.

In the backward deletion procedure, the algorithm minimizes \epsilon by eliminating the basis functions until a suitable model is established. Since the selection of basis functions and location of variable nodes are based on the specific research problem, MARS is an adaptive model.

After deriving the optimal MARS model, all the basis functions involving the same variable were combined. Analysis of variance (ANOVA) was employed to compare the statistical significance of the variables and evaluate the contribution of the input variables. In addition, the maximum number of basis functions is preset in advance. If the preset values were not reasonable, it would result in a final model which is not optimal. Therefore, multiple sets of models can be tested to determine the preset maximum number of basis functions for the final optimal model.

3 Data and variable disjunction

3.1 Data description

The city of Nantong is located at the estuary of the Yangtze River in China. In recent years, two metro lines were planned and constructed in Nantong. Its metro line 1 had 28 subway stations and metro line 2 had 14 stations (Fig.2).

Thick sandy strata are widely distributed in Nantong, and the groundwater level is near the ground level. The typical physical and mechanical properties of the sandy strata are summarized in Tab.1. Notably, thick sandy strata were located above and below layer ④ (silty clay). In some areas, the absence of layer ④ in the formation led to higher requirements for dewatering system design.

The field data from 16 excavations of subway station along metro line 1 were used as the training data set. The field data from 5 excavations of subway stations of metro line 2 were used as the test data set for the model. The earth-retaining structures of these stations were diaphragm walls braced by multi inner struts. In addition, the lateral wall deflections of excavations with construction quality problems (over-excavation, untimely support, wall leaking, etc.) were affected by many factors and differed significantly from those of the excavations under normal construction [48]; therefore, their data were not collected in the database.

3.2 Variable disjunction

Lateral wall deflections are affected by excavation design, construction, and ground conditions (Fig.3). The main affecting factors in excavation design include excavation size (length, width, and depth), and wall penetrating ratios. The soil properties and groundwater conditions at the site where the excavation is located are also important factors. All the subway stations were segmentally excavated using the bottom-up method. Therefore, the length of each segmental excavation also affects the lateral wall deflections. Alternatively, water-rich sandy soils are widely distributed in Nantong; hence, implementation of a dewatering system during the excavation is essential.

Based on the above considerations, 12 affecting factors were identified as input variables for the prediction model. The design factors (X_3, X_4) and the construction factors (X_1, X_2) of the excavation were collected. For the soil properties, the unit weight (X_5, X_9), the compression modulus (X_6, X_{10}), the cohesion (X_7, X_{11}), and the friction angle (X_8, X_{12}) of the upper and lower soils of the excavation base were included in the database as well, see Tab.2. The distributions and properties of the soil layers were variable across the region. To improve the generalizability of the prediction model, the parameters of multiple soil layers within 60 m BGS were averaged. Besides, excavations with large lengths were excavated in sections. The lengths of the sections also affect the excavation performance; thus, the excavation length averaged over a number of sections was introduced to be analyzed. The detailed description of each input variable is listed in Tab.2. The training data from 16 subway stations of metro line 1 for the prediction model are listed in Tab.3 and Tab.4. The data from 5 subway stations of metro line 2 are used as the test data set for the prediction model (Tab.5 and Tab.6).

The training and test data sets were derived from the measured data of the subway excavations. The lateral wall deflections were normalized to account for the effect of excavation depth (Eq. (8)). Since accurate prediction of the lateral wall deflection is difficult, the wall deflections measured at six deflection gauges of the excavation were used to establish six prediction models, respectively. The six prediction models were used to predict the extent of lateral wall deflections. By comparing the difference between the ranges of predicted and measured lateral wall deflections, the prediction performance of the models was evaluated.

(8)Y_{\mathrm{N}}={\delta }_{\mathrm{h}\mathrm{m}}\times {10}^4/H_{\mathrm{e}},

where Y_{\mathrm{N}} is the normalized lateral wall deflection; {\delta }_{\mathrm{h}\mathrm{m}} is the measured maximum lateral wall deflection; H_{\mathrm{e}} is the excavation depth.

4 Prediction of lateral wall deflection

4.1 Parameters for model evaluation

Based on adaptive node selection and combination, the MARS model makes accurate predictions in the forward selection procedure. Whereas, as the backward deletion procedure proceeds, the predictive performance of the MARS model changes. Model evaluation is used to determine the optimal model and parameters.

The parameters of model evaluation include the mean absolute error (MAE) and Pearson correlation coefficient (PCC). Since the lateral wall deflections are affected by multi factors, it is difficult to accurately predict the specific wall deflection of the excavation. Therefore, the maximum and minimum measured wall deflections of the six monitoring points at a subway station were selected to establish the wall deflection interval. The prediction interval is established based on the prediction results of the MARS model. The prediction performance of the model is evaluated by the Prediction interval coverage probability (PICP) and Prediction interval normalized width (PINW).

The MAE is the mean of the absolute difference between the measured and the predicted values. By converting the error into the form of absolute value, the MAE method avoids positive and negative errors canceling each other out. The MAE is expressed in the same units as the variable being measured to facilitate interpretation and comparison between different data sets. The expression of MAE is shown in Eq. (9).

(9)MAE\left(Y,\hat{Y}\right)=\frac{1}{n}\sum _{i=1}^n\left|{\hat{Y}}_i-Y_i\right|,

where Y is the measured value; \hat{Y} is the predicted value; n is the number of Y.

The PCC is a parameter used to determine the correlation between two variables. It is widely used in machine learning and statistical analysis to identify important variables of the model. The PCC is obtained by dividing the covariance of two variables by the product of the variances of each variable, see Eq. (10). The closer the PCC is to 1, the more accurate the prediction is. When the model had a large MAE while the PCC was close to 1, the empirical coefficients could be adopted to improve the accuracy of the model predictions.

(10)PCC=r\left(Y,\hat{Y}\right)=\frac{\mathrm{C}\mathrm{o}\mathrm{v}(Y,\hat{Y})}{\sqrt{\mathrm{V}\mathrm{a}\mathrm{r}[Y]\mathrm{V}\mathrm{a}\mathrm{r}[\hat{Y}]}},

where \mathrm{C}\mathrm{o}\mathrm{v}(Y,\hat{Y}) is the covariance of Y and \hat{Y}. \mathrm{V}\mathrm{a}\mathrm{r}\left[Y\right], \mathrm{V}\mathrm{a}\mathrm{r}[\hat{Y}] are the variances of Y and \hat{Y} respectively.

The PICP is used to evaluate the reliability of the prediction model, which responds to the probability that the measured data are within the prediction interval, with the following expression:

(11)PICP=\left(1-\alpha \right)\times 100\mathrm{\%},

where \alpha is the confidence level.

The PINW is negatively correlated with the accuracy of the prediction results. When the models have the same PICP, the prediction performance of the model with a smaller PINW is better. Notably, the wider the model’s PINW, the closer the model’s PICP is to 1. Models with high PINW have poor prediction accuracy and will be eliminated in model evaluation. The expression of PINW is as follows:

(12)PINW=\left(UB-LB\right)/(1-\alpha ),

where UB is the upper bound of the prediction interval; LB is the lower bound of the prediction interval.

4.2 Predicting lateral wall deflection with multivariate adaptive regression splines

The number of basis functions for the MARS model was determined to be nine through a series of modeling. Based on adaptive node selection and combination, the MARS model was established. The model considers the effects of dewatering, the excavation design, and soil parameters. The expressions of the six models are given below:

(13)\begin{array}{rl}{Y}_1\left(MARS\right)=& 14.2031-1.8580BF1+0.5479BF2-2.3894BF3-3.8654BF4+0.8344BF5\\ & +0.2772BF6+3.9495BF7+21.6261BF8+16.871BF9,\end{array}

(14)\begin{array}{rl}{Y}_2\left(MARS\right)=& 16.7881-2.2858BF1+1.5766BF2+2.1401BF3-5.7573BF4+1.2665BF5\\ & -0.9168BF6+1.0522BF7+9.8832BF8+15.5448BF9,\end{array}

(15)\begin{array}{rl}{Y}_3\left(MARS\right)=& 17.6872-2.8816BF1+1.5090BF2+4.5901BF3-4.5771BF4+1.2506BF5\\ & -0.9967BF6+2.4416BF7+6.8208BF8+7.7829BF9,\end{array}

(16)\begin{array}{rl}{Y}_4\left(MARS\right)=& 19.6384-3.2866BF1+0.6683BF2+3.7332BF3-3.9418BF4+0.8605BF5\\ & -0.5805BF6+2.8319BF7+4.8750BF8+9.8863BF9,\end{array}

(17)\begin{array}{rl}{Y}_5\left(MARS\right)=& 20.4744-3.2183BF1+0.2730BF2+2.9285BF3-3.1532BF4+0.8523BF5\\ & -0.4208BF6+2.0810BF7+9.1120BF8+10.2634BF9,\end{array}

(18)\begin{array}{rl}{Y}_6\left(MARS\right)=& 23.0965-3.2616BF1-1.9412BF2+4.8826BF3+0.4926BF4+0.6351BF5\\ & -0.5639BF6+1.0570BF7+0.8142BF8+0.9185BF9.\end{array}

The expressions of the basis functions were given in Tab.7. Fig.4 plots the relationship between the measured and the predicted normalized lateral wall deflections based on the training and test data, respectively. The validation of the MARS prediction model based on the training data set reveals that the model is not overfitted. The error between the predicted and measured deflections is within 10%. The measured and predicted deflections were compared based on the test data set. It is found that although the MAE of the predicted results is small, the difference between R^2 and 1 is large. The prediction accuracy of the model is poor. The expression of R^2 is as follows:

(19)R^2=1-\frac{{\displaystyle \sum _i{(y_i-{\hat{y}}_i)}^2}}{{\displaystyle \sum _i{(y_i-\overline{y})}^2}},

where R^2 is the coefficient of determination; y_i is the measured value; {\hat{y}}_i is the predicted value; \overline{y} is the average value of the measured data.

To quantitatively evaluate the prediction accuracy of the model, PICP and PINW of the prediction results based on the test data set were calculated (Tab.8). It can be seen from the table that the range of the predicted deflections is larger than that of the measured deflections. The PINW of the prediction results are all greater than 1, which also leads to the PICP of the prediction results being closer to 1. The above analysis indicated that the prediction performance of the models established based on the MARS method was relatively poor. It is necessary to modify the MARS method to improve its prediction accuracy.

4.3 Prediction of lateral wall deflection with multivariate adaptive regression splines

Sample capacity is one of the most important factors affecting the quality of a mathematical model. A model training data set with a low sample capacity leads to an inadequate generalization of the model, and the functional relationship between the input variables and predicted values is also affected. The following methods are often used to deal with the problem.

Data enhancement: expanding the size of the training data set through data transformation or data generation, such as rotating, flipping, scaling, panning, adding noise, etc.

Migration learning: extracting features from a small portion of the data in the training data set by utilizing the feature extraction capability of an established model. Then, adjustments are made on a small portion of the data to improve the generalization ability of the model.

Data synthesis: expanding the size of the data by combining data from different sources. For example, downloading similar data from the Internet.

Regularization: reduce model complexity and avoid model overfitting by adding regularization terms during model training.

Integration learning: combining the outputs of several models to improve the generalization of the model and prevent model overfitting. Examples include voting, averaging, and weighting.

Cross-validation methods are often used to select the optimal hyperparameters for neural network models. In this section, models were established based on the data from 16 stations in the training data set using the leave-one-out cross-validation method. The basis function nodes and MAE of the prediction results of each model were recorded in Tab.9.

In addition, the MAE of each prediction model is calculated. The smaller the MAE of the prediction model, the larger the node weight of the corresponding model. The weights of the nodes were calculated based on the MAE of the 16 prediction models using Eq. (20).

(20)W_f=\frac{{\displaystyle \sum _{g=1}^{16}{N}_f{MAE}_g}}{{\displaystyle \sum _{f=1}^{12}\sum _{g=1}^{16}{N}_f{MAE}_g}},

where g is the number of the prediction models used in the modification of the MARS method; N_f is the number of times the fth node is used in the model; {MAE}_g is the mean of the absolute difference of the gth prediction model.

The determined weight parameters were used for the modification of the MARS method. The process of establishing the MMARS model is as follows.

1) Add node weight matrix in the model building program.

2) Iterate through the model nodes and incorporate the weight matrix into the node matrix.

3) Assign the node weights to the basis functions entering the prediction model and calculate the function coefficient matrix that minimizes the prediction error.

4) Based on the established prediction model, iteratively remove each basis function from the model. Calculated the prediction error of each submodel.

5) Select the submodel with the smallest error as the optimal model and calculate the GCV of the optimal submodel.

6) Repeat steps 4 and 5 for the optimal submodel identified in step 5 until the model has only intercept terms.

7) Select the submodel with the smallest GCV as the final MMARS model.

The number of basis functions for the MMARS model was determined to be nine through a series of modeling. Based on the MMARS method, the prediction model was established. The expressions are given below:

(21)\begin{array}{rl}{Y}_1\left(MMARS\right)=& 8.7600-1.0937BF1-1.4980BF2+5.4978BF3+0.8469BF4+1.2123BF5\\ & +5.3192BF6-1.5977BF7+0.6429BF8+0.7379BF9,\end{array}

(22)\begin{array}{rl}{Y}_2\left(MMARS\right)=& 12.7566-1.5144BF1-1.3712BF2+7.1493BF3+1.3952BF4+0.7373BF5\\ & +2.2907BF6-2.8497BF7+0.3904BF8+0.5706BF9,\end{array}

(23)\begin{array}{rl}{Y}_3\left(MMARS\right)=& 15.7431-2.5749BF1-1.1958BF2+8.0852BF3-1.3859BF4+0.4670BF5\\ & +3.6001BF6-3.0278BF7+0.1347BF8+0.6347BF9,\end{array}

(24)\begin{array}{rl}{Y}_4\left(MMARS\right)=& 18.2952-3.0583BF1-1.4057BF2+6.5530BF3+0.9646BF4+0.6036BF5\\ & +3.6509BF6-3.1029BF7+0.0842BF8+0.2981BF9,\end{array}

(25)\begin{array}{rl}{Y}_5\left(MMARS\right)=& 19.0136-3.1214BF1-1.4492BF2+6.7052BF3+0.9045BF4+0.5033BF5\\ & +2.9528BF6-3.0835BF7+0.1337BF8+0.2901BF9,\end{array}

(26)\begin{array}{rl}{Y}_6\left(MMARS\right)=& 21.5061-2.8419BF1-1.4916BF2+5.1496BF3+0.6107BF4+0.3178BF5\\ & -0.8744BF6+1.7205BF7+0.2120BF8+0.0087BF9.\end{array}

The expressions of the basis functions were given in Tab.10. Fig.5 plots the relationship between the measured and the predicted normalized lateral wall deflections based on the training data and test data, respectively. Validation of the MMARS prediction model based on the training data set reveals that the model is not overfitted. The R^2 of the MMARS prediction model is 0.8766, which is larger than that of the MARS prediction model. The prediction accuracy of the MMARS prediction model is better than that of the MARS prediction model. Notably, the PINW of the MMARS prediction model is significantly smaller than that of the MARS prediction model (Tab.11). The PICP of the prediction results of the NTFR station is small because of the small value of the PINW.

Overall, the prediction performance of the MMARS model is better than that of the MARS model. The prediction model obtained using the proposed MMARS method provides a reasonable prediction of the lateral wall deflections.

5 Discussion

5.1 Sensitivity analysis

The ANOVA method analyzes the level of significance of differences between groups by comparing the magnitude of within-group variance and between-group variance in multiple data sets. ANOVA can be used to evaluate the effect of different parameters on predicted deflections and determine which variables are significant. Basis functions with the same variables were removed from the MMARS model and the lateral wall deflections were calculated. The standard deviation (STD) of the calculated wall deflections was used as an evaluation parameter. The larger the STD, the greater the effect of the removed variables on the predicted deflections. The GCV of the sub-MMARS models formed by removing the basis functions with the same variables were calculated as well. The calculated GCV was used as another evaluation parameter. The larger the GCV, the greater the prediction bias, the more important the variables removed.

The ANOVA method was employed to analyze the sensitivity of the predicted deflections of the six MMARS models to the variables. The six MMARS models share the same 12 basis functions. These basis functions with the same variables were put into one group, totaling 6 basis function groups. Fig.6 plots the sensitivity of the predicted deflections of the six models to the six variables or group of variables. Influenced by the non-uniform distribution of the variables, the STD distribution of different models in Fig.6 varies considerably. By analyzing the distribution of GCV and STD in the figure, it is found that variables 1, 2, and 8 have a significant influence on the model prediction accuracy. To quantify the influence of each variable on the predicted deflections, the GCV of each submodel was used to calculate the relative sensitivity of the predicted deflections to the variables (Fig.7). Dewatering mode (Var 1) has the most significant effect on the lateral wall deflections. The stations collected are located in water-rich sand soils with low cohesion. Consequently, the cohesion of the soils above the excavation base (Var 7) has the least effect on the lateral wall deflections, and the friction angle of soils (Vars 8 and 12) has a greater effect on the predicted deflections than the cohesion. Notably, the effect of the excavation strategy (Var 2) on the lateral wall deflection is significant as well.

5.2 Limitations

The MMARS prediction model was established by analyzing the measured data from 21 deep excavation of Nantong metro lines 1 and 2. Since the excavation depths, ratios of length to depth, wall pentation ratios, and other related design factors of these excavations were very similar, these factors had been removed at the backward deletion procedure. In addition, the excavations collected by the database were located in the water-rich sandy soils with low cohesion. The sensitivity of the lateral wall deflections to dewatering mode was the highest, while the sensitivity of lateral wall deflections to soil cohesion was the least. The proposed MMARS prediction model is suitable for the prediction of lateral wall deflection caused by long and narrow subway excavations in water-rich sands with excavation depths of 16.5–20.8 m. For excavations in other plan-geometric shapes or located in other hydrogeological conditions, a prediction model can be established by referring to the methodology proposed in this paper.

6 Conclusions

The MARS method was introduced in this paper to predict the lateral wall deflections caused by deep excavations in thick water-rich sands. The effects of dewatering, excavation size, wall penetration ratio, construction design, and soil parameters on the lateral wall deflections were considered in the prediction model. Generally, the following major conclusions can be drawn.

1) Compared with other machine learning methods, the MARS method can produce simple, easy-to-interpret models with high computational efficiency, and estimate the contributions of input variables. The proposed MMARS method can explain 35% more variation of the dependent variable than the MARS method. This indicates that the MMARS method is more accurate and suitable for estimating lateral wall deflection caused by deep excavation in water-rich sands [49–51].

2) Since the excavations collected in the database were constructed in the water-rich sands with low cohesion, the dewatering mode has the most significant influence on the lateral wall deflection caused by excavation; while, the cohesion of soils has the least influence. The effects of friction angle of soils and excavation length on e lateral wall deflections are significant as well.

3) The proposed MMARS prediction model is suitable for prediction of lateral wall deflection caused by long and narrow subway excavations in water-rich sandy formations. Excavations in other plan-geometric shapes or constructed in other hydrogeological conditions can be modeled by referring to the methodology proposed in this paper to predict lateral wall deflections.

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