1. Introduction
The compression index of clayey soils is a measure of the soil's ability to compress or consolidate under an applied load [ 1 ]. It is a crucial property for engineers to consider when designing foundations, as it can affect the stability and settlement of the structure. The compression index of clayey soils is typically determined by conducting a series of oedometer laboratory tests on soil samples [ 2 , 3 ]. These tests involve applying increasing levels of stress to the soil and measuring the resulting consolidation or compression. The compression index is then calculated based on the amount of compression that occurs under a given stress level [ 4 ]. In general, clayey soils with a high compression index will be more prone to settlement and instability under load, while those with a low compression index will be more stable [ 5 , 6 ]. Engineers must consider the compression index of the soil when designing foundations to ensure that the structure is stable and will not experience excessive settlement.
Since conducting oedometer tests is time-consuming, costly, and unwieldy, scholars tried to create empirical formulas to predict the compression index [ 7 - 10 ]. However, most empirical formulas are based on the on-site environment and thereby their universality is insufficient. The empirical formula may not account for variations in soil properties and conditions that can affect the compression index. Additionally, empirical models are based on a limited amount of data and may not be accurate for all types of clay soils [ 11 - 13 ].
Encouragingly, with the rapid development of the soft computing technique, many scholars paid attention to its computational efficacy and high accuracy. Since the soft computing technique has been successfully used in different disciplines of civil engineering [ 14 - 21 ], researchers attempted to apply the soft computing technique to establish the relationship between the basic soil properties and the compression index [ 22 , 23 ]. Kurnaz et al. developed an artificial neural network (ANN) model to predict the compression and recompression index. The model was built on a dataset that consists of 246 laboratory oedometer tests, and the model’s inputs (soil properties) included the natural water content, liquid limit, plastic index, and specific gravity of soil particles [ 24 ]. Kordnaeij et al. proposed a group method of data handling (GMDH) type neural network to predict the recompression index. The used dataset, compiled from 344 consolidation tests for soils, included the soil properties such as the liquid limit, initial void ratio, specific gravity, natural water content, plastic index, and dry density [ 25 ]. Nguyen et al. proposed a hybrid ANN model: Biogeography-Based Optimization ANN. They used 188 soil samples to build the hybrid ANN model. The input parameters include the depth of samples, clay, moisture content, bulk density, dry density, specific gravity, void ratio, porosity, degree of saturation, liquid limit, plastic limit, plastic index, and liquid index. The principle component analysis (PCA) was used to reduce the dimension of input parameters [ 26 ]. Benbouras et al. exploited the performance of the multilayer neural network, genetic programming, and multiple regression in predicting the compression index. They used 373 oedometer test samples to develop the machine learning models. The best prediction model was established based on the input variables: wet density, water content, liquid limit, plastic index, void ratio, and fine contents [ 27 ].
Overall, the above-mentioned researches mainly focus on the ANN or ANN-based models. To the best knowledge of the authors, no relevant researches discuss the application prospect of tree-based models in predicting the compression index of clayey soils. Considering the merits of tree-based models, for example, they can handle a large number of features and still maintain good accuracy, and they are easy to be interpreted and explained because they are based on a set of decision trees [ 28 ], we propose a hypothesis: the tree-based model could perform well in this topic. Based on this, we will develop the models for predicting the compression index using the tree-based technique. The developed tree-based models are random forest (RF) and extreme gradient boosting (XGBoost). First, we collected a dataset of clayey soils from a published article (Ref. [ 29 ]) to establish these two tree-based models. Meanwhile, we used the grid search algorithm to seek the optimal hyperparameters of the models. By comparing their performance using some evaluation metrics, we finally determined the best model for predicting the compression index of clayey soils. Our main contribution is: we verified the promising application of tree-based models in predicting the compression index.
The rest of the paper is organized as follows: Section 2 presents the background of the data source; Section 3 describes the principle and implementation of the tree-based models; Section 4 discusses the main results of modeling; Section 5 summarizes the main conclusions.
2. Materials
In the present study, we collected a dataset that includes 391 experimental samples from a previously published article. The dataset is composed of the experimental results of consolidation testes (ASTM D 2435-96) for soils that were sampled at 125 construction sites in the north of Iran [ 29 ]. It mainly contains the physical properties of clayey soils, such as natural water content (ω_{n}), liquid limit (LL), plastic index (PI), initial void ratio (e_{0}), the specific gravity of soil particles (G_{s}), and compression index (C_{c}). Our goal is to build an effective relationship between the compression index and another five physical properties of clayey soils, with the help of the tree-based machine learning models.
Before beginning to develop the tree-based models, we need to do pre-processing on the dataset. Since the experimental tests may be subject to human-induced error, outliers could exist in the dataset, which will harm the performance of tree-based models. Thus, we use the boxplot method to detect the outliers of the dataset—which is a common way in statistics [ 30 ]. Boxplot can show the visualization of the five-number summary: the extreme lower (Min), the extreme upper (Max), the first quartile (Q1), the third quartile (Q3), and the median. Figure 1 shows the data distribution of the physical properties of clayey soils. The box extends from Q1 to Q3 of the data; the red line and rhombus point represent the median and mean values, respectively; and the black circle point denotes the outliers of each variable [ 31 ]. Intuitively, the outliers exist in each variable and should be removed. After removing the outliers, 349 data samples were available to develop the tree-based models. Table 1 presents the statistical indices of each variable in the new dataset. We can find that the range of natural water content is between 12.7% and 42.1%; the range of liquid limit is between 24% and 62%; the range of the plastic index is between 4% and 37%; the range of initial void ratio is between 0.476 and 1.059; the range of specific gravity of soil particles is between 2.5 and 2.77; the range of compression index is between 0.05 and 0.385.
Variables | Symbol | Unit | Min. | Max. | Mean | Std. Dev. |
---|---|---|---|---|---|---|
Natural water content | ω_{n} | % | 12.7 | 42.1 | 27.418 | 5.427 |
Liquid limit | LL | % | 24 | 62 | 38.883 | 8.675 |
Plastic index | PI | % | 4 | 37 | 17.848 | 7.633 |
Initial void ratio | e_{0} | - | 0.476 | 1.059 | 0.739 | 0.115 |
Specific gravity of soil particles | G_{s} | - | 2.5 | 2.77 | 2.64 | 0.054 |
Compression index | C_{c} | - | 0.05 | 0.385 | 0.194 | 0.059 |
When conducting the modeling, a common way is to divide the entire dataset into two parts: the training dataset and the testing dataset. In this way, it can effectively examine the model’s generalization ability and help in avoiding overfitting. Thus, we randomly split the 349 data samples into two sets: one is the training dataset (90% of the entire data) which has 314 samples, and the other one is the testing dataset (10% of the entire data) which has 35 samples. Such a splitting strategy is consistent with the published article [ 29 ], and we anticipate verifying whether our developed models have a better performance compared with the model in that published article. We use the training dataset to establish the tree-based models for predicting the compression index and then use the testing dataset the examine the models’ generalization ability. To make the random division valid, a key rule that should be obeyed is to keep the training and testing datasets have similar statistical properties. Herein, we used the cumulative distribution function to judge whether the training dataset has acceptable statistical similarity with the testing dataset.
A cumulative distribution function can describe the probability distribution of a continuous random variable, and it is a non-decreasing function that ranges from 0 to 1 as the value of the random variable increases from negative infinity to positive infinity [ 32 ]. Figure 2 illustrates the cumulative distribution of variables in the training and testing datasets. We find that the variables: ω_{n}, e_{0}, G_{s}, and C_{c}, in both the training and testing datasets, have similar tendencies and shapes. But for variables: LL and PI, they have slight differences because the line’s position of the testing dataset is below that of the training dataset. This might be because the training dataset involves more instances compared with the testing dataset, which incurs that the variables (LL and PI) have lower cumulative probabilities. Additionally, we also observe that the range of each variable in the training dataset almost covers that in the testing dataset—according to the x-axis in each subplot. This can confirm that the models fitted on the training dataset would show promising performance on the testing dataset. From the above analysis, we believe that the division of the training and testing dataset is reasonable, and they can be used to conduct the modeling accordingly.
Furthermore, we present the linear relationship between the compression index and each variable—only applied to the training dataset, as shown in Figure 3. Intuitively, the initial void ratio (e_{0}) has a relatively strong relationship with the compression index, followed by the natural water content (ω_{n}). As for the liquid limit (LL), plastic index (PI), and specific gravity of soil particles (G_{s}), they all show an insignificant relationship with the compression index. From this point, we believe that a sophisticated model should be constructed to characterize the intrinsic relationship between these variables and the compression index. The subsequent sections will discuss it deeply.
3. Methods
3.1. Random forest (RF)
RF is a supervised machine learning algorithm that is used for both classification and regression tasks [ 33 ]. It is an ensemble model that is composed of multiple decision trees, which are trained on different samples of the training data and then aggregated to make a final prediction. The key idea behind RF is to create a diverse set of decision trees, each of which is trained on a randomly selected subset of the training data and a randomly selected subset of the features. This process, known as bootstrapping, helps to reduce the variance of the model and make it more robust. During the training process, each decision tree in the RF makes a prediction based on the features in its training set. The final prediction of the RF is then made by aggregating the predictions of all the individual decision trees, for example, by taking the average for regression tasks, as shown below:
$\begin{array}{cc}y=\frac{1}{K}{\sum}_{\mathrm{i=1}}^{K}{T}_{i}\left(x\right)& \mathrm{(1)}\end{array}$
where y represents the average of prediction results, K is the number of decision trees, and T_{i}(x) represents the predicted results of a single decision tree.
One of the main benefits of using a random forest model is that it can handle large amounts of data and a high number of features, and it is also resistant to overfitting. Additionally, it is relatively easy to interpret and understand, as the individual decision trees are simple models that can be inspected and analyzed.
RF has two key hyperparameters: the number of trees and the maximum depth, which can highly affect its performance [ 34 ]. The number of trees denoted how many decision trees are in a forest and it dramatically controls the prediction accuracy of the RF model. For another hyperparameter: maximum depth, its role is to reduce the RF model’s complexity to avoid possible overfitting. In the present study, we aim to seek the optimal values of these two hyperparameters and thus construct a high-performance RF model for predicting compression index.
3.2. Extreme gradient boosting (XGBoost)
Extreme gradient boosting (XGBoost) is a supervised machine learning algorithm that is used for both classification and regression tasks [ 35 ]. It is an ensemble model that is composed of multiple decision trees, which are trained sequentially in a way that allows the model to learn and improve from the mistakes made by earlier trees.
XGBoost is a variant of the gradient boosting algorithm, which is a type of boosting algorithm that is based on the concept of boosting weak learners to form a strong learner. Boosting algorithms work by iteratively adding weak models to the ensemble and adjusting the weights of the training data so that the mistakes made by the previous models are emphasized and corrected in the subsequent models [ 36 ]. In the XGBoost model, the decision trees are trained using gradient descent to minimize the loss function, which measures the difference between the predicted values and the true values in the training data. The loss function of the XGBoost model is shown below:
$\begin{array}{cc}{X}_{\mathrm{obj}}={\sum}_{\mathrm{i=1}}^{n}l\left(\mathrm{y,}\stackrel{\u02c6}{y}\right)+{\sum}_{\mathrm{k=1}}^{K}\Omega \left({f}_{k}\right)& \mathrm{(2)}\end{array}$
where X_{obj} represents the objective function, ${\sum}_{\mathrm{i=1}}^{n}l\left(\mathrm{y,}\stackrel{\u02c6}{y}\right)$ represent the predictive loss between the predicted and real values, ${\sum}_{\mathrm{k=1}}^{K}\Omega \left({f}_{k}\right)$ represents the regularization term that is used to avoid overfitting. In general, the technique of minimizing a quadratic function is the way to optimize the objective function [ 37 ].
When constructing the XGBoost model, two key hyperparameters should be considered, that is, the number of trees and the learning rate. The number of trees refers to the maximum number of gradient-boosted trees. It controls the predictive accuracy of the XGBoost model. In general, if its value is too low/high, the model will encounter underfitting/overfitting. The learning rate refers to the step size shrinkage in each iteration. It can make the boosting process more conservative. In the present study, we aim to seek the optimal values of these two hyperparameters and thus construct a high-performance XGBoost model for predicting compression index.
3.3. Evaluation criteria
To quantitatively assess the accuracy of the above-mentioned RF and XGBoost models, some commonly used regression evaluation metrics are utilized, for example, coefficient of determination (R^{2}), root mean squared error (RMSE), mean absolute percentage error (MAPE), and mean absolute error (MAE). The following equations are used to compute these metrics.
$\begin{array}{cc}{R}^{2}=1-\frac{{\sum}_{\mathrm{i=1}}^{N}{({y}_{i}-{\stackrel{\u02c6}{y}}_{i})}^{2}}{{\sum}_{\mathrm{i=1}}^{N}{({y}_{i}-\stackrel{-}{y})}^{2}}& \mathrm{(3)}\end{array}$
$\begin{array}{cc}\mathrm{RMSE}=\sqrt{\frac{1}{N}{\sum}_{\mathrm{i=1}}^{N}{({y}_{i}-{\stackrel{\u02c6}{y}}_{i})}^{2}}& \mathrm{(4)}\end{array}$
$\begin{array}{cc}\mathrm{MAPE}=\frac{1}{N}{\sum}_{\mathrm{i=1}}^{N}\left|\frac{{y}_{i}-{\stackrel{\u02c6}{y}}_{i}}{{y}_{i}}\right|\times \mathrm{100\; \%}& \mathrm{(5)}\end{array}$
$\begin{array}{cc}\mathrm{MAE}=\frac{1}{N}{\sum}_{\mathrm{i=1}}^{N}|{y}_{i}-{\stackrel{\u02c6}{y}}_{i}|& \mathrm{(6)}\end{array}$
where y_{i} denotes the measured compression index, ${\stackrel{\u02c6}{y}}_{i}$ denotes the predicted compression index, $\stackrel{-}{y}$ denotes the average of y_{i}, and N is the number of samples. For the above four metrics, the closer the R^{2} to 1, the better the model’s performance; the smaller the RMSE, MAPE, and MAE, the better the model’s performance.
3.4. Study step
The main step of the research method in the present study are as below:
As mentioned previously, the entire dataset is divided into two parts: the training dataset involving 314 soil samples and the testing dataset involving 35 soil samples. Then, we use the training dataset to establish the RF and XGBoost models, respectively. In this process, the grid search algorithm is employed to seek the optimal hyperparameters of the RF and XGBoost models [ 38 ]. The hyperparameters of the RF model are the number of trees and the maximum depth, and the hyperparameters of the XGBoost model are the number of trees and the learning rate. Meanwhile, a five-fold cross-validation algorithm is used when training the RF and XGBoost models, which aims to help the models avoid overfitting. After determining the hyperparameters of the RF and XGBoost models, we use the testing dataset to examine their generalization ability. Lastly, we also analyze which variable is highly sensitive for predicting the compression index of clayey soils. Figure 4 displays the flowchart of the present study. In the present study, we used two open-source Python libraries: Scikit-learn [ 39 ] and XGBoost [ 35 , 36 ] to develop the RF and XGBoost models, respectively.
4. Results
4.1. Evaluation of model performance
In this section, we trained the RF and XGBoost on the same training dataset and then determined their respective optimal hyperparameters. First, we defined the searching domain of these two models, as shown in Table 2. Regarding the RF model, the searching domain of its hyper-parameters is: the number of trees increases from 50 to 300 with the increment of 10, and the maximum depth increases from 1 to 20 with the increment of 1. Regarding the XGBoost model, the searching domain of its hyperparameters is: the number of trees increases from 50 to 300 with the increment of 10, and the learning rate increases from 0.01 to 0.30 with the increment of 0.01. We then use the mean squared error (MSE) as an evaluation metric to determine the optimal hyperparameters of each model.
Model | Hyperparameter | Searching domain | Increment |
---|---|---|---|
RF | Number of trees | [50, 300] | 10 |
Maximum depth | [1, 20] | 1 | |
XGBoost | Number of trees | [50, 300] | 10 |
Learning rate | [0.01, 0.30] | 0.01 |
Figure 5 illustrates the possible results of hyperparameters of the RF model. We can find that the MSE reached a relatively large value when the maximum depth is less than 7. When the maximum depth is larger than 7, the value of MSE does not fluctuate strongly. Another point is that the maximum depth has a significant influence on the MSE compared with the number of trees because the MSE significantly reduced with the increase of the maximum depth. Conclusively, according to Figure 5 (b), the optimal hyperparameters of the RF model are: the number of trees is 130 and the maximum depth is 10.
Figure 6 illustrates the possible results of hyperparameters of the XGBoost model. We can find that the MSE reached a relatively large value only when the number of trees and learning rate are both in small values. For other cases, the MSE does not have obvious changes. According to Figure 6 (b), we determined the optimal hyperparameters of the XGBoost model, that is, the number of trees is 60 and the learning rate is 0.14.
After figuring out the values of the hyperparameters of each model, we use the obtained hyperparameters to construct the RF and XGBoost models, respectively. Subsequently, we examined their performance on both training and testing datasets. At the same time, we also compared them with the model (ANN) in the published article [ 29 ]. Table 3 shows the performance of the RF and XGBoost models on the training and testing datasets. As a result, the RF model has the lowest error on both training and testing datasets compared with the XGBoost and ANN models. Its performance indices are as follows: R^{2} of 0.928, RMSE of 0.016, MAPE of 7.046%, and MAE of 0.012 on the training dataset; R^{2} of 0.818, RMSE of 0.025, MAPE of 10.082%, and MAE of 0.020 on the testing dataset. Additionally, we can also find that both RF and XGBoost models outperform the ANN model. Thus, we conclude that the tree-based models have a promising prospect of predicting the compression index of clayey soils.
Figure 7 shows the comparison between the experimental compression index and the predicted compression index by the RF model. Intuitively, for the training dataset, almost all the data points are concentrated around the black dashed line. This indicates the compression index predicted by the RF model approximates the experimental compression index. As for the testing dataset, most of the data points are concentrated around the black dashed line, but several data points are not. This indicates although the generalization ability of the current RF model is acceptable, it still needs further improvement. Overall, the developed RF model shows acceptable and effective performance on both the training and testing datasets.
Model | Training dataset | Testing dataset | ||||||
---|---|---|---|---|---|---|---|---|
R^{2} | RMSE | MAPE (%) | MAE | R^{2} | RMSE | MAPE (%) | MAE | |
RF | 0.928 | 0.016 | 7.046 | 0.012 | 0.818 | 0.025 | 10.082 | 0.020 |
XGB | 0.832 | 0.024 | 10.933 | 0.019 | 0.833 | 0.026 | 11.125 | 0.021 |
ANN [29] | - | 0.035 | 13.340 | 0.027 | - | 0.034 | 13.170 | 0.027 |
4.2. Sensitivity analysis
From the above analysis, we have successfully obtained the best tree-based model for predicting compression index, namely, the RF model. In this section, we will figure out which variable shows the highest influence on predicting compression index when using the RF model. The RF model has an intrinsic attribute: feature importance, which can measure the importance of each feature when constructing a split node in a decision tree. The standard for constructing the split node is “squared error” when the prediction is a regression task. Its main principle is to minimize the L2 loss using the mean of each split node [ 40 ]. In short, the more times the feature is used in a split node to minimize the L2 loss, the higher its importance. Based on this, we can obtain the importance of each feature (variable), as shown in Figure 8. Intuitively, for the present engineering instance, the variable e_{0}, i.e., the initial void ratio, shows the highest impact on predicting compression index; the variables G_{s} and ω_{n}, i.e., the specific gravity of soil particles and natural water content, show relatively slight impact on predicting compression index; the variables PI and LL, i.e., the plastic index and the liquid index, show negligible impact on prediction compression index. As a result, we conclude that the initial void ratio should be a significant concern in predicting the compression index.
Further, to identify the specific effect of the initial void ratio (e_{0}) on the compression index, we used the Partial Dependence Plots and Individual Conditional Expectation plots to achieve visualization and analysis of the interaction between the initial void ratio and the compression index. In general, Partial Dependence Plots can show the average (overall) dependence between the target response and the input feature of interest [ 41 ]. Individual Conditional Expectation plots can reflect the individual dependence between the target response and the input feature of interest—based on the selected data samples [ 42 ]. Figure 9 shows the particular effect of the initial void ratio on the compression index. The red dashed line represents the average dependence between the initial void ratio and the compression index. Intuitively, the compression index increases with the increase of the initial void ratio, especially when the initial void ratio is between 0.58 and 0.90. However, when the initial void ratio is between 0.476 and 0.58 as well as 0.90 and 1.059, the compression index is almost unchanged. As for the individual dependence between the initial void ratio and the compression index (all blue lines), most of the data samples present a similar trend to the red dashed line—although few of them are fluctuant. In summary, the relationship between the initial void ratio and the compression index is approximately positive linear, which is beneficial for us to determine the compression index. Some published studies also pointed out that the compression index of clayey soils is highly dependent on the initial void ratio. For instance, Tiwari and Ajmera reported a significant linear relationship between the compression index and the initial void ratio [ 43 ]. Akbarimehe et al. also concluded a valid linear correlation between the compression index and the initial void ratio through the consolidation tests [ 44 ]. Erzin et al. developed an empirical formula based on a robust optimization model and found that the compression index is more sensitive to the initial void ratio [ 45 ].
5. Conclusion
In the present study, we proposed two tree-based models (RF and XGBoost) to predict the compression index of clayey soils. First, a dataset for soil consolidation tests, collected from a previously published work, was utilized to develop the tree-based models. In the tree-based models, the input parameters included natural water content (ω_{n}), liquid limit (LL), plastic index (PI), initial void ratio (e_{0}), and specific gravity of soil particles (G_{s}), whereas the compression index is the target output. Then, we used a grid search algorithm to seek the optimal hyperparameters of the tree-based models. As a result, the optimal hyperparameters of the RF model are: number of trees = 130, maximum depth = 10, and the optimal hyperparameters of the XGBoost model are: number of trees = 60, learning rate = 0.14. By comparing their performance on both the training and testing datasets, we found that the RF model outperformed the XGBoost model. The RF model obtained the lower errors when implementing the task of predicting the compression index of clayey soils, evidenced by R^{2} of 0.928 and 0.818, RMSE of 0.016 and 0.025, MAPE of 7.046% and 10.082%, and MAE of 0.012 and 0.020 on the training and testing datasets, respectively. This confirms that the RF model can help in reducing the cost of implementing laboratory experiments to determine the compression index of clayey soils. Furthermore, according to the feature importance of input parameters in the RF model, we found that the initial void ratio (e_{0}) has a significant impact on predicting the compression index in the present engineering instance. This is beneficial for engineers to understand the compression characteristics of clayey soils—we emphasize an approximately positive linear relationship between the initial void ratio (e_{0}) and the compression index of clayey soils and we recommend the engineers focus on this point when dealing with similar scenarios.
Acknowledgments
The authors would like to appreciate the Faculty of Engineering, Universiti Malaya, and the facilities provided which enabled the study to be carried out.
Funding
This research received no external funding.
Conflicts of interest
The authors declare no conflict of interest.
Authors contribution statement
L.T., B.H., A.G., S.M.H.K.: Conceptualization; L.T., B.H.: Data curation; L.T., B.H.: Formal analysis; L.T., B.H., A.G., S.M.H.K.: Investigation; L.T., B.H.: Methodology; L.T., B.H.: Software; A.G., S.M.H.K.: Supervision; A.G., S.M.H.K.: Validation; L.T., B.H.: Visualization; L.T., B.H., A.G., S.M.H.K.: Roles/Writing - original draft; L.T., B.H., A.G., S.M.H.K.: Writing - review & editing.
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