Compression Index Regression of Fine-Grained Soils with Machine Learning Algorithms

Abstract

Soil consolidation, particularly in fine-grained soils like clay, is crucial in predicting settlement and ensuring the stability of structures. Additionally, the compressibility of fine-grained soils is of critical importance not only in civil engineering but also in various other fields of study. The compression index (C_c), derived from soil properties such as the liquid limit (LL), plastic limit (PL), plasticity index (PI), water content (w), initial void ratio (e₀), and specific gravity (G_s), plays a vital role in understanding soil behavior. This study employs machine learning algorithms—the random forest regressor (RFR), gradient boosting regressor (GBR), and AdaBoost regressor (ABR)—to predict the C_c values based on a dataset comprising 915 samples. The dataset includes LL, PL, W, PI, G_s, and e₀ as the inputs, with C_c as the output parameter. The algorithms are trained and evaluated using metrics such as the coefficient of determination (R²), mean absolute error (MAE), mean squared error (MSE), and root mean squared error (RMSE). Hyperparameter optimization is performed to enhance the model performance. The best-performing model, the GBR model, achieves a training R² of 0.925 and a testing R² of 0.930 with the input combination [w, PL, LL, PI, e₀, G_s]. The RFR model follows closely, with a training R² of 0.970 and a testing R² of 0.926 using the same input combination. The ABR model records a training R² of 0.847 and a testing R² of 0.921 under similar conditions. These results indicate superior predictive accuracy compared to previous studies using traditional statistical and machine learning methods. Machine learning algorithms, specifically the gradient boosting regressor and random forest regressor, demonstrate substantial potential in predicting the C_c value for fine-grained soils based on multiple soil parameters. This study involves leveraging the efficiency and effectiveness of these algorithms in geotechnical engineering applications, offering a promising alternative to traditional oedometer testing methods. Accurately predicting the compression index can significantly aid in the assessment of soil settlement and the design of stable foundations, thereby reducing the time and costs associated with laboratory testing.

1. Introduction

The compressibility of fine-grained soils is critically important across multiple fields of study. In civil engineering, the compression characteristics of fine-grained soils under a load affects the stability of structures such as buildings and roads; therefore, engineers must carefully account for this property to prevent issues like differential settlement and structural damage. In addition, in environmental engineering, understanding soil’s low permeability due to its compressibility helps in designing effective barriers against contaminant migration in groundwater [1,2,3]. This knowledge is vital for safeguarding water resources and managing hazardous waste disposal sites. Moreover, soil compressibility influences soil fertility, water retention, and drainage, directly impacting crop growth in agriculture [4,5,6]. Farmers adapt their irrigation and drainage practices to optimize agricultural productivity and prevent soil erosion. Additionally, in soil science, soil’s compressibility is integral to soil classification systems and understanding soil behavior under different conditions. This knowledge supports land use planning, environmental assessments, and sustainable soil management practices. Overall, a comprehensive understanding of the compressibility of soils enhances our ability to address challenges in infrastructure stability, environmental protection, agricultural productivity, and soil conservation across various disciplines.

Consolidation is a process applied to fine-grained soils. It involves slowly compressing the water in these soils under a load and removing it through drainage. Fine-grained soils, particularly clay, exhibit unique characteristics such as high compressibility, which play crucial roles across various disciplines. In the engineering field, this process increases the load carrying ability of the ground and minimizes long-term settlement, significantly affecting structural strength. Consolidation is particularly critical in fine-grained soils like clay because they react more slowly to the movement of water under a load. The compressibility of soils is quantified using the compression index (C_c), an important measurement for understanding settlement behavior. The compression index indicates the logarithmic amount of settlement per unit load increment and is typically determined in the laboratory through consolidation tests. This parameter is crucial in engineering applications for predicting soil behavior and developing appropriate foundation designs [7].

In geotechnical engineering, the evaluation and prediction of soil behavior under various loading conditions are critical for the stability and longevity of civil engineering structures. Key soil properties influencing such behavior include the liquid limit (LL), plastic limit (PL), water content (w), plasticity index (PI), specific gravity (G_s), and initial void ratio (e₀), all of which are integral to calculating the compression index of soils. The liquid limit (LL) denotes the water content level at which the soil transitions from liquid to plastic, while the plastic limit (PL) marks the shift from plastic to semi-solid, essential for characterizing soil plasticity [8]. Water content directly impacts soil’s consistency and compressibility and is pivotal in soil behavior analysis [9]. The plasticity index, derived from the difference between the LL and PL, provides a comprehensive measure of soil plasticity, influencing its mechanical properties [7]. The initial void ratio (e₀), expressing the volumetric relationship between soil voids and solids, significantly determines the soil compression and settlement potential under a load (Head, 1980). Furthermore, specific gravity (G_s), evaluating the particle density of soil relative to water, is crucial for understanding soil compressibility and stress behavior [10]. These parameters collectively contribute to determining the C_c value, a critical metric for predicting soil compression under loading [11]. The interrelation and influence of these soil properties on the C_c value are fundamental in geotechnical investigations and form the basis of effective soil behavior modeling and predictive analysis in civil engineering. From the 1950s to the present, research on consolidation processes has prompted the development of several important equations. These equations are critical in geotechnical engineering for understanding and predicting the compression behavior of fine-grained soils. These mathematical models are utilized to forecast how soils will respond under a load, offering precise settlement predictions for engineering applications.

Table 1. Evolution of the prediction equations for the compression index (C_c) over history.

No.	Empirical Equation	Soil Type	Reference
(1)	$C_c=0.75\left(e_0-0.5\right)$	Low-plasticity clayey soil	Sowers and Sowers [12]
(2)	$C_c=1.15\left(e-0.35\right)$	Clay (from Rio de Janeiro, Santos, Sao Paulo, Minas Gerais, and Mexico City)	Nishida [13]
(3)	$C_c=0.013\left(\mathrm{LL}-13.5\right)$	Clay	Yamagutshi [14]
(4)	$C_c=0.017\left(\mathrm{LL}-20\right)$	Clay	Shouka [15]
(5)	$C_c=0.009\left(\mathrm{LL}-10\right)$	Normally consolidated clays	Terzaghi et al. [11]
(6)	$C_c=0.83\left(\mathrm{LL}-0.09\right)$	Weald clay London clay Kaolin	Schofield and Wroth [16]
(7)	$C_c=0.0024\mathrm{LL}+0.071$ $C_c=0.261e_0-0.04$	Soft silty clays (from the lowlands of Santos) Clay and silty clays (from Sao Paulo)	Ozdikmen [17]
(8)	$C_c=0.4(e_0+0.1001w_n-0.25$) $C_c=0.006\left(\mathrm{LL}-9\right)$ $C_c=0.01\left(w_n-5\right)$ $C_c=0.4\left(e_0-0.25\right)$	Clay	Azzouz et al. [18]
(9)	$C_c=0.01w_n$ $C_c=0.009w_n+0.005\mathrm{LL}$	Cohesive soils	Koppula [19]
(10)	$C_c=0.2343\left(\mathrm{LL}/100\right)G_s$ $C_c=0.2343w_n{G}_s$	Clay	Nagaraj and Murthy [20]
(11)	$C_c=0.003\mathrm{LL}\left(1+e_0\right)$	Normally consolidated clays	Pandian and Nagaraj [21]
(12)	$C_c=0.01w_n$ $C_c=-0.156+0.411e_0+0.00058LL$	Colloidal-size particle soils	Al-Khafaji and Andersland [22]
(13)	$C_c=0.008\left(\mathrm{LL}-12\right)$ $C_c=0.007\left(\mathrm{PI}+18\right)$ $C_c=0.014\left(\mathrm{PI}+3.6\right)$	Low-plasticity clays Low-plasticity silts Highly plastic silts	Sridharan et al. [23]
(14)	$C_c=0.0103w_n$	Soft clays	Nagaraj and Miura [24]
(15)	$C_c=1.235w_n-5.65$	Low-plasticity clays Highly plastic clays Normally consolidated clays Over-consolidated clays	Lav and Ansal [25]
(16)	$C_c=-0.404+0.341e_0+0.006w_n+0.004\mathrm{LL}$	Marine clays	Yoon and Kim [26]
(17)	$C_c=0.013w_n-0.115$ $C_c=0.49e_0-0.11$ $C_c=2.926\left(\mathrm{LL}/100\right)G_s$ $C_c=0.014\mathrm{LL}-0.168$ $C_c=0.013w_n+0.168$	Clay	Park and Lee [27]

summarizes the evolution of these equations over time.

The use of computational intelligence methods, such as artificial neural networks (ANNs), is extensively combined with traditional statistical techniques like regressions [28,29,30]. These techniques have been widely employed in various geotechnical problems, such as estimating the C_c parameter [31,32,33]. However, different solution approaches for these problems yield varying results. The goal is to identify the algorithms that yield the most accurate predictions. Deep learning and machine learning algorithms, which are sub-branches of artificial intelligence, offer predictive equations.

An area of data science and artificial intelligence called machine learning (ML) makes use of algorithms and statistical models to facilitate the automated acquisition of knowledge by computers via experience so they can perform certain tasks without being explicitly programmed. This process provides the ability to recognize patterns from data and make decisions, allowing machines to solve complex problems and make predictions [34]. In fields such as geotechnical engineering and consolidation, machine learning methods are increasingly used to more accurately predict soil properties and behavior, model soil compaction and consolidation, and optimize engineering designs. These techniques can improve risk assessments, streamline construction processes, and increase the reliability of engineering solutions [35].

Studies have been published in the literature that accurately predict the compression index using various machine learning methods. Utilizing an artificial neural network (ANN), a machine learning technique, researchers predicted the compression index using a dataset that included parameters like w, LL, PI, and G_s [27]. Benbouras et al. [36] used ANN models, genetic programming, and multiple regression analysis to estimate the compression index on a dataset that contained LL, w, FCs (fine contents), P_h (wet density), PI, e₀, and soil types. In addition, Tsang et al. [37] employed machine learning techniques like extreme gradient boosting and random forest models on a dataset comprising e₀, w, LL, PI, and G_s to predict the compression index. Also, Chia et al. [38] utilized random forest (RF) and gradient boosting tree (GBT), tree-based machine learning techniques, with input parameters including LL, e₀, PL, and moisture content (MC) to determine the value of C_c. Lastly, Kim et al. [39] predicted the compression index using an ANN model, a form of deep learning, on a dataset containing w, PI, LL, and the compression index (C_c).

Artificial intelligence methods are increasingly being used in various soil and rock analyses in geotechnical engineering, yielding successful results. Baghbani et al. [40] reported that artificial neural networks (ANNs) are the most widely used method and emphasized that artificial intelligence methods successfully predict complex relationships in geotechnical engineering, particularly in areas such as frozen soils, rock mechanics, and slope stability. Qader et al. [41] analyzed the geotechnical properties of soils in Erbil using geographic information systems (GISs) and artificial neural networks (ANNs), successfully estimating parameters such as SPT-N and the bearing capacity of soils, which were predominantly in the silty clay (CL) class. Their study demonstrated that these techniques provide valuable information for soil analysis. Additionally, Jolfaei and Troncone [42] examined the breakout dimensions caused by drilling in a rock environment using the finite element method. They then developed a model that estimated the breakout dimensions using artificial neural networks, finding that the internal friction angle has the greatest influence on the breakout dimensions.

The primary aim of this work is to conduct a comparative evaluation of various machine learning algorithms for predicting the compression index (C_c) of soils, using soil properties such as LL, PL, w, PI, G_s, and e₀ as the inputs. The main objective is to identify the most effective ML algorithms that accurately predict the C_c value, thereby enhancing our understanding of soil behavior under a load. To achieve this goal, we employed various machine learning approaches in this study, including random forest, gradient boosting, and AdaBoost regressors. Each algorithm was rigorously evaluated using regression metrics such as the coefficient of determination (R²) and error metrics including the mean absolute error (MAE), mean squared error (MSE), and root mean squared error (RMSE). Additionally, this study aims to investigate the impact of each input parameter on the predictive capability of the algorithms and provide insights into their relative effects on soil compressibility. The ultimate objective is to develop a reliable and efficient prediction tool that can assist geotechnical engineers in assessing soil behavior for a wide range of engineering applications, thereby reducing the necessity for extensive and costly laboratory testing.

2. Dataset and Methods

2.1. Dataset

In this study, a new dataset was created by carefully selecting the data used in various studies from the literature [43,44,45,46,47]. The features of the new dataset were standardized by removing the mean and scaling the unit variance. The input parameters of the dataset included the liquid limit (LL), plastic limit (PL), water content (w), plasticity index (PI), specific gravity (G_s), and initial void ratio (e₀), and the output parameter was the compression index (C_c). A classical technique called train–test split, used as an alternative to the cross-validation method, was employed to determine the most appropriate train and test split ratio by experimenting with several different splits. Based on the optimal train–test split ratio, the training and testing sets contained 732 and 183 samples, respectively. As a result of this technique, all the data in the testing set were entirely unseen during the training phase.

Table 2. Statistic results of the dataset variables.

Variable	Minimum Value	Mean	Maximum Value	Standard Deviation
Water content, w	10.20	34.10	99.00	13.64
Plastic index, PI	0.00	24.16	96.60	12.60
Plastic limit, PL	0.00	48.07	103.00	17.16
Liquid limit, LL	0.00	25.13	87.20	7.85
Initial void ratio, e0	0.36	0.96	3.50	0.38
Specific gravity, Gs	2.43	2.67	2.83	0.07
Compression index, Cc	0.05	0.29	1.00	0.17

contains statistical numerical data obtained from estimating the minimum value, arithmetic mean, maximum value, and standard deviation of the dataset variables. In addition, Figure 1 displays the value distributions of the dataset variables.

A statistical tool known as Spearman’s correlation coefficient was used to measure the direction and degree of the relationship between two variables. The coefficient value for these variable spans from −1 to 1, with 1 representing a fully positive monotonic connection, −1 representing a fully negative monotonic relationship, and 0 indicating the absence of a monotonic relationship between the variables. Figure 2 shows a Spearman’s correlation coefficient matrix of all the parameters in the dataset in this study. The p-value was used to assess whether the correlation between two variables was statistically significant. Generally, if the p-value is 0.05 or lower, the relationship is considered non-random and statistically significant. Figure 3 demonstrates the Spearman’s correlation coefficient p-value matrix of all the parameters in the dataset in this study.

Table 3. Spearman’s correlation coefficient and its p-value with compression index.

Variable	Spearman’s Correlation Coefficient with Compression Index	Spearman’s Correlation Coefficient p-Value with Compression Index
Water content, w	0.7717	$1.143\times {10}^{-181}$
Plastic index, PI	0.4914	$8.571\times {10}^{-57}$
Plastic limit, PL	0.4222	$7.209\times {10}^{-41}$
Liquid limit, LL	0.4846	$4.575\times {10}^{-55}$
Initial void ratio, e0	0.8517	$1.492\times {10}^{-258}$
Specific gravity, Gs	0.0504	0.1273

The water content (w) shows a robust positive correlation with C_c, with a value of 0.7717, while the initial void ratio (e₀) shows an even stronger relationship with C_c, with a correlation coefficient of 0.8517. This suggests that increases in the water content and initial void ratio may also increase C_c. However, the p-value of 1.143 $\times {10}^{-181}$ obtained for water content (w) indicates that this correlation may not be statistically significant, so caution should be exercised in interpreting these results. Other parameters also show positive correlations with C_c, with correlation coefficients of 0.4846 for LL, 0.4222 for PL, and 0.4914 for PI. The p-values for these parameters were calculated as 4.575 $\times {10}^{-55}$ for LL, 7.209 $\times {10}^{-41}$ for PL, and 8.571 $\times {10}^{-57}$ for PI, indicating that these correlations are statistically significant. On the other hand, the correlation coefficient of 0.0504 obtained for specific gravity (G_s) indicates that there is almost no relationship with C_c, which is supported by the p-value of 0.1273. This confirms that specific gravity (G_s) does not have a significant impact on the compression index (C_c).

lists Spearman’s correlation coefficient and its p-value for the input parameters LL, w, PL, PI, G_s, and e₀ and the output parameter, C_c.

2.2. Machine Learning Algorithms

The field of machine learning (ML) encompasses the development of algorithms with the capacity to learn and make predictions from data without human intervention. Essentially, machine learning algorithms discover patterns and relationships by analyzing datasets and use this information to make predictions or decisions on new data. Among the most prevalent uses of machine learning is regression. Regression is a machine learning task that aims to predict numerical values. For example, predicting the cost of a house based on its characteristics (e.g., size, location, and age) is a regression problem. The algorithm learns the relationship between features and target values in the given dataset and uses this information to make predictions on new data. Regression is used in a variety of areas, from financial forecasting to risk assessments in the healthcare industry.

The success of machine learning depends on selecting the right datasets, using effective algorithms, and constantly evaluating the performance of the model. This process is becoming increasingly important in more industries due to its capacity to understand and analyze the complexity and diversity of data [48,49].

Machine learning algorithms such as random forest, gradient boosting, and AdaBoost each have their own strengths and weaknesses. The random forest algorithm reduces the risk of overfitting by using multiple decision trees and generally produces more stable results. However, when the model contains too many trees, the training time can be longer, and its performance may slow down on large datasets. The gradient boosting algorithm can provide high accuracy by correcting errors at each step and is particularly effective on complex datasets. However, its downside is that training can be time-consuming, and it may be prone to overfitting. The AdaBoost algorithm focuses on boosting simple models; while it can be effective on low-complexity datasets, it is sensitive to noisy data and may amplify small errors in the dataset. The choice of these models depends on the structure of the dataset and the nature of the problem.

2.2.1. The Random Forest Regressor Algorithm

The random forest regressor algorithm, first created by Breiman [50] and then expanded upon by Geurts et al. [51], is an ensemble learning technique that generates several decision trees during the training process and calculates the average prediction of these trees for regression problems. Each tree is built with randomly selected samples from the training set (e.g., bootstrap sample), and at each node, the best split is selected from a random subset of features, introducing useful randomness into the model [50]. Geurts et al. [51] extended this concept to Extremely Randomized Trees, where the split thresholds for each feature are randomly selected, adding an additional level of randomness. This approach provides a model that is effectively resistant to overfitting, an advantage derived from random feature selection and averaging the predictions from several trees. Although they are capable of modeling complex relationships with high accuracy, random forests can be computationally demanding, especially when composed of many large and deep trees. This method provides improved prediction accuracy and stability in regression tasks due to the combination of decision trees’ simplicity and ensembles’ power.

2.2.2. The Gradient Boosting Regressor Algorithm

Gradient boosting is a popular ML technique for both regression and classification. The model is built incrementally, meaning every new model is constructed with the purpose of rectifying errors caused by the preceding versions. The fundamental principle involves the amalgamation of many weak learners to construct a robust learner, usually decision trees. This algorithm generally starts with a constant, F₀(x), aiming to minimize the loss function. Then, at each stage m up to stage ‘m’, a decision tree h_m(x) is adapted to the loss function’s negative gradient, the basis of the gradient boosting process. The model is iteratively revised with the learning rate v, which scales the contribution of each tree [52,53]. The equation of the model is demonstrated in Equation (1).

$${\mathrm{F}}_{\mathrm{m}}\left(\mathrm{x}\right)={\mathrm{F}}_{\mathrm{m}-1}\left(\mathrm{x}\right)+\mathrm{v}· {\mathrm{h}}_{\mathrm{m}}\left(\mathrm{x}\right)$$

(1)

After ‘m’ iterations, the final model, F(x), emerges as the sum of the initial model and all subsequent learners. Stochastic gradient boosting, an extension of this method, adds randomness by subsampling the training data at each stage, thereby increasing the robustness and generalization ability of the model [53]. As highlighted by [48], this approach is part of a broad ensemble learning framework in which each new tree is built on the sum of all previous trees, minimizing the overall prediction error. This structured methodology allows gradient boosting to effectively handle complex nonlinear interactions, making it a versatile tool for regression and classification in various data types.

2.2.3. The AdaBoost Regressor Algorithm

Conceptualized by Freund and Schapire [54] and further developed by Drucker [55], AdaBoost regression represents a significant advance in ensemble learning techniques. Initially, the algorithm assigns equal weights to all training samples, and these weights are dynamically adjusted in subsequent iterations to emphasize the importance of each sample, a method detailed by Freund and Schapire [54]. The basis of the algorithm lies in sequentially integrating weak learners, usually decision trees; the training of each learner is based on the current weight distribution of the samples. This training allows the model to progressively concentrate on more difficult parts of the data, highlighting samples that are difficult to predict. After training each learner, AdaBoost updates the sample weights, increasing them for incorrectly predicted samples and decreasing them for correctly predicted ones; the amount of updating is determined by the accuracy of the learner.

Drucker’s contribution to regression applications of AdaBoost concerns its adaptation with a customized loss function suitable for regression tasks, which led to the AdaBoost R² algorithm. This adaptation minimizes the specific regression losses, usually by using squared error or absolute error loss functions, thus customizing the algorithm more effectively for continuous outcome prediction. In this context, the overarching objective of AdaBoost is to iteratively enhance the model’s focus on the most difficult-to-predict samples, thus increasing the model’s overall predictive ability across the dataset. The final model, containing the weighted total of all weak learners’ predictions, is robust and accurate, especially in scenarios where the data present complex and nuanced relationships. This adaptive and iterative approach of AdaBoost, especially in its specific form for regression, highlights its effectiveness in handling diverse and complex prediction tasks in various domains.

2.3. Performance Metrics of Regression

The performance metrics of regression are used to measure a model’s predictive ability. The most common error metrics include the mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and R-squared (R²). In Equation (2), MAE is defined as the mean of the differences between the predicted values and the actual values. The MSE, as shown in Equation (3), is the mean of the squared errors, while the RMSE, as shown in Equation (4), measures the magnitude of the prediction errors by taking the square root of the MSE. Lower MAE, MSE, and RMSE values indicate a better model performance. R-squared, also known as the coefficient of determination, shows how much of the data’s variation the model can explain; its value ranges from 0 to 1, with a higher value indicating a better model. These metrics are critical for assessing the accuracy of the model’s predictions and understanding how closely they align with the actual values, as shown in Equation (5).

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}{\displaystyle \sum }_{i=1}^n\left|Actua{l}_i-Pre{d}_i\right|$$

(2)

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}{\displaystyle \sum }_{i=1}^n{\left(Actua{l}_i-Pre{d}_i\right)}^2$$

(3)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum }_{i=1}^n{\left(Actua{l}_i-Pre{d}_i\right)}^2}$$

(4)

$${\mathrm{R}}^2=1-{\displaystyle \frac{SSR}{SST}}=1-{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^n{\left(Actua{l}_i-Pre{d}_i\right)}^2}{{{\displaystyle \sum }}_{i=1}^n{\left(Actua{l}_i-Mean\left(Actua{l}_i\right)\right)}^2}}$$

(5)

2.4. Outline of the Research and Analysis Procedures

We conducted all our research and analysis in this study using the software development environment Visual Studio Code (Version 1.93, Microsoft [56]). The programming language used in this environmental was Python3 (Van Rossum and Drake [57]). We implemented all the machine learning algorithms and calculated the resulting metrics using Scikit-Learn, a Python3 machine learning library (Pedregosa et al. [58]). Figure 4 illustrates the flowchart of the entire research and analysis process.

First, the process of loading the dataset into the software development environment, referred to as reading the dataset, was performed. Then, training and testing sets were created from the dataset. A model was defined according to the parameter set, trained with the training set, and used to predict the outputs with the testing set. Subsequently, the performance metrics of the predicted results were calculated. If the metric results were satisfactory, the most appropriate machine learning model was selected.

3. Results

Machine learning algorithms have variables called parameters, similar to mathematical equations. Just as changing the value of a variable in a mathematical equation alters the result, the same applies to machine learning algorithms. Therefore, each algorithm used in the study was trained and evaluated with the parameters LL, w, PL, PI, G_s, and e₀ and the compression index (C_c) using the same training and testing sets.

One of purposes of this study was to identify the parameter values that yielded the best outcomes by assigning different values to three parameters of the random forest regressor algorithm: the number of estimators, the number of maximum features, and the maximum depth of the tree. For the parameter of the number of estimators, which indicates the number of trees in the random forest, we used integer values of 10, 50, 100, 200, and 400, respectively. For the parameter of the number of maximum features, which determines the number of features to consider when seeking the best split, we used the options no change (1.0), square root, and logarithmic change. For the maximum depth of the tree parameter, we used integer values of 10, 20, 30, 40, and 50, respectively. Figure 5 and Figure 6 show the training and testing outcomes with blue circle symbols of the best parameter search for the random forest regressor algorithm. The optimal training R² value of 0.970 and testing R² value of 0.926 for the random forest regressor were obtained with the parameters number of estimators = 100, number of maximum features = 1.0, and maximum depth = 20.

We aimed to identify the optimal values by experimenting with different values for the loss and learning rate parameters of the gradient boosting regressor algorithm. The learning rate parameter was set to multiples of 10 between ${10}^{-1}$ and ${10}^{-8}$. For the loss function parameter, we employed loss functions including the squared error, absolute error, Huber loss, and quantile. Figure 7 and Figure 8 display the training and testing outcomes with blue circle symbols from the optimal parameter search for the gradient boosting regressor algorithm. The highest training R² of 0.925 and the highest testing R² of 0.930 for the gradient boosting regressor were achieved with the parameters set to a learning rate ${10}^{-1}$ and using a squared error loss function.

Experiments were conducted with various values for three parameters of the AdaBoost regressor algorithm—the learning rate, loss function, and number of estimators—to achieve the optimal performance. The learning rate parameter was adjusted to 1.0, 0, and multiples of 10 between ${10}^{-1}$ and ${10}^{-8}$. We set the parameter of the number of estimators to integer values of 10, 50, 100, 200, and 400. For the loss function parameter, we applied linear, square, and exponential loss functions.

Figure 9 and Figure 10 present the training and testing results with bule circle symbols obtained during the optimal parameter search for the AdaBoost regressor algorithm. The AdaBoost regressor achieved an optimal training R² value of 0.847 and testing R² value of 0.921 with the following parameters: 200 estimators, a learning rate of 1.0, and a square loss function.

Table 4. The R² results obtained from the training and testing of various parameter combinations.

Algorithm	Parameter Combination	Training R2	Testing R2
Random Forest	First selection (100, 1.0, 20)	0.9708	0.9264
	Second selection (100, 1.0, 30)	0.9706	0.9265
	Third selection (100, 1.0, 50)	0.9706	0.9265
Gradient Boosting	First selection (${10}^{-1}$, squared)	0.9250	0.9300
	Second selection (${10}^{-1}$, Huber)	0.9040	0.9220
	Third selection (${10}^{-1}$, absolute)	0.8510	0.8920
AdaBoost	First selection (200, 1.0, square)	0.8470	0.9217
	Second selection (100, 1.0, square)	0.8468	0.9210
	Third selection (10, 1.0, exponential)	0.8468	0.9100

displays the training and testing R² results for the top three selections, which had the highest values, obtained using different parameter combinations with the algorithms used in this study.

We tested all combinations of the six inputs in our dataset one by one for the random forest regressor, gradient boosting regressor, and AdaBoost regressor algorithms. The combinations we tried, totaling 63 all in all, were as follows: single-input combinations ([w], [PL], [LL], etc.), two-input combinations ([w, PL], [w, LL], etc.), three-input combinations ([w, PL, LL], etc.), four-input combinations ([w, PL, LL, PI], etc.), five-input combinations ([w, PL, LL, PI, e₀], etc.), and six-input combinations ([w, PL, LL, PI, e₀, G_s]). Ignoring a few combinations that had very small differences of 0.002, the six-input combination, which included all the inputs in the dataset, showed the most successful results for all three algorithms.

Table 5. The R² results obtained from the training and testing of various input combinations.

Algorithm	Input Combination	Training R2	Testing R2
Random Forest	[w, PL, PI, e0]	0.970	0.928
	[w, PL, PI, e0, Gs]	0.970	0.926
Gradient Boosting	[PL, LL, e0, Gs]	0.916	0.932
	[w, PL, LL, PI, e0, Gs]	0.925	0.930
AdaBoost	[w, PL, LL, PI, e0, Gs]	0.847	0.921

displays only the highest training and testing R² results obtained with different input combinations of the algorithms used in this study.

The random forest regressor with the six-input combination ([w, PL, LL, PI, e₀, G_s]) achieved impressive R² scores of 0.970 for training and 0.926 for testing. The gradient boosting regressor performed slightly better in testing with a four-input combination ([PL, LL, e₀, G_s]), with a training R² score of 0.916 and testing R² score of 0.932, while the six-input combination yielded a marginally higher training R² value of 0.925 but a slightly lower testing R² value of 0.930. The AdaBoost regressor lagged slightly behind when using the six-input combination, with training and testing R² scores of 0.847 and 0.921, respectively.

4. Discussion

When comparing the results obtained in the current study with those from similar studies in the literature, the three different algorithms used in our study stand out.

Table 6. Comparison of the results of the testing R² between this study and previous studies.

References	Used Algorithm	Input Parameters	Testing R2
Park and Lee [27]	Artificial Neural Network	[w, e0, LL, PI, Gs, Wsand, silt, clay]	0.885
Benbouras et al. [36]	Artificial Neural Network	[Ph, e0, w, PI, LL, FC]	0.562
	Genetic Programming	[Ph, w, PI, LL]	0.360
	Multiple Regression Analysis	[Ph, e0, w, PI, LL, FC]	0.409
Tsang et al. [37]	Random Forest	[w, e0, LL, PI, Gs]	0.818
	Extreme Gradient Boosting		0.833
Chia et al. [38]	Random Forest	[e0, LL, PI]	0.860
	Gradient Boosting Tree		0.890
Kim et al. [39]	Artificial Neural Network	[w, LL, PI]	0.905
In this study	Random Forest	[w, e0, LL, PL, PI, Gs]	0.926
	Gradient Boosting		0.930
	AdaBoost		0.921

presents a comparison of this study and other works from the literature based on the testing R² values obtained for predicting results using different algorithms and input parameters. Park and Lee [27] used artificial neural networks (ANNs) and achieved an R² value of 0.885. Benbouras et al. [36] obtained R² values of 0.562 with artificial neural networks, 0.360 with genetic programming, and 0.409 with multiple regression analysis. Tsang et al. [37] achieved R² values of 0.833 with extreme gradient boosting and 0.818 with random forest methods. Chia et al. [38] reported an R² value of 0.860 with random forest and a higher value of 0.890 with a gradient boosting tree, using a smaller number of parameters. Lastly, Kim et al. [39] reported an R² value of 0.905 using artificial neural networks.

In this study, an impressive R² value of 0.926 was achieved with the random forest model, while the gradient boosting and AdaBoost models showed R² values of 0.930 and 0.921, respectively. Overall, the current study achieves superior results compared to those reported in the literature, particularly when using a comprehensive set of input parameters [w, LL, PL, PI, e₀, G_s], highlighting the potential of the methods used in this study to improve the prediction performance.

5. Conclusions

The assessment of fine-grained soils settling under loads, such as buildings, vehicles, and infrastructure, heavily relies on the compression index (C_c). Calculating the C_c parameter is therefore crucial for determining the settlement in soil layers. However, using traditional methods, such as oedometer testing, to measure the value of C_c is time-consuming, requires specialized laboratory equipment, and needs expert staff. Predicting C_c values using additional fine-grained soil characteristics like w, PL, LL, PI, e₀, and G_s can save time and costs compared to oedometer testing. In this study, machine learning algorithms, a subfield of artificial intelligence, were employed to estimate the values of C_c using these parameters. A dataset of 915 samples containing these parameters was used to train and test the model. This approach aims to assist geotechnical researchers in calculating the compression index (C_c) more efficiently, thereby saving time and reducing costs.

The compression index is crucial for soil consolidation computations in geotechnical engineering, especially for fine-grained soils. Various machine learning models have been used to predict this coefficient based on different soil properties, providing valuable insights into their relationships with the compression index. This study evaluated the effectiveness of machine learning techniques such as a random forest regressor, gradient boosting regressor, and AdaBoost regressor in predicting the compression index, highlighting the impact of the findings.

The machine learning algorithms in this study were tested with different parameters and various combinations of input parameters to determine the best results. The random forest regressor model demonstrated a significant coefficient of determination (R²) of 0.970 during training and 0.926 during testing when it was trained with the inputs [w, PL, LL, PI, e₀, G_s], using parameters such as a maximum depth of 20, a maximum number of features of 1.0, and 100 estimators. The gradient boosting regressor model achieved R² values of 0.925 for training and 0.930 for testing when trained with the same inputs and parameters, specifically at a learning rate of 0.1 and with a squared error loss function. The AdaBoost regressor model achieved R² values of 0.847 for training and 0.921 for testing with the inputs [w, PL, LL, PI, e₀, G_s] and parameters including 200 estimators, a learning rate of 1.0, and a square loss function. These results indicate that the input parameters—w, PL, LL, PI, e₀, and G_s—are effective for successfully predicting the compression index (C_c). When compared with similar studies in the literature, this study achieved higher testing R² results.

Traditional methods, such as empirical equations, do not typically report performance metrics like R² because they do not directly focus on the data fit or model optimization. As a result, R² values can only be calculated and compared for data-driven modeling techniques, such as machine learning algorithms. Machine learning algorithms reduce costs and shorten project durations by making fast and accurate predictions in geotechnical engineering. Compared to the traditional methods, these approaches provide reliable results in complex ground conditions, detect potential risks in advance, and enable safer designs. In this way, structural problems can be prevented in the long term.

This study demonstrated that machine learning models offer valuable insights into understanding the relationships between soil properties and the compression index (C_c). However, it also highlights the risk of overfitting, particularly when there is a significant disparity between the training and testing data. This underscores the importance of exploring other machine learning models or input combinations to improve the compression index predictions. Addressing these model limitations and exploring alternative methodologies can enhance the accuracy and reliability of soil consolidation predictions relevant to geotechnical engineering.

In future work, we plan to explore dimensionality reduction techniques, such as Principal Component Analysis (PCA) or t-SNE, to improve the efficiency of our machine learning models when applying them to high-dimensional geotechnical datasets. The current literature highlights that these methods reduce the computational complexity while preserving the data structure. For example, the book by Lespinats et al. [59] and the work by Van Der Maaten et al. [60] discuss the potential of dimensionality reduction techniques to improve the model performance and enhance our understanding of soil behavior in detail. Incorporating such methods may provide valuable insights to further improve the estimation of the compaction index (C_c).