1. Introduction

2. Data-Driven Modeling

Figure 4. Classification of the AI modelling techniques (Adapted from Giustolisi et al. [41] and Zhang et al. [42]).

Figure 4. Classification of the AI modelling techniques (Adapted from Giustolisi et al. [41] and Zhang et al. [42]).

2.1. Support Vector Machine (SVM)

In supervised learning model (SVM) method, initially developed by Vapnik [43], weights are calculated based on input data, known as training data, in order to learn the governing function from a set of inputs. SVM selects a limited number of input sample vectors, which are always a fraction of the total number of samples. These input vectors are referred to as support vectors. Using these input vectors, parameter values of the method for minimizing error are calculated. As a result, SVM requires much less data than similar methods such as ANN, and as a result, it takes much less time and money.

The goal of this method is to find a classifier that separates the data and maximizes the distance between these two classes. Figure 5 shows a plane cloud that separates two sets of points.

Figure 5. The hyperplane H that separates the two groups of points.

Figure 5. The hyperplane H that separates the two groups of points.

According to Figure 6, the closest points used alone to determine the hyperplane are called support vectors. As is clear, there are many hypermaps that can divide samples into two categories. The principle of SVM is to choose something that maximizes the minimum distance between the hypermap and the training samples (that is, the distance between the hypermap and the support vectors), this distance is called the margin.

Figure 6. Support vectors.

Figure 6. Support vectors.

Kernel function, as an important part of SVM method, receives data as input and transforms them into the required form. Various functions are provided for this purpose. Some of these functions, including linear, nonlinear, polynomial, radial basis function (RBF), and sigmoid, are given in equations 8 to 12. According to Equations 8 to 12, the kernel function is a series of mathematical functions that provide a window for data manipulation. With the help of this transformation, a complex and non-linear level of decision-making becomes a linear equation, but in a larger number of dimensional spaces.

$$\mathrm{Polynomial}\mathrm{kernel}:k=k\left(x_i,x_j\right)={\left(x_i.x_j+1\right)}^d$$

(8)

$$\mathrm{Gaussian}\mathrm{kernel}:k=k\left(x_i,x_j\right)={exp\left(-\frac{{‖x_i-x_j‖}^2}{2{\sigma }^2}\right)}^{}$$

(9)

$$\mathrm{Radial}\mathrm{basis}\mathrm{function}\left(\mathrm{RBF}\right):k=k\left(x_i,x_j\right)={exp\left(-{\gamma ‖x_i-x_j‖}^2\right)}^{}$$

(10)

$$\mathrm{Hyperbolic}\mathrm{tangent}\mathrm{kernel}:k=k\left(x_i,x_j\right)={tanh\left({k\text{'}x}_i.x_j+c\right)}^{}$$

(11)

$$\mathrm{Sigmoid}\mathrm{kernel}:k=k\left(x_i,x_j\right)={tanh\left(\alpha x^{T}y+c\right)}^{}$$

(12)

where x_i and x_j are vectors in input space, γ and σ parameters define the distance influence of a single sample, d is degree of polynomial, c ∈ R is the relative position to the origin, κ’ is a real-valued positive type function/kernel, α, T and c are certain values.

In this study, two important functions, i.e., radial basis function (RBF) and Gaussian kernel, which have been successful in past research, were evaluated along with two other functions, i.e., Sigmoid kernel and Polynomial kernel functions.

3. Database Collection and Processing

3.1. Experiment and Data Collection

The database was prepared based on the results of two studies [4,40]. A 3D representation of the collected database for CBR is shown in Figure 1. The existing database considers nine input parameters, including liquid limit (LL) and plasticity index (PI) of soil, the sludge content, compaction number of blows, optimum moisture content (OMC) and maximum dry density (MDD) of soil and mixture and bulk modulus Gs of soil. CBR is the only output.

Figure 1. Database used for modelling based on the CBR.

Figure 1. Database used for modelling based on the CBR.

Figure 2 shows the effect of different input parameters, such as compaction number of blows, sludge content, maximum dry density and optimum water content, on CBR. According to the results in Figure 2a, with an increase in the number of compression hammer blows, CBRs increased dramatically as the number of compression hammers increased. Figure 2b illustrates the effect of sludge content on CBR. It can be seen from Figure 2b that as the sludge content increased, CBR increased significantly at first. When the sludge content reached around 8%, CBR began to decrease as the sludge percentage increased. Figure 2c shows the effect of increasing the maximum dry density (MDD) of the soil on the CBR of the soil. According to the results, it can be concluded that the CBR increased as the maximum dry density of the soil increased. In addition, Figure 2d shows the effect of optimum moisture content (OMC) on CBR. Results depict that CBR decreased as optimum water content increased.

Figure 2. Effect of (a) compaction number of blows, (b) sludge content, (c) MDD and (d) OMC on the CBR in collected database.

Figure 2. Effect of (a) compaction number of blows, (b) sludge content, (c) MDD and (d) OMC on the CBR in collected database.

In order to provide a more detailed analysis of the database, Table 1 shows the statistical information including the minimum, maximum, mean and standard deviation.

3.2. Preparation of the Data for AI Modelling

3.2.1. Normalization

In a database, each input or output variable has a specific unit. The data normalization function can eliminate the weight of the units, reduce network errors, and increase training speed by adjusting the data between zero and one. The linear normalization function utilized in this study is outlined below.

$$X_{norm}=\frac{X-X_{min}}{X_{max}-X_{min}}$$

(1)

The four terms in this equation are X_max, X_min, X, and X_norm, which correspond to maximum, minimum, actual, and normalized values, respectively.

3.2.2. Testing and Training Databases

There are two types of databases required for the implementation of mathematical models, namely training databases and testing databases. For all three mathematical models, training and testing were randomly divided. Figure 3 shows the main division of the database, where 80% of the main database was used for training and 20% for testing.

A random database division resulted in two test and experiment databases that were considered fixed and were used in all three mathematical models. Statistical parameters are shown in Table 2 and Table 3, respectively, for the training and test databases. The statistical information, including the minimum, maximum, mean, and standard deviation, is very similar for both databases. This issue can increase the accuracy of the network in predicting the output.

Figure 3. Distribution of (a) training and (b) testing databases.

Figure 3. Distribution of (a) training and (b) testing databases.

3.2.3. Statistical Parameters

The performance of a network can be evaluated through a variety of parameters such as coefficient of determination (R²) and mean absolute error (MAE). Equations 2-7 show the definitions of Mean Absolute Error (MAE), Mean Square Error (MSE), Root Mean Square Error (RMSE), Mean Squared Logarithmic Error (MSLE), Root Mean Squared Logarithmic Error (RMSLE), and Coefficient of Determination (R²).

$$MAE=\frac{\sum _N\left(X_m-X_p\right)}{N}$$

(2)

$$MSE=\frac{\sum _N{\left(X_m-X_p\right)}^2}{N}$$

(3)

$$RMSE=\sqrt{\frac{\sum _N{\left(X_m-X_p\right)}^2}{N}}$$

(4)

$$MSLE=\frac{\sum _N{\left(\log {(X}_m+1)-\log \left(X_p+1\right)\right)}^2}{N}$$

(5)

$$RMSLE=\sqrt{\frac{\sum _N{\left(\log {(X}_m+1)-\log \left(X_p+1\right)\right)}^2}{N}}$$

(6)

$$R^2={\left[\frac{{\sum }_{i=1}^N(X_m-\overline{X_m})(X_p-\overline{X_p})}{{\sum }_{i=1}^N{(X_m-\overline{X_m})}^2{\sum }_{i=1}^n{(X_p-\overline{X_p})}^2}\right]}^2$$

(7)

Where N is the number of datasets, X_m and X_p are actual and predicted values, and $\overline{X_{\mathrm{m}}}$, $\overline{X_{\mathrm{p}}}$ are the average of actual and predicted values, respectively. Ideally, the model should have a R² value of 1 and a MAE, MSE, RMSE, MSLE, RMSLE value of 0.

4. Results

5. Discussion

6. Conclusions

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