## 4 Proposed ANN with Hybridization of ABC and LM Algorithms for Compression Coefficient Estimation  
This section presents the proposed ANN model trained by a hybrid method of ABC and LM algorithms. This model is trained with the aforementioned experimental dataset and is specifically designed for the task of CC estimation. The proposed model, namely, ABC-LM-ANN, is demonstrated in Figure 5. It is noted that the model relies on the MATLAB Statistics and Machine Learning Toolbox [66] to perform the LM algorithm. In addition, the ABC algorithm and the integrated ABC-LM-ANN model were coded by the authors in MATLAB programming environment. To construct the ANN model used for CC estimation, it is required to appropriately set the two-weight matrices of \(\mathrm{W_{1}}\) and \(\mathrm{W_{2}}\), as well as the two bias vectors of \(\mathrm{b_{1}}\) and \(\mathrm{b_{2}}\). It is worth reminding that the size of the matrix \(\mathrm{W_{1}}\) is \(\mathrm{N_{R}}\) x \(\mathrm{N_{DX}}\), where \(\mathrm{N_{R}}\) and \(\mathrm{N_{DX}}\) are the numbers of neurons in the hidden layer and the number of input neurons, respectively. Herein, \(\mathrm{N_{DX}}=12\), which is the number of CC's conditioning factors. Meanwhile, based on the suggestion of [PERSON] [67] and several trial-and-error experiments, the value of \(\mathrm{N_{R}}\) used in this study is selected to be 10 and the size of \(\mathrm{W_{2}}\) is \(\mathrm{N_{O}}\) x \(\mathrm{N_{R}}\) where \(\mathrm{N_{O}}\) is the number of output neurons. Herein, \(\mathrm{N_{O}}=1\), which is the targeted output of CC. The size of the two bias vectors of \(\mathrm{b_{1}}\) and \(\mathrm{b_{2}}\) is \(\mathrm{N_{R}}\) and 1, respectively. Since it is required that a solution of ABC is presented in the form of a vector, the ANN structure including \(\mathrm{W_{1}}\), \(\mathrm{W_{2}}\), \(\mathrm{b_{1}}\), and \(\mathrm{b_{2}}\) are vectorized and  
Figure 5: The proposed artificial bee colony Levenberg–Marquardt artificial neural network (ABC-LM-ANN) for compression coefficient (CC) estimation. concatenated to formulate a solution or food source. On the basis of the above explanation, the length of a vector representing a solution is \(\rm{N_{R}}\times\rm{N_{DX}}+\rm{N_{O}}\times\rm{N_{R}}+\rm{N_{R}}+1\). To train the ANN model using the ABC-LM algorithm, the following cost function (CF) is used to minimize the discrepancy between the observed and the predicted output of CC:  
\[\rm{CF}=\frac{\rm{MSE_{TR}}+\rm{MSE_{VA}}}{2} \tag{9}\]  
where \(\rm{MSE_{TR}}\) and \(\rm{MSE_{VA}}\) represent the mean squared error (MSE) of the data samples in the training and validating datasets, respectively. Herein, the training set and the validating set occupy 80% and 10% of the whole dataset, respectively. The rest of the original dataset (10%) is reserved for testing purposes. The MSE is given as follows:  
\[\rm{MSE}=\frac{\rm{\sum_{i=1}^{N_{D}}\left(Y_{A,i}-Y_{P,i}\right)^{2}}}{N_{D}} \tag{10}\]  
where \(\rm{Y_{A,i}}\) and \(\rm{Y_{P,i}}\) denote the actual and the predicted CC of the ith data instance. \(\rm{N_{D}}\) is the number of data instances. The inclusion of the validating data is to safeguard against overfitting during the ABC-LM training process. Overfitting typically occurs when the constructed ANN model can provide a perfect fit to the training data, nevertheless, the model has an inferior fit when predicting CC values for novel soil samples. The inclusion of both training and validating datasets is to force the ABC-LM to seek for a network structure that can deliver good predictive accuracy and guarantee generalization property. During the model training phase, ABC acts as a global optimizer to guide the population of honeybees to better solutions that represent network structures. Moreover, to accelerate the optimization process, the LM backpropagation algorithm is activated once in ten iterations on a randomly selected solution.