2. Key Laboratory of Loess Earthquake Engineering, Gansu Earthquake Agency, Lanzhou 730000, China;
3. Lanzhou Institute of Seismology, China Earthquake Administration, Lanzhou 730000, China
Soil compressibility is an important parameter in the field of geotechnical research. Due to the fact that soil particles are incompressible, the change of soil volume is thus equal to the change of the entire pore volume. Besides, soil compressibilty is not a constant and is associated with the stress variation. A majority of previous studies focus on changing several influencing factors to obtain the corresponding compressibility parameters of silty clay. Limited to the difference in test conditions and test methods, the conclusions are also various. As a result, there is currently no wellacknowledged model for silty clay compressibility parameters that fits perfect with the actual condition. Because of the strong ability of nonlinear fitting and approximation, artificial neural network can make up for the defects of traditionally empirical model, attracting more attention from domestic and foreign scholars (Wang Jiaxin et al., 2018; Goh A.T.C., 1995; Orbanic P. et al., 2003; Hinton G. et al., 2012; Hinton G.E., 2007; Smith S.M. et al., 2015; Cai Run et al., 2018). With new insights for modeling the unknown nonlinear systems, neural network has been successfully applied in the field of geotechnical engineering. (Wang Caijin et al., 2019; Wang Maohua et al., 2019; Huang Faming et al., 2018; Gao Jie et al., 2018; Liu Houxiang et al., 2018). In detail, using ANFID neural network, Zhu Shimin et al., (2019) propose a creep model for red clay with the consideration of confining pressure and deflection stress. Also, a BP network model for strength prediction of steel slag is established by Li Xinmin (Li Xinmin et al., 2017). Based on the fieldmeasured data and parameter sensitivity analysis, the key parameters of stratum are determined, and the dynamic construction inversion analysis method based on parameter sensitivity analysis and BP neural network is proposed by Xiao Mingqing (Xiao Mingqing et al., 2017). In the neural network model, generalization ability is the most important index to measure the network performance, and the network without generalization ability has no practical significance. The generalization ability of neural network is related to the training sample and network structure. The limitations of the traditional BP neural network (i.e., learning algorithm is LMBP) has the following disadvantages (Cai Run, 2018): (1) It is highly sensitive to the initial weights; (2) the learning and memory of the network is unstable; (3) the BP algorithm easily makes the weights fall into local minima; (4) the learning of traditional neural networks is prone to over fitting. Moody claims that the main reason for over fitting is the redundancy of network structure (Moody J.E., 1992). Therefore, some scholars indicate some improved methods. Taking the advantage of genetic algorithm for weight optimization, Rao Yunkang et al., (2019) establish a prediction model for maximum dryness density, providing a reference for compaction quality control as well as gravel soil selection that both meet the requirements of engineering compaction performance. Furthermore, by optimizing the relevant parameters of the output layer and hidden layer of the neural network, the model built by this method can effectively reduce the time cost and improve the accuracy of geological disaster monitoring and early warning (Wang Fang et al., 2019). Wang Shudong et al., (2019) combine the Artificial Fish School Algorithm (AFSA) and the Elman network and then apply them to the slope stability judgement, obtaining expected results that have higher accuracy, better convergence, and more suitable slope displacement prediction. When the traditional BP neural network is used to calculate the compressibility parameters of silty clay, the output results are discrete. With the robust theoretical basis, Bayesian regularization makes full use of the relationship between input and output. This method introduces a priori distribution in the model parameters and optimize the solutions by constraining the output function. Importantly, it can prevent overfitting by modifying the loss function. It also solves the problem of high calculation and local extreme values, stabilizing the output model and thus determining the soil compressibility parameters more accurately.
This paper builds an output model based on correlation analysis (SPSS) and Bayesian regularization algorithm to improve the BP neural network (BRBP). Firstly, the main factors that affect the compressibility of the sample are checked by correlation analysis, for the purpose of avoiding the lengthy input information caused by subjective factors, thereby reducing the calculation dimension and improving the convergence performance of the network. Secondly, by optimizing the parameters of Bayesian regularization BP neural network, the defect that its calculation result is easy to fall into the local minimum value is effectively solved, thereby improving the output accuracy. Finally, the paper compares the established output model and the traditional LMBP neural network output model through the test data of a project in Jianyang City. The results show that our improved model can significantly improve the output accuracy and can be used as a powder in the region. To conclude, it can be used as a feasible technical process and method for the output of clay compressibility parameters.
1 METHOD 1.1 Onedimensional Compressibility IndexThe deformation characteristics of soil are very complicated, not only closely related to the basic properties of the soil, such as composition, state, and structure, but also related to the stress conditions (stress level, deformation conditions) of the soil. The compressibility of soil refers to the characteristic that the volume of soil shrinks under the action of vertical pressure when the pressure is equal in all directions or the lateral limit. At present, the most commonly used indoor test for determining the soil compressibility parameters is the lateral compression test, also known as the consolidation test. The commonly used compression indicators are the compression coefficient a and the compression modulus E_{s}. The limits of the indicators are shown in Table 1:
The Back Propagation neural network, which is proposed by Rumelhart and McClelland in 1986, is a multilayer forward neural network (Peng Lishun et al., 2019). The neural network model of BP algorithm has reached 80%90% according to Cai Run et al., (2018). The learning process of BP neural network mainly includes two processes: forward propagation of information and error back propagation. Fig. 1 shows a threelayer BP neural network, including the input layer, hidden layer and output layer. Each neuron in the input layer is responsible for receiving the input information and passing it to each neuron in the middle layer; The hidden layer is the internal information processing layer, which is responsible for information transformation. At present, the most common method for determining the number of nodes J in the hidden layer is the trial and error method; the output layer is responsible for outputting the information of each neuron. The forward process includes the input information passing through the hidden layer to the output layer from the input layer and completing a forward learning process through the operation of the function. If the actual output does not match the expected output, the backpropagation phase of the error is entered. The error signal is returned along the original input route. Meanwhile, the error passes through the output layer, updating the weights and biases of each layer through error gradient descent algorithm. Such process continues until the sum of the error squares and the expected value reach the size of the set error value or the preset learning times. Forward and reverse processes are used repeatedly to determine the final output value size.
The training performance function of artificial neural network adopts loss function (mean square error function)E_{e}, which is defined as:
$ {{E_e} = \frac{1}{2}\sum\limits_{k = 1}^k {\sum\limits_{k = 1}^n {{{\left({{y_k}  {A_k}} \right)}^2}} } } $  (1) 
In formula(1): n is the number of samples, k is the number of neural network output, y_{k} is the expected output value, and A_{k} is the actual output.
1.3 Bayesian Regularization MethodThe Bayesian neural network method combines Bayesian method with neural network method. This paper introduces the Bayesian principle into the weight and the threshold of the neural network. The posterior probability of the weight is used as the optimization objective function, and the weight of the neural network is obtained by maximizing the posterior probability of the weight, solving the problem that the BP algorithm easily leads to the weight to fall into the local minimum value. For ordinary neural networks, the loss function needs to be designed, and the optimal parameters can be optimized by SGD, Adam and other optimizers to minimize the error. However, for Bayesian neural networks, it is the KL divergence that needs to be minimized. The regularization theory modifies the loss function for the purpose of preventing the occurrence of overfitting, improving the generalization ability of the network. Overfitting may occur if regularization terms are not added into the formula (Mackay D.J.C., 1992; Wong M.L. et al., 2004). In this paper, we adds a penalty term E_{w} to the Bayesian regularization optimization algorithm to achieve a suitable fit, and then alleviates the problem of slow convergence and low learning efficiency of traditional neural network algorithms. The network performance function is as follows:
$ {F = a{E_{\rm{w}}} + b{E_{\rm{D}}}} $  (2) 
In the formula: E_{w} is the square sum of all network weights; E_{D} is the error value between the output value and true value of each layer; a and b are regularization coefficients of performance function, also called hyperparameters which control the distribution of weights and thresholds of the neural network, thus affecting the training effect. When a≤b, as the number of training increases, the error of training samples becomes smaller, resulting in the network overfit phenomenon; When a≥b, as the number of training increases, the weights become smaller, smoothing the network output. It will also produce the phenomenon of underfitting.
Under the Bayesian theory, we can obtain the formula as follows:
$ {a = \frac{\gamma }{{2{E_{\rm{w}}}}}, b = \frac{{n  \gamma }}{{2{E_{\rm{D}}}}}} $  (3) 
In the formula:γ is the effective weight number, which represents the number of connection weights of the neural network that can reduce the training error, n is the total number of neural network parameters,
The sampling location, Longya Village, is situated in the south of Jianyang City, Sichuan Province. This area belongs to eroded shallow hill landform, most of which are agricultural. The microgeomorphic units are shallow hillside landform and shallow hill valley landform (Fig. 3). According to the results of drilling, insitu testing and indoor geotechnical tests, the strata in the site are mainly Quaternary artificial accumulation layers (Q_{4}^{ml}) and Quaternary slope alluvial layers(Q_{4}^{dl+pl}).The sampling depth of silty clay is 0.4 m18.3 m, the measured compression coefficient is 0.220.56, and the compression modulus is 3.277.75. Among them, 40 sets of samples are used as test objects, 20 sets of samples are used as training samples of the Bayesian regularization neural network, 10 sets of samples are used as verification samples of the Bayesian regularization neural network, and the remaining 10 sets of samples are used as test data.
Since there are numerous factors affecting the compressibility index of silty clay, and the coupling relationship between the compressibility index and the test conditions is complex, this paper selects the physical property parameters of the soil as the input of the model, and analyzes the correlation between the physical property parameters and the compressibility parameters to obtain the correlation coefficient (Pearson) of each parameter with the compressibility parameters, the results are shown in Table 2.
From the table above, we can see that the correlation between each influencing factor and compressibility is different. According to the actual situation, we choose highly correlated and moderately correlated indicators, respectively (the absolute value of the correlation coefficient is greater than 0.5). No consideration will be given to lowlevel related factors. Finally, the most relevant indexes such as density ρ(g/cm^{3}), water content ω(%), porosity e, porosity n(%), and liquidity index I_{L} are selected as the input of the model.
The great change in all collected data is not conducive to the training of neural networks. In order to avoid neuron saturation during training, the sample data is normalized and the output results are denormalized. The larger input values should still fall when the transfer function gradient is large. There are two purposes of doing this. Firstly, it is easy to facilitate data processing; Secondly, it can speed up the convergence of the program when it runs (Peng Lishun etal., 2019). Assuming that the data X=(X_{i}) is normalized to X′=(X′_{i}), the calculation formula of the normalization process is as follows:
$ {X' = \left({{Y_{{\rm{max}}}}  {Y_{{\rm{min}}}}} \right)\frac{{{X_i}  {X_{{\rm{min}}}}}}{{{X_{{\rm{max}}}}  {X_{{\rm{min}}}}}} + {Y_{{\rm{min}}}}} $  (4) 
Among them:Y_{max} and Y_{min} are artificially specified upper and lower bounds, in this paper, we set Y_{max}=1 and Y_{min}=0.
X_{max} and X_{min} correspond to the maximum and minimum values in the sample data, and satisfies the requirement X_{min}≤X′≤X_{max}.
Normalized and antinormalized handler code in MATLAB are:
$ \begin{array}{l} \left[ {\rm{y}1, {\rm{PS}}} \right] = {\rm{mapminmax}}\left({{\rm{x}}, 0, 1} \right){\rm{ }}\\ {\rm{y}}2 = {\rm{mapminmax}}\left({'{\rm{reverse}}', {\rm{y}}1, {\rm{PS}}} \right) \end{array} $  (5) 
The training sample data are shown in Table 3:
The neural network model has five input neurons. The tansig function is selected from the input layer to the hidden layer, and the purelin function is selected from the hidden layer to the output layer. Comparing the convergence curves of the two training functions (Fig. 4): When the error performance of the BRBP neural network model is reduced to the target value, the training is stopped. However, because the training speed of the LMBP neural network model is too fast, each time the training is stopped, the network error performance drops far below the target value, which is prone to overfitting. This is also the reason why LMBP neural network has poor promotion ability compared to BRBP neural network.
This paper selects a hidden layer node number J according to the empirical formula J=2×n+1, in which n is the number of input units. By continuously changing the number of hidden layer neurons, we observe the test error value until the neural network converges. This paper also uses a trial and error method to determine the number of hidden layer nodes, tested J=10, 11, 12, 13, 14, 15 are finalized the number of hidden layer nodes is 14.
In order to maintain the efficient training capabilities of the neural network, we set the related parameters reasonably. The other parameter settings of the neural network model are shown in Table 4. The data samples are trained according to the network structure set by the parameters, and the training stops when the target accuracy is reached.
The method to calculate the influence degree of each input variable on the weight of the output variable uses the weights proposed by Garson. It calculates the weights of compression coefficient and compression modulu respectively(Garson G.D., 1991).The calculation formula is as follows:
$ {I_h} = \frac{{\sum\limits_{m = 1}^{{N_j}} {\left({\frac{{\left {\omega _{hm}^{ij}} \right}}{{\sum\limits_{k = 1}^{{N_i}} {\left {\omega _{km}^{ij}} \right} }} \times \nu _{mn}^{jt}} \right)} }}{{\sum\limits_{k = 1}^{{N_i}} {\left\{ {\sum\limits_{m = 1}^{{N_j}} {\left({\frac{{\left {\omega _{km}^{ij}} \right}}{{\sum\limits_{k = 1}^{{N_i}} {\left {\omega _{km}^{ij}} \right} }} \times \nu _{mn}^{jt}} \right)} } \right\}} }} $  (6) 
In formula(6):I_{h} is the influence weight of the h input parameter on the n output variable; N_{i} and N_{j} are the number of nodes in the input layer and hidden layer; ω^{ij}is the weight of the input layer to the hidden layer; ν^{jt} is the weight of the hidden layer to the output layer; The subscripts h, m and n represent the corresponding neurons in the neural network model. The larger the value of I_{h}, the greater the proportion of its influence weight, and the greater its impact on the output.
After calculation, the final weights and thresholds of the neural network model based on BRBP are obtained in Table 5 and Table 6. The weight of the input variable to the hidden layer and the hidden layer to the output variable indicate the relative magnitude of the influence of the input variable on the output variable.
Garson weight coefficient calculated by formula(6)is shown in Fig. 5. Among the five selected influencing factors, the influence degree on the compression coefficient is relatively uniform, among which the water content and liquidity index are relatively prominent. However, among the influence degree from the five factors on the compression modulus, the proportion of porosity and water content is larger, and the proportion of density is smaller. This shows that in this model the density has the smallest effect on the output of the compression modulus, and the porosity and water content have a greater effect.
Using the established BRBP neural network model and the traditional LMBP neural network model, the verification data (Table. 7) are used to verify the compression coefficient and compression modulus respectively. The fitting results are shown in Fig. 6 and Fig. 7:
It can be seen from Fig. 5 and Fig. 6 that the two neural network models have a good generalization ability for the mapping relationship between the compressibility parameter and the nonlinear function of the selected parameter, but there is still a certain gap between the two in terms of output accuracy. In the compression coefficient model, the correlation coefficient between the output value of the Bayesian regularized neural network and the target training value reaches 0.999, while the correlation coefficient between the output value of the traditional LMBP neural network and the target training value is 0.956. In the compression modulus model, the correlation coefficient between the output value of the Bayesian regularized neural network and the target training value reaches 0.997, while the correlation coefficient between the output value of the traditional LMBP neural network and the target training value is 0.905. Therefore, compared with the traditional LMBP neural network model, the output of the BRBP neural network model is more stable and the correlation is higher.
The output error curve of the validation sample data is shown in Fig. 8:
After the network training is completed, the established Bayesian regularized neural network is simulated through the measured data. Meanwhile, the output of the network is obtained by using the simulation function. By comparing whether the error between the output value and the measured value meets the requirements, the network's generalization ability is checked. The test data are shown in Table 8:
The output results of the two neural networks are shown in Table 9 and Table 10. It can be seen from the tables that the maximum absolute error and relative error of the compression coefficient output by the LMBP neural network model are 0.16 and 64%, respectively. The average value of the relative error is 15.54%; The maximum absolute error and relative error of the compression coefficient output by the BRBP neural network model are 0.08 and 29.6%, respectively. The average relative error is 6.15%; the maximum absolute error and relative error of the compression modulus output by the LMBP neural network model are 2.28 and 39.0%, respectively. The average relative error is 6.07%; The maximum absolute error and relative error of the compression modulus output by the BRBP neural network model are 1.70 and 29.1%, respectively. The average relative error is 4.62%. Such results show that the model is more stable and has better advantages in solving nonlinear problems such as high dimensionality.
In order to reflect the accuracy of the output results of the two neural networks more intuitively, the output results of the two neural network models are superimposed and compared, as shown in Fig. 9 and Fig. 10:
The errors of the output results of the two models according to Fig. 9 and Fig. 10 are both acceptable, indicating the applicability and effectiveness of the compressibility parameter study. LMBP neural network has excellent autonomous learning ability and strong generalization ability. This paper utilizes the advantages of LMBP neural network and solves the problems existing in traditional LMBP neural network, then proposes a Bayesian regularization BP neural network output model, and finally compares it with the traditional BP neural network output model. Overall, the traditional BP neural network output model is prone to fall into local minimization, while the BP neural network output model after Bayesian regularization improves the local minimization problem obviously. The relative error of the output is small. The BP neural network output model after Bayesian regularization is more stable than the traditional BP neural network output results, indicating that the BRBP neural network is more suitable for the output of the compressibility parameters of silty clay in the study area.
3 DISCUSSION AND CONCLUSIONSThe compressibility parameters of silty clay are affected by various factors, and the nonlinear mapping between them and variables is difficult to be represented by an accurate mathematical function model. Therefore, based on the experimental data, the paper uses correlation analysis (SPSS) and Bayesian regularization neural network to establish the output model, the following conclusions are obtained:
(1) Most of the silty clay in the study area is medium compressive, and there is also high compressive silty clay. However, the accidental results of the test are not ruled out. As can be seen from the output of the paper, the test result is a sample of highly compressive silty clay. Furthermore, the output result based on the BRBP neural network model is still highly compressive silty clay, indicating that the results of this sample are not associated with test errors.
(2) The correlation analysis method is used to select the main influencing factors with high correlation, and replace all the original variables as input parameters to build the model. This method effectively reduce the dimensionality, eliminate the information redundancy between variables, and significantly improve the training efficiency and output accuracy.
(3) According to Garson theory, among the factors affecting the compression coefficient of silty clay in this area, ω and I_{L} are relatively prominent. Among the influencing factors of compression modulus, e and ω are larger, while the proportion of ρ is smaller.
(4) The inspection of the measured data and the comparison with the unimproved BP neural network model are both excellent, indicating a relatively stable output of the model. In detail, the average value of the relative error of the compression coefficient is reduced from 15.54% to 6.15%, the average value of the relative error of the compression modulus is reduced from 6.07% to 4.62%, and the accuracy of the output results is significantly improved. When this method is applied in practice, the workload can be effectively reduced compared to the experimental workload. Besides, the efficiency is high and the accuracy can also meet the relevant requirements. It shows that the research method has good theoretical significance and practical value, and provides a new technical method for obtaining the compressibility parameters of silty clay in this area.
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