# Landslide Displacement Prediction Method Based on GA-Elman Model
## 2 Method  
### GA-Elman Displacement Prediction Model  
#### 2.1.1 Elman Neural Network  
Elman neural network is a feedback recurrent neural network with local memory function [24]. The least square method is used to realize the minimum mean square deviation of the actual output and the expected output of the neural network, which can better reflect the dynamics of the system. The Elman neural network also has the input layer, hidden layer and output layer similar to the neural network, and also has a special structure layer: the context layer (state layer). The significance of the context layer is to memorize the output value of the hidden layer at the previous moment, which can be considered as a one-step delay operator and enhance the internal feedback mechanism of the network. It can make the neural network have better nonlinear and time-varying characteristics, and effectively improve the global stability of the neural network. The learning process of the neural network consists of two parts: the forward propagation of the signal and the back propagation of the error. Through the repeated training of these two parts, the weight value and threshold value between the input layer and the hidden layer and between the hidden layer and the output layer can be continuously developed to the direction of error reduction, making the prediction results more close to the measured results [26]. The structure of Elman neural network is shown in Figure 1.  
The formation mechanism of landslide disaster is complex and there are many factors leading to landslide deformation, including rainfall infiltration, groundwater activity, earthquake, human engineering excavation and so on. Therefore, these factors should be fully considered in the design of input layers for reasonable selection and the neural network model should be optimized to improve the practicability of the model. According to the neural network model, the mathematical model of Elman neural network can be expressed as:  
\[\left\{\begin{array}{l}y(k)=g(w^{2}x(k))\\ x(k)=f(w^{1}x_{c}(k)+w^{2}(u(k-1)))\\ x_{c}(k)=x(k-1)\end{array}\right. \tag{1}\]  
\(u(k)=[u_{1}(k),u_{2}(k),\cdots,u_{n}(k)]\) is the input vector. The function of the input vector is to transmit data to the neural network.  
\(y(k)=[y_{1}(k),y_{2}(k),\cdots,y_{n}(k)]\) is the output vector. The output vector plays a linear weighting role in the network structure.  
\(x(k)=[x_{1}(k),x_{2}(k),\cdots,x_{n}(k)]\) is the output vector of hidden layer. The hidden layer is used as a transfer function.  
Figure 1: Structure of Elman neural network.  
\(x_{c}(k)=[x_{c1}(k),x_{c2}(k),\cdots,x_{cn}(k)]\) is the output layer of context layer. The context layer is used to memorize the output value of the previous time of the hidden layer and return the input to the network.  
\(w^{1},w^{2},w^{3}\) are weight matrices from context layer to hidden layer, input layer to hidden layer and hidden layer to output layer, respectively.  
In this study, \"tansig\" nonlinear function is used as the transfer function of the hidden layer and the transfer function \(f(\cdot)\) of the hidden layer is a nonlinear function. In addition, the maximum training times and accuracy of the array will directly affect the reliability and accuracy of the neural network. When designing the number of hidden layers, the running time and accuracy of the neural network model should be considered. Due to the context layer, the Elman neural network has the so-called dynamic memory function. Its function is to memorize the output value of the hidden layer unit at the previous moment and return it to the input vector. In this paper, the transfer function \(g(\cdot)\) of the output layer adopts the S-type logarithmic \"logsig\" function, which mainly plays a role of linear weighting.