# Landslide Prediction with Model Switching +
## 3 Model Switched Landslide Prediction System
### Prediction Model Design  
To predict the future FS, a feed-forward Back-Propagation Neural Network (BPNN) is employed as the prediction model. A BPNN is a powerful computation system, created by generalizing the Widrow-Hoff learning rules [10] into a multi-layer with a non-linear differential transfer function network. The complex network connections imply that the high learning and reasoning ability of BPNN can be applied to deal with problems with high complexity. Figure 2 shows the framework of a BPNN-based prediction model. Training Method for Time-Series Based BPNN  
The basic computational procedure of a BPNN is explained to provide a basic description of the type of ANN that is implemented here. Figure 3 shows the basic structure of a time-series based BPNN. There are three types of layers: input, hidden, and output layers. In the input layer, time-series input for time slots \(t-n\) to \(t\), corresponding to a specific feature, such as factor of safety, rainfall, and soil moisture, is taken as input data. Each pair of nodes in the adjacent layers are linked by a weight. The values in all nodes in a previous layer and the weights are multiplied and accumulated as the input for a node of the next layer. The inputs are, then, given to an activation function to calculate the output value of the node. Repeating the above operations, layer by layer, from input layer to output layer, the final output can be derived. Figure 2: Back-Propagation Neural Network (BPNN) prediction model framework design. Algorithm 1 has a complete description as a prediction model; the following equations explain the functions that are used in this algorithm. At first, the previous FS is used to predict the future FS by a BPNN model. To dynamically determine suitable weights for different FS, a back-propagation method is applied to train and update the weights before prediction, in order to completely illustrate the details of back propagation method. Given the training sample, \(T_{data}=(x_{i},p_{i}),i=1,...,N\), where \(x_{i}=[x_{i1},x_{i2},...x_{in}]\in R^{n}\) is the impact factor and \(p_{i}=[p_{i1},p_{i2},...p_{ik}]\in R^{k}\) is the training target, the general mathematical model of a standard single hidden layer feed-forward network with \(\bar{N}\) hidden neurons is shown in Equation (2). \[o_{j}=\sum_{i=1}^{\bar{N}}g(w_{i}\cdot x_{i}),\qquad j=1,...,N, \tag{2}\]  
where \(o_{j}=[o_{j1},o_{j2},...o_{\bar{j}k}]\) is the \(j\)th output of the BPNN, \(w_{i}\) is the weight of the connection from the input neurons to the \(i\)th hidden neuron, and \(g(x)\) is an activation function that represents how much adjustment the output should be from the neuron, based on the sum of the input. The activation function used in our BPNN model is depicted in Equation (3); namely, the sigmoid function. The sigmoid function, also called the logistic function, is a commonly-used activation function which has an output range from 0 to 1. \[g(x)=\frac{1}{1+e^{-x}}. \tag{3}\]  
The difference between the prediction result and the actual result is called the prediction error. In order to reduce prediction error, the weights need to be updated. The Levenberg-Marquardt (LMA) [11] method has the fastest convergence and the lowest mean square error. Therefore, LMA is selected as a training function to calculate the output network error \(\delta\) and adjustment weight \(W\). It is depicted in Equation (4):  
\[W_{k+1}=W_{k}-[J^{T}J+uI]^{-1}J^{T}\delta, \tag{4}\]  
where \(W_{k}\) represents the weight matrix in the \(k\)th iteration, \(J\) is the Jacobian matrix [12] that contains the network error for weight and the first-order differential of weight, \(I\) is the unit matrix, \(\delta\) is the  
Figure 3: A fully connected feed-forward Back-Propagation Neural Network with time-series. network output error, and \(u\) is a constant. LMA can dynamically adjust the constant \(u\) to reduce the network output error \(\delta\).