. Materials and Methods

2.1 Time Series Theory

Understanding the dynamics of landslides is crucial for effective risk mitigation measures. The displacement of a landslide, which is influenced by both internal and external factors, follows a time series model. This section describes these factors in detail, with a focus on how they affect the evolution of the displacement function.

2.1.1 Internal Factors

Internal factors play a significant role in shaping the displacement of a landslide. They include terrain, geological structure, rock mass or soil type, and mechanical characteristics. These factors influence the slope stability, which directly affects the rate of landslide displacement. For instance, an unstable slope will exhibit a higher degree of displacement compared to a stable one.

2.1.2 External Factors

External forces also have a considerable impact on landslide displacement. Two primary external factors are rainfall and run-off water levels (RWL). Rainfall causes the soil to become moist, which can lead to increased shear stress on the slope. RWL, on the other hand, affects the flow of water through the slope, which in turn influences the hydraulic pressure and deformation process.

The displacement function under the influence of these internal variables typically exhibits a monotonic increase with time, indicating an overall increasing tendency towards landslide displacement. This pattern is known as a linear growth trend or "monometric" growth.

However, when external forces such as rainfall and WLR are considered, the displacement function becomes more complex. It often follows a roughly periodic function due to seasonal changes in rainfall and WLR. This type of pattern is known as a "seasonal" growth trend or "quasi-periodic" growth.

It is important to note that the evolution of landslide displacements over time exhibits a step pattern, demonstrating that the displacement is an unstable development with a specific growth trend. This pattern suggests that there may be opportunities for intervention strategies aimed at stabilizing the slope before it reaches its peak displacement.

In summary, understanding the time series nature of landslide deformation processes involves analyzing both internal and external factors that shape the displacement function. By examining these factors, researchers and practitioners can develop more effective risk mitigation strategies to prevent or minimize the potential impacts of landslides.

The calculation of the cumulative displacement (CD) of a landslide can be done using the TSM after analyzing the monitoring data. The CD can be divided into three types: TTD, PTD, and random term displacement (RTD). The calculation formula for CD based on the TSM is shown in equation (1):

C D = TTD + PTD + RTD

However, RTD is often caused by factors such as wind loads and vehicle loads, which are difficult to monitor effectively and gather relevant data for. As a result, RTD is excluded from landslide deformation prediction. In this case, the equation for calculating CD can be rewritten to exclude RTD, as shown in equation (2):

C D = TTD + PTD

To extract the TTD values, the moving average method is chosen. The expression for TTD is given as:

X = x1, x2, ..., xi, ..., xn

The time series of the CD is then determined by finding the average value of X over time. This average is represented by the value TTD_t ̄ in equation (4):

TTD_t ̄ = k*(x_t+x_t−1+...+x_t−k+1)/k

Where X is the time series of the CD, x_i is the monitored data of the CD at moment i, and k is a constant used to calculate the moving average.

.1. TSM Algorithm for Landslide Displacement Prediction

The TSM (Time-Series Model) algorithm is used to predict the landslide displacement by utilizing the historical data of landslide events. The TSM algorithm is based on the statistical characteristics of the time series data and uses the method of exponential smoothing to fit the data. The exponential smoothing model is represented as:E(Xt)=μ+Xt-1*α*exp(-β*(t-t0))where Xt is the time series data value at time t, μ is the initial value of the mean, α is the smoothing parameter, β is the decay parameter, and t0 is the starting time of the series. After fitting the data using the exponential smoothing model, a new value can be predicted for any given time t using the following equation:Yt=E(Xt)+εwhere Yt is the predicted value for time t+1, ε is a small random number added to ensure that the prediction will converge to the actual value. By using the TSM algorithm with different smoothing parameters and decay parameters, multiple predictions can be made for future landslide displacements. However, it should be noted that these predictions are not guaranteed to be accurate, as they depend on the quality and quantity of historical data available.

The TSM algorithm has several advantages over other traditional methods for landslide displacement prediction. It takes into account the temporal relationship between different events, which helps to avoid overfitting in the case of short-term or isolated events. Additionally, since it does not rely on external knowledge or expert judgment, it can be applied to a wide range of landslide scenarios regardless of their location or magnitude. Moreover, it can provide a continuous and systematic analysis of landslide displacement patterns, which can help to identify potential risk areas and develop effective preventive measures. In this study, we will apply the TSM algorithm to predict landslide displacement in Zhangjiajie National Forest Park based on historical data. Specifically, we will extract the time series data from CDs and use them to train an artificial neural network model for predicting landslide displacement. We will also compare our results with those obtained using the PSO algorithm and other traditional methods.

LSSVM算法是一种支持向量机(SVM)的变体，它使用最小二乘法来计算最优权重向量。LSSVM算法的主要思想是将原始数据投影到一个新的坐标系中，使得在新坐标系下的误差最小化。这个新坐标系是由原始数据中的一些线性无关的特征向量和它们之间的正交关系组成的。

LSSVM算法的优点是它不需要对原始数据进行任何预处理，因此可以很容易地应用于各种类型的数据。此外，LSSVM算法还可以用于非线性分类问题，因为它可以使用核函数将非线性可分问题转换为线性可分问题。

LSSVM is an improved version of the Support Vector Machine (SVM) algorithm, which is based on statistical theory. The primary difference between LSSVM and traditional SVM lies in how it handles the inequality constraint. In contrast to SVM's approach of transforming this constraint into a nonlinear equation, LSSVM employs a different technique.

LSSVM uses the Sum Squares Error loss function as the empirical loss for training data. This change allows it to convert the quadratic programming problem that arises in traditional SVM into a linear programming problem that can be solved more efficiently. By doing so, LSSVM reduces the computational complexity involved in training the model.

To understand how LSSVM works, let's consider the samples of training data. Each sample is represented by the input vector x_i, where x_i is a vector in R representing the feature vector for the i-th sample. Similarly, y_i represents the target value associated with the i-th sample, and l is the sample space.

The expression of the LSSVM model in the feature space is given by:

y_x = ωT * φ(x) + b

where φ(x) is a dimensional transformation function that maps each input sample from its feature space to the high-dimensional feature space; ω is a weight vector; and b is a bias term.

By applying this model to new data points, we can solve for their corresponding output values. The process involves transforming the input data using the dimensional transformation function and then applying weights and biases to obtain the final output.

One key advantage of using LSSVM over other machine learning algorithms is its ability to handle larger feature spaces. This is achieved through efficient optimization techniques that enable the model to learn complex relationships between input features and output labels. As a result, LSSVM can be particularly effective in applications where traditional models struggle to capture such relationships.

..

(8) 优化问题可以通过将其转换为拉格朗日函数来求解。表达式如下：

L(ω， B, ξ， λ) = J(ω， ξ) - ∑i=1^l λ_i * wT * φ_x_i + b + ξ - y_i (9)

其中，λ_i 是拉格朗日乘子。偏导数简化为 KKT 条件，使用具有简单结构和强大泛化能力的 RBF 高斯核函数来解决问题。C 是与 RBF 函数相关的参数，决定了样本的映射范围。C 值越大，映射维度越高，训练效果越好，泛化能力越低。只需要优化核参数 C 和惩罚系数γ。LSSVM 的非线性函数方程为：y_x = ∑i=1^l λ_i * φ_x_i^T * φ_x_i + b (10)

2.4. PSO-LSSVM 算法

Figure 1 shows the overall process of the PSO-LSSVM algorithm. In this method, the performance of LSSVM, particularly the kernel parameter C and the penalty factor γ， is crucial to its accuracy. Therefore, a Particle Swarm Optimization (PSO) approach is employed to optimize these parameters. The specific steps involved in the PSO algorithm are outlined in Figure 1b. These steps include initializing parameters such as population size m, number of iterations k, learning factor c, initial position x, initial velocity v for the particles, etc.

The second step involves predicting the learning samples using the particle vectors in the LSSVM. A prediction error of the current position of each particle is calculated and used to determine its fitness value. Each particle's fitness value is compared with its optimal fitness value. If the particle's current fitness value is better than its optimal fitness value, the particle's current position is considered as an optimal position.

In the third step, the particles are updated based on their fitness values. The particles with higher fitness values have a higher chance of escaping from their current positions and exploring new regions, while particles with lower fitness values have a low probability of exploration. This process continues until a maximum number of iterations has been reached or a stopping criterion is met.

Finally, the optimized parameters C and γ are used to train an LSSVM model. By utilizing the PSO-LSSVM method, we can improve the accuracy of the LSSVM model by optimizing its hyperparameters through an iterative optimization process.

以下是根据您提供的内容重构的内容。请注意，我已经尽量保留原始的格式和结构，并对其中的数学公式进行了适当的调整：

1. 初始化粒子群优化算法，包括粒子数、迭代次数等参数；

2. 对于每个粒子，计算其适应度值(与种群最优位置的适应度值进行比较);

3. 如果某个粒子的适应度值优于其最优位置，则将其最优位置替换为种群最优位置；

4. 对于每个粒子，计算惯性权重并根据方程(5)和(6)更新其位置和速度；

5. 判断是否达到最大迭代次数或满足精度要求。如果满足任一条件，则结束过程并找到最优解。相反，步骤(b)将继续执行，并进行新一轮搜索。

在训练预测模型后，使用决定系数R2、均方根误差(RMSE)和平均绝对误差(MAE)分析预测结果。R2表示可解释变异占总变异的百分比。具有较高R2的模型更接近真实模型。

希望这个重构的内容能够满足您的需求！如果您有其他问题或需要进一步的帮助，请随时告诉我。

Reconstructed text:

The closer the model is to the data, the better its performance results. The Root Mean Square Error (RMSE) is used to measure the deviation between the predicted value and the measured value. The Mean Absolute Error (MAE) reflects the actual situation of prediction error. The smaller the RMSE and MAE are, the higher the accuracy of the model. These are calculated using the following formulae:

1. RMSE calculation:

RMSE = ∑ i = 1 N (Oi − O−) · (Pi − P−) ∑ i = 1 N (Oi − O−)2 · (Pi − P−)2

2. RMSE definition:

(11) RMSE = ∑ i = 1 N Pi − Oi2 N (12)

3. MAE definition:

(13) MAE = 1 N ∑ i = 1 N Pi − Oi (where Oi is the measured value, Pi is the predicted value, O− and P− represent their average values respectively, and N is the total number of samples.