# Displacement Patterns and Predictive Modeling of Slopes in the Bayan Obo Open-Pit Iron Mine
## 2 Data and Methodology
### Methodology
#### 2.3.2 The Decomposition Principle and Process of PSO-VMD  
Particle Swarm Optimization (PSO) [37] is a swarm intelligence-based optimization algorithm proposed by [PERSON] and [PERSON] in 1995, inspired by bird flock foraging and fish swarm predation behaviors. PSO seeks optimal solutions by simulating information sharing and collaboration among individuals (particles). It features simplicity, ease of implementation, and fast convergence, making it widely applicable to optimization problems such as parameter optimization, neural network training, and path planning. The basic principle involves randomly generating a particle swarm, with each particle's initial position \(\mathrm{x_{i}}\) and velocity \(\mathrm{v_{i}}\) randomly initialized. Each particle records its individual best position \(\mathrm{pbest_{i}}\) (the best position in the history of the particle) and the global best position \(\mathrm{gbest_{i}}\) (the best position in the history of the entire swarm). The individual best and global best are updated using the following velocity update Equation (8) and position update Equation (9):  
\[\mathrm{v_{i}^{t+1}}_{\mathrm{i}}=\omega\cdot\mathrm{v_{i}^{t}}+c_{1}\cdot \mathrm{r_{1}}\cdot\left(\mathrm{pbest_{i}}-\mathrm{x_{i}^{t}}\right)+c_{2} \cdot\mathrm{r_{2}}\cdot\left(\mathrm{gbest-x_{i}^{t}}\right) \tag{8}\]  
Figure 5: K-distance plot. Figure 6: Silhouette coefficient heatmap. The red star indicates the optimal parameter. where \(\omega\) is the inertia weight, balancing global convergence and convergence speed; \(c_{1}\) and \(c_{2}\) are individual and social learning factors, respectively; and \(r_{1}\) and \(r_{2}\) are random numbers in \([0,1]\) to enhance exploratory randomness. \[x^{t+1}_{i}=x^{t}_{i}+v^{t+1}_{i} \tag{9}\]  
If the current fitness value surpasses the individual best, \(\text{pbest}_{i}\)=\(x^{t+1}_{i}\) is updated. If a particle's fitness exceeds the global best, \(\text{gbest}\) is updated. These steps are repeated until the termination criteria (fitness convergence) are met. VMD is an adaptive signal decomposition method. Its core idea is to decompose raw signals into multiple Intrinsic Mode Functions (IMFs) with specific central frequencies by constructing a variational optimization problem. Unlike heuristic methods, such as EMD, VMD employs a rigorous mathematical framework solved iteratively via the Alternating Direction Method of Multipliers (ADMM). Assuming a signal f(t) can be decomposed into K modes \(u_{k}(t)\), each with a central frequency \(\omega_{k}\), VMD aims to minimize the total bandwidth of all modes while ensuring their sum equals the original signal. The constrained variational problem is formulated as (Equation (10)):  
\[ \begin{split}\min_{\{u_{k},\omega_{k}\}}\sum\limits_{k=1}^{K} \left\|\partial_{t}((\delta_{t}+\frac{i}{\pi t})*u_{k}(t))\mathrm{e}^{-j \omega_{k}t}\right\|_{2}^{2}\\ \text{s.t.}\sum\limits_{k=1}^{K}u_{k}(t)=f(t)\end{split}  \tag{10}\]  
where \(u_{k}\) are mode components, f is the original signal, t is the time window, \(\delta_{t}\) is the Dirac function, \(\omega_{k}\) values are central frequencies,\(\mathrm{e}^{-j\omega_{k}t}\) represents the estimated central frequency of analytic signals, k is the number of modes, j is the imaginary unit, and \(*\) denotes convolution. By introducing a quadratic penalty term \(\alpha\) and Lagrange multiplier \(\lambda(t)\), the constrained problem is transformed into an unconstrained optimization (Equation (11)):  
\[ \begin{split} L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum \limits_{k=1}^{K}\left\|\partial_{t}((\delta_{t}+\frac{j}{\pi t})*u_{k}) \mathrm{e}^{-j\omega_{k}t}\right\|_{2}^{2}+\left\|f(t)-\sum\limits_{k=1}^{K} u_{k}(t)\right\|_{2}^{2}+\left\langle\lambda(t),f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\right\rangle \end{split}  \tag{11}\]  
The ADMM algorithm alternately updates \(u_{k}\), \(\omega_{k}\), and \(\lambda\) until convergence. The update equations for \(u_{k}\) and \(\omega_{k}\) are given in Equations (12) and (13):  
\[