## 2 Methodology
### VMD Module  
Variational Mode Decomposition (VMD), first introduced in 2014 by [PERSON] and [PERSON], is a technique used to decompose nonlinear, non-stationary data into multiple Intrinsic Mode Functions (IMFs), separating the local characteristics of the data [36]. VMD decomposes the time series data into Intrinsic Mode Functions (IMFs), each representing a different frequency component. High-frequency IMFs capture short-term fluctuations, while low-frequency IMFs reveal long-term trends. This decomposition helps to isolate noise and enhance the detection of underlying periodic patterns, improving the accuracy of passenger flow predictions. The criteria for identifying an IMF can be summarized as follows:  
(1) The number of extrema and the number of zero-crossings must differ at most by one throughout the entire sequence;  
(2) At any point, the mean of the envelope defined by the local maxima and the envelope defined by the local minima is zero. VMD serves as a key step in the subway passenger flow prediction model proposed in this paper. The VMD of a signal \(X\) can be represented as follows:  
\[X=\sum_{k=1}^{K}u_{k}(t) \tag{2}\]  
where \(\mathrm{u_{k}(t)}\) are the IMFs, and k represents the number of Intrinsic Mode Functions. The purpose of VMD is to decompose complex subway passenger flow data into a finite set of IMFs, which each represent different frequency components. These IMFs are then used to reveal deeper characteristics of the subway passenger flow data. The  
Figure 2: Diagram of sliding window. objective of VMD is to minimize the bandwidth of the IMFs by solving the following optimization problem:  
\[min_{\{u_{k}\}}\sum_{k=1}^{K}\left\|\partial_{t}\Big{[}\big{(}u_{k}(t)-\hat{u}_{ k}(t)\big{)}^{2}\Big{]}\right\| \tag{3}\]  
where \(\hat{u}_{k}(t)\) are the Hilbert transforms of the IMF \(\hat{u}_{k}(t)\), and the operator \(\partial_{t}\) refers to the partial derivate with respect to time. The Hilbert transform is used for the instantaneous frequency analysis of the decomposed Intrinsic Mode Functions (IMFs), allowing for a more accurate capture of the nonlinear characteristics in the data. The optimization problem aims to minimize the total bandwidth of the IMFs by solving this equation iteratively. This ensures that the IMFs generated are as narrowband as possible, capturing distinct frequency components from the signal. In VMD, each IMF is modeled as a mode function \(u_{k}(t)\) that is the output of a bandpass filter with a fixed center frequency \(\omega_{k}\). The mode function can be expressed as follows:  
\[u_{k}(t)=A_{k}(t)cos(\phi_{k}(t)) \tag{4}\]  
where \(A_{k}(t)\) is the amplitude envelope, and \(\phi_{k}(t)\) is the instantaneous phase. VMD is a non-recursive, adaptive, and data-driven method that can decompose nonlinear and non-stationary data into IMFs with limited bandwidth. The center frequencies \(\omega_{k}\) and bandwidths are iteratively updated by solving the constrained optimization problem to ensure accurate decomposition. The parameters of these mode functions, including their center frequencies and bandwidths, are determined by solving a constrained optimization problem that minimizes the bandwidth of each IMF and makes their sum as close as possible to the original signal. As shown in Figure 3, the VMD module in this paper executes the following steps:  
Step 1: Data Preprocessing. The original metro passenger flow data are normalized to ensure consistent scaling, reducing the influence of different data scales on the analysis and aiding convergence in subsequent optimization. The preprocessed data \(X(t)\) are input into  
Figure 3: The procedure of VMD. the VMD module, where they are first transformed into a variational optimization problem. The original time series \(X(t)\) can be represented as follows:  
\[X(t)=\sum_{k=1}^{K}u_{k}(t) \tag{5}\]  
Step2: Transformation to an Unconstrained Optimization Problem.