# Research on Price Prediction of Construction Materials Based on VMD-SSA-LSTM
## 2 Methodology  
### Variational Mode Decomposition (VMD)  
[PERSON] et al. [31] applied Variational Mode Decomposition (VMD) and integrated it with LSTM to develop a novel wind speed prediction model, VMD-LSTM. The proposed model significantly improved both the prediction accuracy and performance. VMD is a powerful signal processing tool that decomposes a complex signal into multiple Intrinsic Mode Function (IMF) components with different center frequencies. Figure 1 is a detailed flowchart of the VMD. \begin{table}
\begin{tabular}{c c c c c} \hline
**Category** & **Method** & **Characteristics** & **Limitations** & **Submitter** \\ \hline \multirow{4}{*}{Statistical models} & \multirow{4}{*}{ARIMA} & Time series analysis, suitable for linear and stationary data & Assumes linearity and stationarity, cannot handle nonlinearity & [PERSON] et al. [18]; [PERSON] et al. [19, 20] \\ \cline{2-5}  & & GM (1,1) & Grey forecasting, analyzes small sample data & Sensitive to input data quality, poor for long-term forecasting & [PERSON] [17] \\ \hline \multirow{4}{*}{Machine learning} & Support Vector Machine (SVM) & Regression modeling for small to medium datasets & Sensitive to parameter tuning, requires thorough preprocessing & [PERSON] et al. [28] \\ \cline{2-5}  & Artificial Neural Networks (ANN) & Mimics brain structure, strong nonlinear modeling & Prone to overfitting, demands large datasets and computational resources & [PERSON] et al. [15]; [PERSON] et al. [12] \\ \cline{2-5}  & Long Short-Term Memory (LSTM) & A type of recurrent neural network, captures long-term dependencies & Sensitive to noise and non-stationary data, complex optimization & [PERSON] et al. [23]; [PERSON] et al. [24]; \\ \hline \multirow{4}{*}{Hybrid approaches} & GM (1,1) + BP & Combines grey forecasting and neural networks for improved accuracy & Higher computational complexity & [PERSON] et al. [26] \\ \cline{2-5}  & LSTM + Data Preprocessing & Combines interpolation, noise reduction, normalization with LSTM & Complex preprocessing, high implementation requirements & [PERSON] et al. [22] \\ \cline{2-5}  & LSTM +VMD & Signal decomposition (VMD) + LSTM & High computational cost, complex implementation and validation & [PERSON] et al. [31] \\ \cline{2-5}  & LSTM + SSA & Optimization (SSA) + LSTM & High computational cost, complex implementation and validation & [PERSON] et al. [32] \\ \hline \multirow{4}{*}{Optimization algorithms} & genetic algorithm (GA) or particle swarm optimization (PSO) & Optimizes neural network parameters & Prone to local optima, sensitive to initial parameters & [PERSON] et al. [27]; [PERSON] et al. [25]; [PERSON] et al. [28] \\ \hline \end{tabular}
\end{table} 
Table 1: Comparison table of different construction material price forecasting methods. The center frequency and bandwidth of each mode function are determined adaptively without the need to pre-set the basis functions, making VMD suitable for processing non-stationary signals and signals with complex frequency characteristics. VMD uses the alternating-direction multiplier method (ADMM) to decompose \(f(t)\) into K subsequences. The specific steps and formulas are given in Equations (1)-(4). Step 1: for each \(u_{k}\), the associated analytic signal is computed using the Hilbert transform and the spectrum is constructed. \[\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum_{i\
eq k}\hat{u}_{i}^{n+1 }(\omega)+\hat{\lambda}^{n}(\omega)/2}{1+2\alpha(\omega+\omega_{k}^{n})^{2}} \tag{1}\]  
Step 2: the modal spectrum is shifted to the baseband using the respective estimated center frequencies, which are calculated as follows  
\[\omega_{k}=\frac{\int_{0}^{\infty}\omega{\left|\hat{u}_{k}(\omega)\right|}^{2 }d\omega}{\int_{0}^{\infty}{\left|\hat{u}_{k}(\omega)\right|}^{2}d\omega} \tag{2}\]  
Step 3: Introducing a quadratic penalty term and the Lagrange multiplier \(\lambda\) makes the problem unconstrained; the bandwidth is estimated by the Gaussian smoothness of the demodulated signal, i.e., the \(L^{2}\)-parameter of the gradient.