## 3 Adaptive Time-Frequency Analysis
### Variational Mode Decomposition and Optimization Models
#### 3.2.1 Variational Mode Decomposition  
Definition  
Variational mode decomposition (VMD) is a signal processing method proposed by [PERSON] et al. in 2013 [65]. VMD decomposes a signal into several intrinsic mode functions (IMFs), each of which possesses a specific center frequency and bandwidth. The core idea of VMD is to achieve signal decomposition by minimizing the solution to a variational problem, ensuring that each IMF is as concentrated as possible within a certain frequency band [69]. Application  
Variational mode decomposition (VMD) has demonstrated remarkable performance in machinery fault diagnosis, particularly in extracting fault features from rotating machinery such as bearings and gearboxes. By decomposing vibration signals, VMD effectively isolates fault characteristic frequencies and, when combined with methods like envelope spectrum analysis, achieves precise fault diagnosis. For instance, a method for estimating spalling size in rolling bearings based on signal processing techniques has been proposed [70]. Another example is a natural gas pipeline leak detection method based on optimized VMD and the ConvFormer model [71]. Furthermore, VMD has been applied to the prediction of structures such as bridges and buildings. For example, by decomposing the vibration response signals of structures, VMD can identify modal changes caused by damage, providing a basis for predictive assessment. To accommodate different prediction scenarios, researchers have also optimized the VMD model in various ways. One study introduced a thermal load time series decomposition and prediction model based on VMD and fast independent component analysis (FastICA) [72]. Another proposed a short-term offshore wind speed prediction method based on a hybrid VMD-PE-FCGRU model [73]. Additionally, a lithium-ion battery health state estimation method based on the MEWOA-VMD and Transformer architecture has been developed [74]. There is also a concrete dam displacement prediction model based on VMD, TSVR, and GRU [75]. In another approach, complex power load time series are decomposed into multiple relatively simple subsequences using VMD, and by integrating the ICSS algorithm, abrupt and irregular changes in the time series are identified and separated. Finally, neural networks in deep learning are used to predict these subsequences, and the final prediction results are obtained through ensemble learning, significantly improving the accuracy and stability of predictions [76]. VMD is also widely applied in signal denoising and feature extraction. Thanks to its inherent proficiency in handling noise, VMD excels in these applications. Moreover, researchers are dedicated to optimizing and upgrading VMD using innovative methods. One such method is a signal preprocessing approach based on VMD technology and sample entropy. This method decomposes milling robot response signals into multiple modal components using VMD and then calculates the sample entropy of each component to assess its complexity and irregularity. By setting appropriate thresholds, modal components containing amplitude and frequency modulation (AM-FM) harmonic components can be identified and eliminated, achieving the goals of denoising and signal optimization [77]. Another method combines VMD, empirical wavelet transform (EWT), and continuous wavelet transform (CWT) to improve the signal-to-noise ratio of acoustic signals and accurately detect pipeline leak locations. By analyzing and processing signals in different frequency domains, noise can be effectively suppressed, and useful information can be extracted, significantly enhancing the accuracy of leak detection [78]. A joint denoising method based on VMD parameter optimization has been proposed for removing noise from the acoustic signals of high-voltage shunt reactors [79]. This method integrates VMD, complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), and an improved continuous wavelet transform (ICWT) structure.