## 3 Adaptive Time-Frequency Analysis
### Variational Mode Decomposition and Optimization Models
#### 3.2.3 Variational Mode Extraction  
Definition  
Variational Mode Extraction (VME) is a signal processing method based on variational Bayesian inference. In VME, the modal functions of the signal are treated as latent variables, while the observed signal data are used to estimate the distributions of these latent variables. By maximizing the likelihood function, VME can find the modal function parameters that best fit the signal data. Application  
This method converts time lags into phase period differences through a variational mode extraction algorithm, enhancing the accuracy and automation of frequency domain analysis. Its effectiveness has been verified in the actual SHM system of the Shanghai Tower [98]. In terms of innovations, first, deep learning networks have been integrated. For instance, an intelligent fault diagnosis method for rolling bearings based on VME and an improved one-dimensional convolutional neural network (I-1 DCNN) has been proposed. By combining VME and I-1 DCNN, useful features of bearing vibration signals are extracted from noisy environments, improving the accuracy and computational efficiency of fault diagnosis [99]. Another method [100] involves rolling bearing fault diagnosis based on power spectral density-variational mode extraction (PSD-VME), multiclass relevance vector machines (mRVM), and Dempster-Shafer evidence theory. Features are extracted through PSD-VME and classified using mRVM and DS evidence theory, verifying the method's high classification accuracy with a small amount of sample data. Furthermore, innovations have been made by accurately grasping the advantages of VME. In the multi-fault diagnosis of rolling bearings, a variational mode extraction method based on the convergence trend of variational mode decomposition (VMD) has been applied [101]. The initial center frequency is adaptively determined through the VMD convergence trend graph, and kurtosis is used as an evaluation index to optimize the balance factor, enhancing the accuracy and computational efficiency of fault feature extraction. Additionally, parameter optimization algorithms have been employed. A parameter-adaptive variational mode extraction method based on particle swarm optimization (PSO) has been proposed for bearing fault diagnosis [102]. Researchers have constructed a comprehensive index, L-KCIE, to optimize VME parameters, and they verified its effectiveness and superiority in fault feature extraction. Strength and Weakness
* Extracts specific frequency components by setting the modal center frequencies. First, the VME method has a clear objective, enabling it to extract signals targeting specific desired modal center frequencies, thereby successfully obtaining valuable intrinsic modal functions and accurate fault characteristic frequencies. This makes VME more targeted and effective in signal processing. Second, VME offers high flexibility, allowing users to extract specific frequency components in signals by setting the desired modal center frequencies. This flexibility enables VME to adapt to different signal characteristics and analysis requirements. Third, VME demonstrates strong robustness when processing signals containing noise. By setting parameters reasonably, it can effectively extract useful information from signals while suppressing noise interference. Finally, compared to some traditional signal processing methods, VME may have certain advantages in terms of computational efficiency thanks to the optimization of its algorithm design and the simplification of the iteration process. However, VME also has some disadvantages. First, it cannot adaptively decompose multi-component signals and usually can only extract one component from a desired modal center frequency, which means that multiple processing steps or combining other methods may be required when dealing with complex signals. Second, the performance of VME can be significantly affected by initialization parameters. Improper parameter settings may lead to inaccurate extraction results or convergence to incorrect modes.