## 3 Adaptive Time-Frequency Analysis
### Variational Mode Decomposition and Optimization Models
#### 3.2.4 Adaptive Variational Mode Decomposition  
Definition  
Adaptive variational mode decomposition (AVMD) is a signal decomposition method based on variational mode decomposition, but it possesses higher adaptability and applicability. AVMD adaptively selects the number of mode functions and the weighted average standard deviation through an online hyperparameter estimation method in order to minimize reconstruction error. Based on the selection results, adaptive basis functions and mode functions are computed, and signal separation is achieved. Application  
In the literature, researchers have focused on the fundamental research concerning AVMD. In robotic machining, chatter detection constitutes a complex task owing to the significant yet subtle variations in system stiffness. Consequently, an adaptive variational mode decomposition (AVMD) method grounded in kurtosis and instantaneous frequency has been proposed. This method aims to circumvent the issue of mode mixing inherent to the variational mode decomposition (VMD) process and facilitate the adaptive selection of decomposition parameters. The approach leverages a genetic algorithm to optimize decomposition parameters, significantly reducing optimization time by refining the discrete step size of these parameters. Approximate entropy, energy entropy, and a novel chatter feature based on entropy, termed the entropy drift coefficient, are extracted to differentiate between chatter and stable machining states [103]. Additionally, an adaptive noise reduction method based on automatic variational mode decomposition (AVMD) has been introduced for liquid pipeline leak detection [104]. This method enhances the VMD framework to effectively extract signal modes in environments with low signal-to-noise ratios. Furthermore, numerous models employ intelligent algorithms and local optimization techniques. A differential evolution-based adaptive variational mode decomposition technique (DAVMD) has been proposed [105]. This technique continuously adjusts the number of decompositions and the penalty factor of VMD through the differential evolution (DE) algorithm, optimizing the decomposition results in terms of energy distribution, frequency characteristics, and other aspects. By optimizing the number of decompositions and the penalty factor, the performance of variational mode decomposition is enhanced. Moreover, an adaptive variational mode decomposition (VMD) denoising scheme based on the whale optimization algorithm (WOA) has been developed for denoising ground-penetrating radar (GPR) data [106]. This scheme optimizes the denoising parameters of VMD using WOA, significantly improving the signal-to-noise ratio (SNR). Strength and Weakness  
* Adapts mode count
* Has strong noise resistance  
AVMD possesses unique advantages. First, AVMD has the merit of adaptive determination of the number of modes, making it a model of adaptive signal processing within the realm of signal processing. Traditional variational mode decomposition (VMD) is a fully non-recursive decomposition model, but it requires the predefinition of the number of modes, which limits the method's adaptability. Deviations in the predefined number of modes can lead to mode loss or mixing, affecting the accuracy of signal decomposition. In contrast, AVMD automatically determines the number of modes by analyzing signal characteristics (such as permutation entropy and kurtosis criteria), thereby avoiding the issues of mode loss or aliasing caused by inaccurate predefined mode numbers in VMD. For example, during the processing of electromechanical oscillation signals in power systems, AVMD can adaptively select the number of modes based on the spectral distribution, significantly improving decomposition accuracy [107]. Second, AVMD exhibits strong noise resistance and performs exceptionally well in noisy environments, effectively separating noise components from signals.