## 3 Adaptive Time-Frequency Analysis
### Empirical Mode Decomposition and Optimization Models
#### 3.1.1 Empirical Mode Decomposition
have proposed a hybrid scheme for rolling bearing fault prediction based on empirical mode decomposition (EMD) and kernel principal component analysis (PCA). This hybrid scheme combines complete ensemble empirical mode decomposition with multiscale interpolation (CEEMD-Msl) and PCA to construct a nonlinear health indicator, and it uses a dual-channel transformer network and convolutional block attention module to extract multi-domain features for rolling bearing fault prediction [42]. Strength and Weakness  
No need to preset the number of modes or frequency ranges  
Capable of capturing complex patterns and transient features  
Empirical mode decomposition (EMD) is the most classic adaptive time-frequency analysis technique. Compared to the aforementioned non-parametric dependent time-frequency analysis techniques, it can decompose signals based on their inherent characteristics without the need to preset the number of modes or frequency ranges. This method is particularly suitable for processing signals with complex nonlinear and multiscale characteristics. Furthermore, the decomposition process of EMD is driven by data iteration, based on the local maxima and minima of the signal. During this process, the signal is naturally decomposed into a series of intrinsic mode functions (IMFs) without the need for additional prior knowledge or model parameter adjustments. Compared with traditional Fourier transform and wavelet transform, the mode-mixing phenomenon in EMD is reduced. Each IMF represents a local manifestation of the signal within a specific frequency range. Although they are not completely orthogonal, they ensure a certain degree of frequency-time resolution of the signal. In terms of handling boundary effects, EMD has also undergone certain optimizations. By specially processing the signal boundaries, the influence of boundary effects on the decomposition results is reduced. In contrast, the aforementioned time-frequency analysis methods may encounter issues such as reduced resolution or spectrum leakage at signal boundaries. Additionally, the IMFs obtained after EMD decomposition can be reconstructed through simple addition operations with high reconstruction accuracy. This characteristic gives EMD advantages in applications such as signal denoising, feature extraction, and pattern recognition. It is particularly noteworthy that EMD is especially suitable for processing nonlinear and non-stationary signals and is capable of capturing complex patterns and transient features in the signal. In contrast, Wavelet transform and short-time Fourier transform may require more complex multiscale analysis when processing nonlinear signals. Finally, regarding the adaptability of EMD, it can process any type of signal, including process control [43; 44], modeling [45; 46], surface engineering, wind speed prediction [47],system identification [48], bearing diagnosis [49], etc. This excellent adaptability makes its application versatile. Of course, as the most basic adaptive time-frequency analysis technique, EMD still has much room for improvement. Its disadvantages include endpoint effects and energy non-conservation, which are inherent to the EMD algorithm itself. Since IMFs are not strictly orthogonal, the total energy of the decomposed IMFs may not match the energy of the original signal, leading to energy non-conservation in some cases. For example, in the study by [PERSON] [35], simulations and analyses of IMFs revealed distortions at both ends of the IMFs, and the decomposition results also exhibited distortions at both ends. Mode mixing, although less severe than in traditional methods, remains an issue, with single IMFs containing oscillations of different scales, or oscillations of similar scales being distributed across different IMFs. This is caused by the intermittency of the signal. As stated by [PERSON] et al. [34], intermittency can lead to severe mixing of time-frequency distributions, making the physical meaning of individual IMFs ambiguous.