## 3 Adaptive Time-Frequency Analysis
### Empirical Mode Decomposition and Optimization Models
#### 3.1.1 Empirical Mode Decomposition  
Definition  
Empirical mode decomposition (EMD), proposed by [PERSON] et al. [34], is an adaptive signal processing method that decomposes a signal into a set of intrinsic mode functions (IMFs), each representing a natural oscillation mode in the signal. The EMD method extracts different frequency components from the signal through an iterative process based on the signal's local characteristic time scale. An IMF, which is the basic function obtained during the EMD decomposition process, must satisfy the following two conditions: (1) Relationship between the number of extrema and the number of zero-crossings: They must be equal or differ by at most one. (2) Mean of the local envelope: At any point, the mean of the envelope defined by the local maxima and minima is zero [35]. It is worth noting that IMFs are determined by the signal itself rather than predefined kernel functions, making EMD an adaptive signal processing method. The formula for the EMD method can be expressed as  
\[x(t)=\sum_{i=1}^{N}C_{i}(t)+R(t) \tag{4}\]  
where  
* \(x(t)\) is the original signal,
* \(C_{i}(t)\) is the \(i\)-th IMF,
* \(R(t)\) is the residual term, representing the part of the signal not decomposed into IMFs,
* \(N\) is the number of IMFs. Application  
In recent years, researchers have applied empirical mode decomposition (EMD) in conjunction with various network models. A multitude of methods have emerged for sequence prediction by integrating EMD with neural networks. Typically, EMD is used to decompose signals into modal functions, which are then analyzed using classical networks. One approach combines EMD with long short-term memory neural networks to estimate the state of health (SOH) and predict the remaining useful life (RUL) of lithium-ion batteries. EMD is utilized to decompose the voltage signals of lithium-ion batteries and extract intrinsic mode functions (IMFs). These IMFs are then processed by an LSTM network to predict the battery's SOH and RUL [36]. Another method involves a model that combines EMD with gated recurrent unit (GRU) neural networks. EMD is used to decompose time series data and extract IMFs, which are subsequently processed by a GRU network to enhance prediction accuracy and robustness [37]. Additionally, there is a predictive model that combines EMD with least squares support vector regression (LSSVR). EMD is used to decompose time series data and extract IMFs and residuals, which are then individually predicted by LSSVR. The final prediction is obtained by aggregating these individual predictions [38]. Furthermore, a hybrid model that integrates three classical models--EMD, LSTM, and autoregressive incorporated moving average (ARIMA)--is used for wind speed prediction. EMD is used to decompose wind speed data and extract IMFs, LSTM processes the time series data,and ARIMA captures the linear trends in the data. The overall model improves prediction accuracy and stability [39]. Moreover, research on noise reduction methods is also making progress. One noise reduction method combines uniform-phase empirical mode decomposition (MUPEMD) with a genetic algorithm-dynamic zero-phase interval filtering algorithm (GA-DZPIF) to improve the processing efficiency of ship-radiated noise signals and mitigate modal aliasing issues. MUPEMD is used to decompose the signals, while GA-DZPIF is used to filter the single-mode components, with structural similarity (SSIM) being used as an evaluation metric [40]. Additionally, a method based on improved ensemble noise reconstruction empirical mode decomposition (IENEMD) and adaptive threshold denoising (ATD) has been proposed. This method combines IENEMD and ATD to extract weak fault features of rolling bearings. IENEMD is used to decompose the signals, while ATD is used for denoising. The effectiveness and robustness of this method have been verified through simulation experiments and practical cases [41]. Furthermore, [PERSON] et al.