# Hybrid Empirical and Variational Mode Decomposition of Vibratory Signals
## 2 Evolution of the Hilbert-Huang Transform
This method has been applied in various fields, such as biomedical signal analysis and vibration monitoring [46; 47; 48; 49]. In more recent developments, Variational Mode Decomposition (VMD) has been introduced as an advanced technique for adaptive signal decomposition, addressing mode mixing through a variational approach [20]. This technique represents a significant evolution in signal analysis methods as illustrated in the Figure 1, offering a robust alternative to EMD for analyzing nonlinear and nonstationary signals. Since its introduction, the HHT has been applied in diverse fields, including environmental and geophysical sciences, neurology, and technological innovation. In environmental science, it has facilitated the analysis of PM10 particle concentration fluctuations in bulk loading ports [50] and has played a crucial role in identifying areas prone to landslides in Taiwan [51]. In the context of natural disaster prevention, its use in analyzing infrasound signals for early warning systems for tsunamis highlights the potential of the HHT to contribute to safety and preparedness [52]. In neurology, the HHT has enabled advances in understanding cerebral blood flow regulation [53] and the analysis of electroencephalographic signals, opening up new possibilities in the study of brain activity and the development of advanced brain-computer interfaces [54; 55]. In technological innovation, the HHT has spurred advancements in structural health monitoring and fault diagnosis in machinery, proving exceptional for detecting structural damage [56] and diagnosing faults in induction machine bearings [16; 57]. In engineering and materials science, the HHT has been applied to innovative solutions such as gas detection using EFPI fiber signals and stochastic analysis of vibratory signals from induction motors for fault detection [58; 59]. The integration of the HHT with artificial intelligence techniques for structural health monitoring and the improvement in feature selection for fault diagnosis in ball bearings underscores its synergy with advanced statistical methods [50; 60]. In geophysics and seismology, the HHT has enhanced our understanding of complex natural phenomena, from magnetic properties linked to density irregularity structures to the study of blasting vibration effects in tunnels [61; 62]. Its application in estimating  
Figure 1: Timeline of signal analysis techniques, from FFT to VMD. Advantages are broken into lines for improved readability. lightning current parameters and calculating seismic intensity parameters for designing reinforced concrete structures showcases its contribution to safety and effectiveness in analyzing natural phenomena and their effects on infrastructure [63,64]. Since its introduction, the HHT has undergone numerous studies aimed at enhancing its precision and effectiveness. For example, the Cole-Hopf transformation has been pivotal in linking linear and nonlinear equations in Banach spaces, providing a basis for understanding the relationship between the evolution operator and its infinitesimal generator within a nonlinear semigroup framework [65]. The trend toward big data analysis has become apparent, with the HHT being applied more extensively thanks to its integration with big data technologies, improving the detection of complex patterns and prediction in various fields [66,67]. The algorithms underlying EMD and its variants, such as EEMD and CEEMDAN, have been refined to improve their efficiency and precision, including the development of adaptive approaches for sifting and parameter selection. Comprehensive reviews of the evolution, challenges, and solutions surrounding the HHT highlight its versatile applications in fields ranging from mechanical engineering to biomedicine [68,69]. These studies emphasize how the HHT has evolved to address complex signal analysis problems by adapting to the inherent characteristics of the analyzed signals. The HHT continues to evolve, addressing challenges and finding applications in a wide range of fields.