Improvement of Signal Decomposition Variational Mode Decomposition Method Based on African Vulture Optimization Algorithm

In the field of signal processing, the Variational Mode Decomposition (VMD) [1] algorithm has garnered widespread attention for its exceptional adaptability and efficiency. This algorithm effectively decomposes complex signals into multiple modal components with distinct frequency and temporal characteristics, providing a more precise time-frequency analysis method compared to traditional Fourier transform, especially when dealing with non-stationary signals. However, the selection of parameters in the VMD algorithm significantly impacts the results, and optimizing these parameters for various practical engineering problems presents a meaningful engineering challenge [2-5].

Optimization algorithms, as powerful mathematical tools, play a crucial role in various fields such as industrial manufacturing, transportation, finance, and artificial intelligence. In the realm of industrial manufacturing, these algorithms enhance processes, logistics routes, and production plans, effectively increasing efficiency and reducing costs. In the transportation sector, they refine public transit routes, flight schedules, and shipping paths, markedly improving operational efficiency [6-9]. As an advanced form of heuristic algorithms [10, 11], metaheuristic algorithms emulate the evolutionary processes found in nature. They significantly enhance flexibility and evolutionary capabilities by adaptively learning and adjusting predefined heuristic methods through higher-level adaptive strategies and meta-knowledge. This enables metaheuristic algorithms to handle complexity more effectively than traditional heuristic algorithms and to generalize well across a variety of problem domains [12-18].

The African Vulture Optimization Algorithm (AVOV), inspired by the hunting behavior of African vultures, is a notable metaheuristic algorithm [19]. Since its introduction in 2021, it has been recognized for its ability to utilize individual and social intelligence to find optimal solutions. The algorithm consists of three stages: reconnaissance, foraging, and aggregation. During the reconnaissance phase, potential areas are searched; in the foraging phase, individuals explore these areas with random movements; and during the aggregation phase, individuals interact and cooperate to effectively discover and exploit promising areas.

In recent years, researchers have made significant progress in the improvement and application of the AVOA algorithm. For instance, Zhang et al. [20] proposed a novel optimized design of a hybrid AlexNet/Extreme Learning Machine (ELM) network for providing an optimal identification tool for Proton Exchange Membrane Fuel Cells (PEMFCs). Gürses et al. [21] investigated the optimization problem of shell-and-tube heat exchangers using the AVOA. Kumar and Mary [22] improved the AVOA algorithm based on the Newton-Raphson method to accurately predict photovoltaic power output and determine the optimal model. Furthermore, other specific algorithms based on AVOA have been developed for different domains: Alanazi et al. optimized photovoltaic systems; Khodadadi et al. [23] utilized multi-objective version of AVOA to solve industrial engineering problems; Balakrishnan et al. [24] performed feature selection and sentiment analysis on movie reviews. In addition, the AVOA has been combined with other algorithms to create hybrid versions. For example, Xiao et al. [25] integrated it with the Ant Colony Optimization algorithm, while Liu et al. [26] combined it with the Honey Badger Optimizer. Furthermore, researchers have also incorporated AVOA with other optimization algorithms, such as Differential Evolution and Harmony Search, to address various practical problems. These successful enhancements and applications have expanded the potential of AVOA, providing effective solutions across multiple fields.

In this study, the sensitivity of AVOV to the location of convergence points was investigated, and an Enhanced Convergence (EC) strategy was proposed to address this issue. An in-depth analysis of the AVOV algorithm's performance on various test functions revealed significant sensitivity to the position of the optimal solution, particularly when the optimal solution was not located at the origin. Under such conditions, the convergence speed and precision of the algorithm were notably reduced. To mitigate this limitation, an Enhanced Convergence strategy was introduced, which optimized the core formulas of the algorithm to maintain efficient convergence performance even when the optimal solution was not at the origin.

Experimental results indicated that the improved EC-AVOV algorithm exhibited higher convergence accuracy and stability across different offset conditions. Compared to the original AVOV algorithm, the EC-AVOV demonstrated the ability to rapidly converge to the global optimal solution when dealing with complex functions that featured non-zero offsets, while effectively avoiding local optima. Additionally, the EC-AVOV algorithm achieved significant enhancements in global search capability and robustness, showing good adaptability and convergence effects in the optimization of both unimodal and multimodal functions.

The AVOA, a newly proposed metaheuristic in 2021 by Abdollahzadeh et al. [19], mimics the competition and navigation behaviors of African vultures. African vultures are intelligent and resilient creatures due to their unique physical features. The rate of starvation for the i-th vulture at iteration k is denoted by S_i,k and computed using Eqs. (3) and (4):

${{S}_{i,k}}=(2\times {{r}_{1}}+1)\times z\times \left( 1-\frac{k}{K} \right)+s$                         (3)

where, s denotes the disturbance term affecting the starvation rate which can be calculated as follows:

$s=h\times \left( {{\sin }^{w}}\left( \frac{\pi \times k}{2\times K} \right)+\cos \left( \frac{\pi \times k}{2\times K} \right)-1 \right)$                         (4)

where, random values for the variables h,r1, and z are selected from the intervals [-2, 2], [0, 1], and [-1, 1], respectively. In AVOA, the parameter w is set to 2.5, k denotes the current iteration number, and K represents the maximum number of iterations. Depending on the value of starvation rate S_i,k, AVOA updates the positions of vultures using different formulas.

To showcase the key characteristics of vultures, the AVOA selects the first or second-best vulture as the lead vulture through Eq. (5):

${{R}_{i,k}}=\left\{ \begin{array}{*{35}{l}}   B{{V}_{1}} & \text{if  }p>\text{rand}  \\   B{{V}_{2}} & \text{otherwise}  \\\end{array} \right.$                         (5)

where, R_i,k represents the randomly selected lead vulture, whereas BV₁ and BV₂ represent the vultures ranked first and second respectively. The value of constant p is set to 0.8 in AVOA.

3.1 Exploration phase

When |S_i,k| is larger than 1, vultures explore the entire solution space randomly. The exploration process employs two strategies based on the foraging behavior of vultures guarding their food. The mathematical model can be described by the Eqs. (6)-(8):

${{P}_{i,k+1}}=\left\{ \begin{array}{*{35}{l}}   \text{Eq}.(\text{5}) & \text{if }{{P}_{1}}\ge ran{{d}_{{{P}_{1}}}}  \\   \text{Eq}.(6) & \text{otherwise}  \\\end{array} \right.$                         (6)

${{P}_{i,k+1}}={{R}_{i,k}}-{{D}_{i,k}}\times {{S}_{i,k}}\quad$                         (7)

${{P}_{i,k+1}}={{R}_{i,k}}-{{S}_{i,k}}+ran{{d}_{1}}\times {{U}_{range}}(lb,ub)$                         (8)

where, P_i,k+1 denotes the newly generated position, P₁ is set to 0.6, rand_P1 represents a random number between 0 and 1, D_i,k represents the random distance between the current vulture and the selected lead vulture, $D_{i, k}=\left|X \times R_{i, k}-P_{i, k}\right|$ represents the random distance between the current vulture and the lead vulture, X is chosen randomly from the range [0, 2], while ub and lb denote the upper and lower bounds respectively. rand₁ is a random number between 0 and 1, and $U_{\textit{range}}(l b, u b)$ generates a uniformly distributed random number in the interval [lb, ub].

3.2 Exploitation phase

When |S_i,k| is less than 1, the vultures enter the development phase which comprises two stages. The first stage commences when |S_i,k| lies between 0.5 and 1, as demonstrated in Eq. (9):

${{P}_{i,k+1}}=\left\{ \begin{array}{*{35}{l}}   \text{Eq}.(8) & {{P}_{2}}\ge rand{{P}_{2}}  \\   \text{Eq}.(9) & {{P}_{2}}<rand{{P}_{2}}  \\\end{array} \right.$                         (9)

During this stage, vultures compete for food where they perform rotating flights, which are modeled by Eqs. (10) and (11), respectively. Here, P₂ is set to 0.4, and randP₂ is a random number between 0 and 1.

${{P}_{i,k+1}}={{D}_{i,k}}\times ({{S}_{i,k}}+ran{{d}_{2}})-({{R}_{i,k}}-{{P}_{i,k}})$                         (10)

$\begin{align}  & {{P}_{i,k+1}}={{R}_{i,k}}- ({{R}_{i,k}}\times \frac{{{P}_{i,k}}}{2\pi }\times (ran{{d}_{3}}\times \cos ({{P}_{i,k}})+ran{{d}_{4}}\times \sin ({{P}_{i,k}}))) \\\end{align}$                         (11)

where, rand₂, rand₃ and rand₄ are all random numbers between 0 and 1.

During the second stage, when |S_i,k| < 0.5, the AVOA algorithm simulates two vulture behaviors: accumulation around the food source and aggressive competition for food. This process is described by Eq. (12) , while Eqs. (13) and (14) illustrate how vultures move around the food source. Eqs. (15)-(17) model the aggressive behavior of vultures towards the food source.

${{P}_{i,k+1}}=\left\{ \begin{array}{*{35}{l}}   \text{Eq}.\text{  }\left( 11 \right) & {{P}_{3}}\ge ran{{d}_{{{P}_{3}}}}  \\   \text{Eq}.\text{  }\left( 13 \right) & {{P}_{3}}<ran{{d}_{{{P}_{3}}}}  \\\end{array} \right.$                         (12)

${{P}_{i,k+1}}=\frac{{{A}_{1}}+{{A}_{2}}}{2}$                         (13)

$\left\{ \begin{array}{*{35}{l}}   {{A}_{1}}=B{{V}_{1}}(t)-\frac{B{{V}_{1}}(t)\times {{P}_{i,k}}}{B{{V}_{1}}(t)-P_{i,k}^{2}}\times {{S}_{i,k}}  \\   {{A}_{2}}=B{{V}_{2}}(t)-\frac{B{{V}_{2}}(t)\times {{P}_{i,k}}}{B{{V}_{2}}(t)-P_{i,k}^{2}}\times {{S}_{i,k}}  \\\end{array} \right.$                         (14)

${{P}_{i,k+1}}={{R}_{i,k}}-|{{D}_{i,k}}|\times {{S}_{i,k}}\times Levy(d)$                         (15)

$Levy(d)=0.01\times \frac{u}{|v{{|}^{\frac{1}{\beta }}}}\text{ }u\sim(0,\sigma _{u}^{2}),v\sim(0,\sigma _{v}^{2})$                         (16)

${{\sigma }_{u}}={{\left( \frac{\Gamma (1+\beta )\times \sin (\frac{\pi \beta }{2})}{\Gamma \left( \frac{1+\beta }{2} \right)\times \beta \times {{2}^{\frac{\beta -1}{\beta }}}} \right)}^{\frac{1}{\beta }}}$                         (17)

In Eq. (12), P₃ is set to 0.4, and randP₃ is a random number within [0,1].

The current study aims to validate that the new algorithm maintains high convergence accuracy even when the optimal solution is not at the origin. To achieve this, we employed the EC-AVOA algorithm to compute eight functions listed in Table 2, with offset values of δ set to 1, 10, and 100, respectively. Subsequently, these computational results were compared with those obtained using the AVOA algorithm. The iterative convergence curves are depicted in Figures 2, with detailed data recorded in Table 3.

Based on the data in Table 3 and Figures 2, the following conclusions can be drawn: as the offset δ increases, the convergence accuracy of the AVOA algorithm significantly decreases. This implies that as the offset increases, the difficulty for these algorithms to find the optimal solution relatively increases. In contrast, the convergence accuracy of the EC-AVOA algorithm also slightly declines, but its performance surpasses that of the original algorithm. The EC-AVOA algorithm maintains relatively stable convergence accuracy under different offsets. It enhances the global convergence and mitigates the impact of the offset by introducing a forced convergence mechanism, thereby improving the performance of the AVOA algorithm. This modification effectively enhances the algorithm's global search capability and robustness, achieving good convergence performance across various offset values. Moreover, except for the F3 function, the EC-AVOA algorithm demonstrates satisfactory convergence performance for both unimodal and multimodal functions.

The algorithm exhibits stability in handling offsets and has been proven to be more suitable and reliable for solving practical problems. In practice, many problem-solving processes involve certain offsets. The EC-AVOA algorithm effectively addresses these offsets and maintains high convergence accuracy, making it a reliable choice for optimizing various practical problems. Whether dealing with unimodal or multimodal functions, the EC-AVOA algorithm shows satisfactory convergence performance, thus serving as a powerful tool for solving complex problems. Therefore, when addressing real-world problems that require consideration of optimal solutions not located at the origin, the EC-AVOA algorithm is trustworthy and worth adopting.

Table 3. Comparison of sensitivity analysis of new algorithms and AVOV on 8 offset functions using δ = 1, 10, and 100

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Figure 2. The convergence curves of new algorithms and AVOV on 8 offset functions using δ = 0, 1, 10, and 100