## 4 Vibration Signal Denoising
### Ivy Optimization Algorithm  
The Ivy optimization algorithm (IVY), proposed by [PERSON] et al. in 2024 [29], is an innovative intelligent optimization algorithm. This algorithm emulates the growth patterns of ivy plants, achieving optimization by orchestrating the orderly population growth and the diffusion and evolution processes of ivy [30; 31]. IVY leverages knowledge from neighboring ivy plants to determine growth directions and self-improves by selecting the nearest and most influential neighbors, thereby continuously enhancing solution accuracy. The algorithm principle is as follows:  
1. Population Initialization: The Ivy algorithm models the population as a group of ivy plants, where each individual represents a potential solution. During initialization, the ivy population can be expressed as \(I=(I_{1},I_{2},\ldots,I_{N_{\text{pop}}})\), where the \(i\)-th population is defined as \(I_{i}=(I_{11},\ldots,I_{ID})\), \(i=1,2,\ldots,N_{\text{pop}}\). Each individual in the population (i.e., ivy plant) has a randomly generated position, representing the potential solution in the search space. The position is calculated as follows: \[I_{i}=I_{\text{min}}+\text{rand}(1,\text{D})\odot(I_{\text{max}}-I_{\text{min}})\] (12) where \(\text{rand}(1,\text{D})\) is a vector of dimension \(D\) with elements uniformly distributed in the range \([0,1]\); \(I_{\text{min}}\) and \(I_{\text{max}}\) are the lower and upper bounds of the search space, respectively; and \(\odot\) represents the Hadamard product (element-wise multiplication) between two vectors. 2. Coordinated and Ordered Population Growth:This algorithm uses a differential equation to simulate the growth rate \(Gv(t)\) of ivy. It considers the relationship between the growth rate and time, and adjusts the growth rate and correction factors \(\Psi\) and \(\varphi\) to simulate the ivy's adaptation to the environment and resource acquisition strategy. The population growth rules are as follows:  
\[\frac{dGv(t)}{dt}=\psi\cdot Gv(t)\cdot\varphi(Gv(t)) \tag{13}\]  
\[\Delta Gv_{i}(t+1)=\text{rand}^{2}\odot(N(1,\text{D})\odot\Delta Gv_{i}(t)) \tag{14}\]  
the vectors \(\Delta Gv_{i}(t)\) and \(\Delta Gv_{i}(t+1)\) represent the growth rates at time \(t\) and \(t+1\), respectively. rand is a random real number within the interval \([0,1]\). rand\({}^{2}\) is a random number with a probability density function equal to \(\frac{1}{2\sqrt{x}}\). \(N(1,\text{D})\) denotes a random vector that follows a normal distribution, where \(1\) is the mean and D is the dimension. 3. Path to Sunlight and Growth Strategy: In natural environments, it is crucial for ivy to find a surface to attach to in order to climb toward sunlight. In forests, young ivy typically prioritizes nearby trees as growth targets, which are often already occupied by mature ivy. Through this strategy, ivy gradually expands its coverage, eventually forming a vast, interconnected area across the forest. The young ivy clings to the mature ivy for climbing, and this competitive mechanism ultimately ensures that only the strongest individuals survive within the entire ivy population, regardless of their age. The following equation describes how a young ivy \(I_{i}\) utilizes an older ivy \(I_{ii}\) to climb and logically move in the direction of the light source: \[I_{i}^{\text{new}}=I_{i}+|N(1,\text{D})|\odot(I_{i}-I_{i})+N(1,\text{D})\odot \Delta Gv_{i}\] (15) \[\Delta Gv_{i}=\left\{ \begin{array}{cc}I_{i}\otimes(I_{\text{max}}-I_{\text {min}}),&\text{Iter}=1;\\ \text{rand}^{2}\odot(N(1,\text{D})\odot\Delta Gv_{i}),&\text{Iter}>1,\end{array} \right.\] (16) the absolute value of \(|N(1,\text{D})|\) is the absolute value of \(N(1,\text{D})\), and \(u\otimes v\) is the Hadamard division of vector \(u\) by vector \(v\). 4. The Propagation and Evolution of Ivy Plants: After the young ivy \(I_{i}\) has explored the search space to find its closest and most significant neighbor \(I_{ii}\), it enters a subsequent phase.