# Rotate Vector Reducer Fault Diagnosis Model Based on EEMD-MPA-KELM
## 3 Theoretical Basis
### Mpa  
The Marine Predators Algorithm (MPA) is a novel and efficient meta-heuristic algorithm proposed by [PERSON] [20] in 2020, which was mainly inspired by the motion laws of predators and prey in the ocean and the optimal encounter rate strategies for biological interactions between predators and prey. The optimization process is divided into three stages based on the speed scores of the predator and the prey, with the predator following a regular levy motion or Brownian motion. At the same time, the prey also acts as the predator identity while being hunted, making the algorithm more realistic in principle. In addition, considering the Marine environmental factors, it can reduce the phenomenon of predators falling into the local optimal value. The detailed steps of MPA algorithm implementation are as follows:  
1. Initialize the Elite matrix and Prey matrix. Each element in the Prey matrix is initialized by a formula. \[X_{ij}=l_{j}+rand\big{(}u_{j}-l_{j}\big{)}\] (16)In formula, _uj_ and _lj_ are the upper and lower limits of the search space in the j dimension; rand is a random vector subject to uniform distribution [0,1], and _Xij_ represents the j-dimensional spatial position of the \(i\) prey. The final Prey matrix is obtained:  
\[\textit{Prey}= \begin{bmatrix}X_{1,1}&X_{1,2}&\dots&X_{1,d}\\ X_{2,1}&X_{2,2}&\dots&X_{2,d}\\ \vdots&\vdots&\ddots&\vdots\\ X_{n,1}&X_{n,2}&\dots&X_{n,d}\end{bmatrix} _{n\times d} \tag{17}\]  
In formula, \(n\) is the population size and \(d\) is the dimension value. The fitness value of each predator was calculated, and the predator with the best fitness was selected to make \(n\) copies to form the Elite matrix:  
\[\textit{Elite}= \begin{bmatrix}X_{1,1}^{I}&X_{1,2}^{I}&\dots&X_{1,d}^{I}\\ X_{2,1}^{I}&X_{2,2}^{I}&\dots&X_{2,d}^{I}\\ \vdots&\vdots&\ddots&\vdots\\ X_{n,1}^{I}&X_{n,2}^{I}&\dots&X_{n,d}^{I}\end{bmatrix} _{n\times d} \tag{18}\]  
In formula, \(n\) is the population size and \(d\) is the dimension value. The Elite matrix has the same dimension as the Predator matrix. 2. Optimization process. In the optimization process, there are three stages. Stage 1: In the early iteration, the first third of the total iteration, the prey moves faster than the predator in this stage. The predator is mainly in the exploratory phase, and the optimal strategy is to be completely motionless. The mathematical model of this stage is:  
\[ \begin{cases}s_{i}=R_{B}\otimes(E_{i}-R_{B}\otimes P_{i})\\ P_{i}=P_{i}+0.5R\otimes s_{i}\end{cases} \quad i=1,2,\dots,n;\ t<\frac{1}{3}T \tag{19}\]  
In formula, \(s_{i}\) is the moving step; \(R_{B}\) is a random vector with Brownian motion based on normal distribution; \(\otimes\) is term by term multiplication; \(R\) is a uniform random vector in [0,1]; \(t\) is the number of current iterations; \(T\) is the maximum number of iterations. Stage 2: In the middle of the iteration, one-third to two-thirds of the way through the iteration, prey and predator are moving at the same speed. Predators are mainly in the exploration and development transition phase, so half of the predators are for exploration and the other half for development. At this point, the prey uses Levi's Walk for exploitation and the predator uses Brownian motion for exploration. The mathematical model of this stage is:  
\[ \begin{cases}s_{i}=R_{L}\otimes(E_{i}-R_{L}\otimes P_{i})\\ P_{i}=E_{i}+0.5R\otimes s_{i}\end{cases} \quad i=1,2,\dots,\frac{u}{2};\ \frac{1}{3}T<t<\frac{2}{3}T \tag{20}\]  
\[ \begin{cases}s_{i}=R_{B}\otimes(R_{B}\otimes E_{i}-P_{i})\\ P_{i}=E_{i}+0.5C_{F}\otimes s_{i}\end{cases} \quad i=\frac{u}{2},\dots,n;\ \frac{1}{3}T<t<\frac{2}{3}T\]  
\[ \begin{cases}s_{i}=R_{L}\otimes(R_{L}\otimes E_{i}-P_{i})\\ P_{i}=E_{i}+0.5C_{F}\otimes s_{i}\end{cases} \quad i=1,\dots,n;\ t>\frac{2}{3}T \tag{21}\]  
3. Fish Aggregation Device Effect (FADs). In order to avoid the stagnation of local optimal value, MPA considers the influence of environmental factors such as fish aggregation device, and the mathematical model of FADs is as follows:  
\[P_{i}=\left\{