# A Novel Hybrid High-Dimensional PSO Clustering Algorithm Based on the Cloud Model and Entropy
## 2 Multi-Dimensional Cloud Model
As a result, the possible clustering partition is regarded as a particle of the population, so that the hybrid PSO can realize the function of the optimal clustering partition and search efficiently and escape from local optima. On the basis of the above analysis, we constructed a fitness function similar to the multi-objective PSO algorithm based on the objective function and constraints of the multi-objective optimization problem. In the given multi-objective interval PSO algorithm, initial conditions and fitness function particle search space that are optimal variables and encoded as the position. During the iteration of multi-objective interval particle swarm optimization algorithm, particles track their own optimal position (_pbest_) and the group optimal position (_gbest_). Consequently, the tracking equation of multi-objective PSO algorithm can be expressed as follows. \[\left\{ \begin{array}{l}v_{i_{d}}^{k+1}=v_{i_{d}}^{k}+c_{1}r_{1}(pbest_{i_{d}} ^{k}-x_{i_{d}}^{k})+c_{2}r_{2}(gbest_{i_{d}}^{k}-x_{i_{d}}^{k})\\ x_{i_{d}}^{k+1}=x_{i_{d}}^{k}+t_{0}*v_{i_{d}}^{k+1}(t_{0}=1)\end{array} \right. \tag{12}\]  
Among them, \(c_{1}\) and \(c_{2}\) are the acceleration constants for multi-objective interval PSO algorithm to adjust the maximum step size of particles to reach the optimal position (_pbest_) and the optimal position (_gbest_). Where \(x_{i_{d}}^{k+1}=x_{i_{d}}^{k}+v_{i_{d}}^{k+1}\). Each particle is assigned an initial position and initial velocity as the optimal initial state. \(r_{1}(x)\) and \(r_{2}(x)\) are uniformly distributed random numbers to simulate the group behavior of the multi-objective interval PSO algorithm. In order to improve the convergence speed and solution quality of multi-objective interval particle swarm optimization algorithm, the basic PSO iterative equation is improved using the following expression. \[v_{i_{d}}^{k+1}=wv_{i_{d}}^{k}+c_{1}r_{1}(pbest_{i_{d}}^{k}-x_{i_{d}}^{k})+c_ {2}r_{2}(gbest_{i_{d}}^{k}-x_{i_{d}}^{k}) \tag{13}\]  
Along with these parameters, \(w\) is the inertial weight coefficient. Larger inertia weight can enhance the global search ability of the multi-objective particle swarm optimization algorithm. With the reduction of inertia weight, the multi-objective PSO algorithm realizes the local search and finally gets the optimal solution. Finally, we propose to introduce a compression factor into the multi-objective PSO algorithm to ensure the convergence equation of the algorithm. Through the above analysis, the convergence equation of the algorithm can be expressed as follows. \[v_{i_{d}}^{k+1}=\chi\Big{\{}v_{i_{d}}^{k}+c_{1}r_{1}(\textit{pbest}_{i_{d}}^{k}-x_{ i_{d}}^{k})+c_{2}r_{2}(\textit{gbest}_{i_{d}}^{k}-x_{i_{d}}^{k})\Big{\}} \tag{14}\]  
where \((\varphi=c_{1}+c_{2})\) is the compression factor of the multi-objective particle swarm optimization algorithm. \(\chi=2/\left|2-\varphi-\sqrt{\varphi^{2}-4\varphi}\right|\). Through research of the PSO and multi-objective PSO algorithm, we can conclude that the average focusing distance of the algorithm particles \(\widetilde{d}\), the maximum focusing distance between particles \(d_{\max}\), and the change rate of the particle focusing on distance \(K\) are shown as follows. \[\left\{ \begin{aligned} \widetilde{d}=\frac{\sum\limits_{i=1}^{n} \sqrt{\sum\limits_{i=1}^{n}\left(p_{\textit{gm}}-x_{id}\right)^{2}}}{d_{\max} =\max\Bigg{(}\sqrt{\sum\limits_{i=1}^{n}\left(p_{gm}-x_{id}\right)^{2}}\Bigg{)} }\\ K=\frac{\left(\max\Bigg{(}\sqrt{\sum\limits_{i=1}^{n}\left(p_{ gm}-x_{id}\right)^{2}}\Bigg{)}-\frac{\sum\limits_{i=1}^{n}\sqrt{\sum\limits_{i=1} ^{n}\left(p_{\textit{gm}}-x_{id}\right)^{2}}}{n}\Bigg{)}}{\max\Bigg{(}\sqrt{ \sum\limits_{i=1}^{n}\left(p_{gm}-x_{id}\right)^{2}}\Bigg{)}}\end{aligned} \right. \tag{15}\]  
Through in-depth research on the multi-objective PSO algorithm, the following functions are used to explore the properties and related mechanisms of the multi-objective PSO algorithm and to perform the following simulation experiments.