# Improved Particle Swarm Optimization for Sea Surface Temperature Prediction
## 3 The Proposed DSL Method
### Parameter Optimization Using LSPSO
The Beta distribution function is defined as:  
\[\beta(x;m,n)=\frac{x^{m-1}(1-x)^{n-1}}{B(m,n)},0<x<1. \tag{10}\]  
Figure 2: Flow chart of LSPSO. The denominator is defined as follows:  
\[B(m,n)=\int_{0}^{1}t^{m-1}(1-t)^{n-1}dt \tag{11}\]  
Pareto dominance relationships are used in many optimization algorithms to measure the pros and cons of a solution, usually defined as follows:  
**Definition 1**. : _(non-dominated relationship). Let \(x\), \(y\) be the two decision variables in equation (12), if \(x\) dominates \(y\), then if and only if:_  
\[\forall i\in\{1,2,\ldots,n\},\exists j\in\{1,2,\ldots n\};f_{i}(x)\leq f_{i}(y )\cap f_{j}(x)<f_{j}(y). \tag{12}\]  
**Definition 2**. : _(non-dominated solution). Assuming that there is a solution set P, where individual \(q\) is not dominated by any other individual, then \(q\) is a non-dominated individual in P. The subset of non-dominated individuals of P is called the non-dominated set (NDset) of P._  
\[NDset=\{q\}\big{|}q\in P\text{ and does not exist }p\in P,\text{ make }p\text{ dominate }q\}\]  
Since only the non-dominated solution in the population P needs to be selected, in the process of comparing different particles, once the particles are dominated by other particles, they are not compared with other particles, which reduces the complexity of the algorithm. Through the non-dominated relationship, a non-dominated solution set S can be obtained. The solutions contained in the set are non-dominated solutions, and some better solutions can be found around the non- dominated solution. The local search ability of the algorithm is enhanced by the local search strategy, which is beneficial to improve the local search ability of the algorithm. For the non-dominated solution set S, one of the individuals is \(x_{ijk}=\left(x_{1,ijk},x_{2,ijk},\ldots x_{n,j,k}\right)^{T}\), \(i\) represents the \(i\)-th individual in S, \(k\) is the number of evolutions, and \(n\) represents the dimension of the variable. The search for the \(m\)-th variable of the individual \(x_{i,k}\) in two directions can be expressed below as \(L_{m,i,t}\) and \(R_{m,i,t}\). \[L_{m,i,t}=x_{m,i,t}-c\times\big{|}P_{m,i,t}-G_{m,i,t}\big{|} \tag{13}\]  
\[R_{m,i,t}=x_{m,i,t}+c\times\big{|}P_{m,i,t}-G_{m,i,t}\big{|} \tag{14}\]  
where P and G are randomly selected from the non-dominated solution set S, and c is the interference coefficient. New individuals can be generated by Equations (13) and (14). Algorithm 2 expresses the complete process of local search. The non-dominated solution set S obtained from each evolution, the local searched population S' and the external archive D, are merged into a new external archive D. The non-dominated solutions in D are used as external archives to ensure the diversity of the population. As the number of iterations increases, the size of the external archive will gradually increase too. It is unrealistic to include all non-dominated solutions in D. At this time, it is feasible to obtain a finite non-dominated subset that can represent the solution space. In order to obtain a more uniform non-dominated solution set, this paper uses the crowded distance to remove redundant individuals and controls the number of non-dominated solutions in D.  
After updating Gbest and Pbest, we use Formulas (7) and (8) to update the velocity and position of the particle. Since the particles will cross the boundary when flying, the particles will be mutated for the particles whose particle velocity exceeds the constraint range. The formula is as follows:  
\[M_{n,i,t}=E_{n,i,t}\times(1-rand\ (-0.2,0.2)) \tag{15}\]where E is randomly selected from the external archive D, and M is the mutated particle. When the flying speed of the particle exceeds half of the constraint range, but is still below the constraint range, the particle velocity is set to half of the constraint range to prevent the particles from crossing the boundary.