## 4 Discussion
### Multi-Parameter Estimation Based on Particle Swarm Optimization  
Equation (13) establishes the maximum likelihood function model of the multi-stress acceleration model. Since the multi-stress acceleration model has multiple parameters, there are nine parameters when the three stresses are coupled. The traditional Newton iteration method requires the gradient and Hessian matrix of Equation (13) and needs to provide the initial parameter estimation value, which has the disadvantages of a large amount of calculation and poor solution accuracy. The least squares method also needs to provide theinitial parameter estimation value when fitting the nonlinear model of Equation (13), and the selection of the initial value has a significant impact on the optimal solution of the two models. This paper introduces the particle swarm algorithm to transform the parameter estimation problem into the optimal solution problem corresponding to the maximum likelihood function. Particle Swarm Optimization (PSO) is a swarm intelligence optimization algorithm inspired by the collective behavior of biological groups such as bird flocks or fish schools. It was proposed by [PERSON] and [PERSON] in 1995 and was originally used to solve optimization problems. Later, it was widely used in various fields, such as engineering optimization, machine learning, and data mining. The algorithm has the characteristics of a simple and flexible algorithm structure, good globality, and fast convergence speed. It is suitable for continuous function extreme value problems and has strong global search capabilities for nonlinear and multi-peak problems. First, define the variables in the particle swarm. In the multi-parameter optimization of the multi-stress acceleration model, the objective function value is the likelihood function, and the optimization goal is to find the maximum likelihood function. The distribution of the parameters is (\(\text{ln}\alpha 0\), \(-\alpha 1\), \(-\alpha 2\), \(-\alpha 3\), \(-\alpha 4\), \(-\alpha 5\), \(-\alpha 6\), \(-\alpha 7\), \(-\beta\)), which can be abbreviated as (a, b, c, d, e, f, g, h, \(\beta\)), where \(\beta\) is the shape parameter of the Weibull distribution. The process of particle swarm optimization is as follows:  
1. The positions and velocities of all particles are randomly initialized in the constraint space of the variables, ensuring that the constraint shape parameter \(\beta>0\) to ensure that the characteristic life is greater than 0;
2. Calculate the objective function value of each particle, that is, the log-likelihood value of the test data under all stress combinations;
3. For each particle, its current objective function value is compared with the optimal objective function value (\(p_{best}\)) in the particle's history and the larger value of the two is selected and redefined as \(p_{best}\);
4. For each particle, its current objective function value is compared with the historical optimal objective function value (\(g_{best}\)) searched by all particles, and the larger value of the two is selected and redefined as \(g_{best}\);
5. Calculate the iteration speed and iteration position of each particle. The expressions are Equation (16). \[ \begin{split}& v_{i}^{t+i}=\omega v_{i}^{t}+c_{1}r_{1}\big{(}p_{ best_{i}}-x_{i}^{t}\big{)}+c_{2}r_{2}\big{(}g_{best_{i}}-x_{i}^{t}\big{)}\\ & x_{i}^{t+1}=x_{i}^{t}+v_{i}^{t+1}\end{split}  \tag{16}\]  
Among them, \(v_{i}^{t+1},x_{i}^{t+1}\) are the speed and position of the \(j\) particle at the \(k\) th iteration, independently generating two random numbers \(r_{1}\) and \(r_{2}\) between 0 and 1, and \(w\) is the inertia coefficient representing the relative weight of the current speed. When running the initial iteration, a larger w value is selected to avoid falling into a local optimal solution due to a small step size. In the later stage of the iteration, the w value needs to be reduced to speed up the convergence and avoid curve oscillation.