## 2 Materials and Methods
### Multi-Stress Acceleration Model Considering Generalized Coupling
#### 2.1.2 Prerequisites of the Model  
A failure mode of LED is caused by \(N\) different stresses. For example, when \(S_{1},S_{2},...,S_{N}\) is higher than normal use and storage pressure, \(S\) can accelerate the aging process of the product. At this time, one or more stresses \(S\) can be selected. The stress level setting of N accelerated stresses cannot change the failure mechanism of a product during accelerated degradation testing; that is, the failure mechanism of the LED in the accelerated life test of these \(S\) accelerated stresses should be consistent with the failure mechanism of the LED during normal use. The model needs to meet the following three assumptions:  
1. Under each stress level combination, \(n\) samples are randomly selected and put to the test for a certain period. Assume that the LED failure life is \(t\); they are independent of each other and obey the Weibull distribution. Under the accelerated life test, the cumulative distribution function of the life is Equation (6). \[F(t,\eta,\beta)=1-\exp\left[-\left(\frac{t}{\eta}\right)^{\beta}\right] \tag{6}\]  
where \(\beta\) is the shape parameter and \(\eta\) is the scale parameter. In LEDs, \(\beta\) determines the trend of failure rate over time. When \(\beta>1\), it usually indicates degradation failure; that is, the failure rate will increase over time. When \(\beta<1\) indicates early failure, and when \(\beta=1\), it means that the failure rate is constant and is not affected by time. \(\eta\) represents the characteristic life; that is, at this point in time, 63.2% of the samples will fail. Under the action of various stress combinations, the degradation process of LEDs may accelerate, thereby changing \(\beta\). Under stresses of higher temperature, higher current, or high humidity, \(\beta\) may increase, indicating that the failure rate of LEDs will rise faster over time, reflecting accelerated degradation failure; similarly, \(\eta\) under stress combinations such as high temperature, high current, or high humidity tends to accelerate the degradation of LEDs, resulting in a decrease in \(\eta\), which reflects that LEDs are more likely to fail under these stresses. The effect of different stress combinations can be quantified by the change in \(\eta\): the higher the stress, the lower the \(\eta\) value, which means that the LED has a shorter life under this stress combination. Accelerated testing will cause the failure of the LEDs to occur earlier to a certain extent, while the exponential distribution of constant failure rate will lead to excessive deviation in the later stage. Therefore, the Weibull distribution with increasing failure rate over time is more suitable for LEDs [26; 27]. 2. In the test, the multi-stress and life satisfy a certain relationship, and it is necessary to ensure that the accelerated stress level does not exceed the maximum stress level that the LED can withstand. For the Weibull distribution, its characteristic life satisfies the multi-stress acceleration model in Equation (7). \[\ln\eta=f(\xi) \tag{7}\]  
where \(\xi\) is the standardized multi-stress vector, \(\xi=(\xi_{1},\xi_{2},..\xi_{N})\), and \(\xi_{i}\) is the standard mode of the \(i\) stress \(S_{i}\), which can be expressed as Equation (8). \[\xi_{i}=\xi(S_{i})=\frac{S_{i}-S_{i0}}{S_{iH}-S_{i0}}\;1\leq i\leq N \tag{8}\]  
In the formula, \(S_{i0}\) is the \(i\) stress \(S_{i}\) normal stress level, and \(S_{iH}\) is the \(S_{i}\) accelerated stress level. Take the logarithm of both sides of equation X and standardize them according to Equation (8). The multi-stress acceleration model in which the life obeys the Weibull distribution under the combined action of three stresses is Equation (9), and the detailed calculation process is shown in Appendix A. \[