## 3 LED Model Parameter Extraction Methods
### Modified Nelder-Mead Method
\begin{table}
\begin{tabular}{c c} \hline \hline
**Number of Vectors** & **Success Rate [\%]** \\ \hline N = 10 & 51.4 \\ (original method) & 86.8 \\ N = 30 & 92.4 \\ N = 40 & 95.8 \\ N = 50 & 98.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dependence of the success rate of the modified Nelder–Mead algorithm on the size of the simplex. Figure 4: The difference in the value of the objective function during the extraction of the parameters of the optical branch of the multi-domain LED model (involving 9 parameters) for different simplex sizes, compared to the best case with 50 points. Cases using 40, 30, 20, and 10 data points (original Nelder–Mead algorithm) are shown. (**Left**) side: Full scale. (**Right**) side: A maximum limit of 10% is applied. The baselines are shifted by \(-1\%\) each for visual clarity. As an illustration, the excess error with respect to the best case of \(5\cdot(\mathrm{N}+1)=50\) points is shown for the parameters of the optical branch of the multi-domain model. For the extraction of the parameters of the optical branch, the number of objective function evaluations for the original Nelder-Mead simplex with \(\mathrm{N}+1=10\) data points in the parameter space was cca. 14,000 and for the 5-fold increased simplex size of \(5\cdot(\mathrm{N}+1)=50\), the number of evaluations of the objective function increased to cca. 52,000, slightly less than the expected linear increase. The success rate of the Nelder-Mead algorithm depends on the actual initial simplex. Therefore, it is essential to investigate how the initial simplex (\(x_{0}\)) influences the success rate. This dependence was investigated by increasing and decreasing the values of the initial parameters by an order of magnitude, except the values of the series resistances and the ideality factors in order to avoid non-physical solutions or no solution for Equation (4). The initial values of the series resistances can be reduced to 0.1 and 0.05, respectively, for electric and optic branches. Ideality factors must be greater than 1. Decreasing the initial simplex with the above limitations has no adverse effect on the extraction process, other than a slight increase in computational resources, due to 20-30% more iterations being needed to achieve a global optimum. Increasing the parameters, however, greatly increases the probability of the successful execution of the new parameter extraction procedure. These statements are based on our initial numerical experiments using 100+ sets of archived LED measurement data detailed in Table 2.  
Reducing the values of the parameter values forming the initial central vector in the parameter space had an adverse effect on the success rate if the parameter extraction process had a 10-fold decrease moderated to a 5-fold reduction. These results are also summarized in Table 2. These experimental results suggest that there are many local minima of the objective function at parameter sets with reduced values. Decreasing the parameter values defining the central vector also narrows the possible range within which additional vectors in the parameter space can be randomly selected. Additionally, overly aggressive upscaling the parameter values results in a decreased success rate. This indicates that using typical parameter sets of the chip-level multi-domain LED model as initial values ensures a high likelihood of success in the extraction process. However, the tested heuristics are not sufficiently robust against higher errors in the measurement data or against unusual LED characteristics. Further steps need to be taken to make our solver robust enough for unsupervised working conditions. The conclusions drawn from our numerical experiments regarding the heuristic modifications of the original Nelder-Mead method are as follows:  
* Including more random vectors in the simplex increases the success rate of the parameter extraction process, but involves an increased computational cost.