## 3 LED Model Parameter Extraction Methods
### Modified Nelder-Mead Method  
In a 9-dimensional space, 10 points can easily approach a false minimum (i.e., a local minimum), or the vertex can contract too much into a set of points that are too close to each other, resulting in the premature termination of the optimization process. A possible approach to overcome this problem is to increase the number of data points in the parameter space from N + 1 to a higher value. As found in the literature [36; 37; 38], a potential solution is to combine the Nelder-Mead method with particle swarm optimization (PSO) or with other methods, as described in other papers [39; 40]. These approaches certainly provide better results, but the computational effort for this problem is very high, as the number of iterations, which involves solving implicit equations, increases linearly with the number of trial data points used within the Nelder-Mead method. In our present approach, the Nelder-Mead algorithm is modified to use more points in the simplex. We found that this increased the success rate of the optimization, i.e., the termination of the algorithm with convergence to a true optimum. The initialization (step 1 described in Section 3.3) is modified as follows: The vectors of measured data included in the simplex are chosen randomly from a growing hyper-volume around the initial vector \(\mathbf{x}_{0}\):  
\[\mathbf{x}_{i}[k]=\left(1.001+\frac{i}{10\cdot\text{N}}rand\left(-\frac{1}{2},\frac {1}{2}\right)\right)\cdot\mathbf{x}_{0}[k] \tag{12}\]  
where \(rand()\) is a random value from a uniform distribution. The initialization ensures that the points are not too close to each other. The resulting initial simplex is shown in Figure 3.  
Figure 3: Selection of vectors for the initial simplex for the modified Nelder-Mead optimization method. Increasing the number of vectors in the simplex will not greatly increase the number of cost function evaluations, as only one vector changes over an iteration, except for contraction (see step 3b in Section 3.3), where all vectors change, and the cost function must be evaluated for all of them. Therefore, this modification does not significantly slow down the original Nelder-Mead method. Although the number of cost function evaluations per iteration does not change significantly, the overall optimization's convergence becomes slower since the central point cannot move as quickly as it could with only \(\mathbb{N}+1\) elements in the simplex. The number of iterations is assumed to increase linearly with the number of vectors in the simplex. The convergence probability of the method is defined as the ratio of successful evaluations to all evaluations. The extraction was performed 100 times with different random simplexes on 10 different LED samples of various kinds for testing purposes. The success rate is defined as the ratio of successful extractions (finding the global minimum, obtained by an exhaustive search) to the total number of attempts. The best heuristic modifications and initial parameter settings of the Nelder-Mead algorithm were identified through experiments on more than 100 sets of measured LED characteristics obtained from the archived data of the Delphi4 LED project. First, we investigated how the Nelder-Mead algorithm behaved with the randomly selected initial \(\mathbb{N}+1\) points (for the parameter extraction of the electrical branch \(\mathbb{N}=9\)). The success rate of the algorithm was approximately 50%, even if we followed the original extraction strategy. Increasing the number of points in the simplex resulted in an increased success rate, reaching 96% when the number of points was \(4\cdot(\mathbb{N}+1)=40\). Further increasing the number of points provided even better success rates; we found that \(5\cdot(\mathbb{N}+1)=50\) yielded a success rate of 98%. The results are summarized in Table 1 and shown in Figure 4, where we can see that the error with the parameters from an unsuccessful extraction is also reduced as the number of vectors increases.