## 3 Proposal and Verification of Improved LoTFWA
### Validation and Analysis of ILoTFWA  
In order to verify the effectiveness of the ILoTFWA proposed in this paper, 24 functions in the CEC2013 test function set were used for simulation experiment analysis, of which 2 were unimodal functions (_f\({}_{1}\)_-_f\({}_{2}\)_), 14 were multimodal functions (_f\({}_{3}\)_-_f\({}_{16}\)_), and 8 were composite functions (_f\({}_{17}\)_-_f\({}_{24}\)_). The specific names of functions are shown in Table 1. The overall improvement effect and superiority were tested among the ILoTFWA, LoTFWA, and several other improved FWAs. The parameter setting of the ILoTFWA proposed in this paper is the same as that of the LoTFWA: the number of fireworks \(m\) = 5, the initial value of the parameter of the number of explosive sparks generated \(\hat{\lambda}_{i}^{0}=300\), dynamic amplitude scaling factor \(C_{n}=1.2\), \(C_{r}=0.9\), scaling factor \(S_{A}=1.2\), \(S_{c}=0.8\) for adaptive adjustment of explosion spark quantity parameters, Gaussian variation adaptive control parameters \(\sigma=0.2\), and the dimension of search space of all test functions \(d=30\). The maximum evaluation time is 300,000, and it runs 51 times independently on each function. The ILoTFWA was also compared with the classical FWA and other improved FWAs that have performed well in recent years. The three comparison algorithms include: the adaptive fireworks algorithm (AFWA) [42], cooperative fireworks algorithm (CoFWA) [43], and guiding fireworks algorithm (GFWA) [44]. The parameters of these algorithms were set to the values recommended in the original paper and were consistent with the corresponding parameters of the algorithm proposed in this paper. Their means and standard deviations are shown in Table 2, and the bold part indicates that the mean was the best among all comparison algorithms. It can be seen from Table 2 that the existing algorithms, the AFWA and GFWA, performed best in the optimization of unimodal functions. Among the multimodal functions (f\({}_{3}\)-f\({}_{16}\)) and composite functions (f\({}_{17}\)-f\({}_{24}\)), the error of the ILoTFWA was the smallest; that is, its overall search performance was the best among all the fireworks algorithms compared, and the LoTFWA algorithm was the second. It can also be seen that the ILoTFWA was significantly better than the existing improved FWAs in most multimodal functions. The above experimental results show that the improved mapping rules designed by the ILoTFWA based on the LoTFWA algorithm are feasible and effective. They can further improve the search efficiency of the algorithm in complex optimization problems such as multimodal functions and composite functions, so as to effectively promote the accuracy of solving problems such as multivariable feature extraction in complex physical processes. \begin{table}
\begin{tabular}{c c l} \hline \hline
**Function Type** & **No.** & **Function Name** \\ \hline \multirow{3}{*}{Unimodal function} & 1 & Rotated Discus Function \\  & 2 & Different Powers Function \\ \hline \multirow{8}{*}{Multimodal function} & 3 & Rotated Rosenbrocks Function \\  & 4 & Rotated Schaffers F7 Function \\  & 5 & Rotated Ackleys Function \\  & 6 & Rotated Weierstrass Function \\  & 7 & Rotated Grievanks Function \\  & 8 & Rastrigins Function \\  & 9 & Rotated Rastrigins Function \\  & 10 & Non-Continuous Rotated Rastrigins Function \\  & 11 & Schwefel’s Function \\  & 12 & Rotated Schwefel’s Function \\  & 13 & Rotated Katsuura Function \\  & 14 & Lunacek Bi Rastrigin Function \\  & 15 & Rotated Lunacek Bi Rastrigin Function \\  & 16 & Expanded Griewanks plus Rosenbrocks Function \\ \hline \multirow{8}{*}{Composite function} & 17 & Composition Function 1 (Rotated) \\  & 18 & Composition Function 2 (Unrotated) \\  & 19 & Composition Function 3 (Rotated) \\  & 20 & Composition Function 4 (Rotated) \\  & 21 & Composition Function 5 (Rotated) \\  & 22 & Composition Function 6 (Rotated) \\  & 23 & Composition Function 7 (Rotated) \\  & 24 & Composition Function 8 (Rotated) \\ \hline \hline \end{tabular}
\end{table}