# Analysis and Improvement of Fireworks Algorithm
## 3 Analysis and Improvement of Fireworks Algorithm
### Opposition-Based Learning Population Initialization
The \"+\" indicates the rejection of the null hypothesis at the 5% significance level, and the \"-\" indicates accept the null hypothesis at the 5% significance level. Table 4 indicates that IFWA showed a large improvement over FWA, EFWA, dynFWA and SPSO2011 in most functions. Figure 4: The FWA, EFWA, dynFWA, SPSO2011 and IFWA run-time cost (Continued). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Functions** & & **SPSO2011** & **FWA** & **EFWA** & **dynFWA** \\ \hline f1 & \(p\)-value & NaN & 0 & 0 & NaN \\  & significance & \(-\) & + & \(+\) & \(-\) \\ f2 & \(p\)-value & 0.0045 & \(1.102\times 10^{-201}\) & \(5.186\times 10^{-22}\) & \(1.0556\times 10^{-37}\) \\ f2 & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f3 & \(p\)-value & \(7.9152\times 10^{-9}\) & \(1.352\times 10^{-145}\) & \(0.0666\) & \(0.9222\) \\ f4 & significance & \(+\) & \(+\) & \(-\) & \(-\) \\ f5 & \(p\)-value & 0 & 0 & \(4.251\times 10^{-102}\) & \(3.192\times 10^{-202}\) \\  & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f6 & significance & \(-\) & \(+\) & \(+\) & \(-\) \\ f7 & \(p\)-value & 0.04621 & \(6.677\times 10^{-35}\) & \(1.9078\times 10^{-40}\) & \(0.0140\) \\  & significance & \(-\) & \(+\) & \(+\) & \(+\) \\ f8 & significance & \(-\) & \(-\) & \(+\) & \(+\) \\  & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f9 & significance & \(+\) & \(+\) & \(+\) & \(-\) \\ f10 & \(p\)-value & \(2.006\times 10^{-109}\) & \(3.799\times 10^{-314}\) & \(1.247\times 10^{-150}\) & \(5.2876\times 10^{-15}\) \\  & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f11 & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f12 & \(p\)-value & \(1.5512\times 10^{-7}\) & \(9.2258\times 10^{-64}\) & \(6.3131\times 10^{-94}\) & \(0.0209\) \\  & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f13 & significance & \(+\) & \(+\) & \(+\) & \(+\) \\ f14 & significance & \(-\) & \(+\) & \(+\) & \(-\) \\ f15 & \(p\)-value & \(0.8234\) & \(6.6992\times 10^{-20}\) & \(1.9623\times 10^{-7}\) & \(0.3320\) \\  & significance & \(-\) & \(+\) & \(+\) & \(-\) \\ f16 & significance & \(-\) & \(+\) & \(+\) & \(-\) \\ f17 & significance & \(+\) & \(+\) & \(+\) & \(-\) \\ f18 & significance & \(+\) & \(+\) & \(-\) & \(-\) \\  & significance & \(+\) & \(+\) & \(-\) \\ f19 & \(p\)-value & \(9.27

## 6 Conclusions

Based on the analysis of the FWA, an improved fireworks algorithm (IFWA) is proposed in this paper. IFWA firstly puts opposition-based learning into FWA to initialize the population. Moreover, aiming at the shortage of explosion amplitude in FWA, a new explosion amplitude mechanism is proposed. Then, the adaptive \(t\)-distribution mutation is proposed for non-optimal fireworks, and elite opposition-based learning for optimal firework. At last, a new selection mechanism is proposed, which reduces the run-time of algorithm. We apply the CEC2013 standard functions to examine and compare the proposed algorithm IFWA with SPSO2011, FWA, EFWA and dynFWA. The results clearly indicate that IFWA can perform significantly better than FWA, EFWA, dynFWA and SPSO2011 in terms of solution accuracy. Overall, the research demonstrates that IFWA performed the best for both solution accuracy and run-time cost. The authors are thankful to the anonymous reviewers for their valuable comments to improve the technical content and the presentation of the paper. This paper is supported by the Liaoning Provincial Department of Education Science Foundation (Grant No. L2013064) and AVIC Technology Innovation Fund (basic research) (Grant No. 2013S60109R). [PERSON] participated in the draft writing. [PERSON] participated in the concept, design and perform the experiments and commented on the manuscript. [PERSON] participated in the data collection, and analyze the data. The authors declare no conflict of interest. Figure 11: Cont. Figure 11: The FWA, EFWA, dynFWA, SPSO2011 and IFWA searching curves.