**7. Conclusions**

To improve the robustness of the forecasting model, the paper provides a novel correntropy-based ELM called the ECC-ELM. It uses a multi-dimension correntropy criterion and the evolved cooperation method to adaptively generate the parameters for kernels. In the proposed algorithm, SDPSO is integrated by minimizing the MIE to determine the proper bandwidths and their corresponding influence coefficients to estimate the probability distributions of the residual error of the model. A novel training process was developed based on the properties of the multi-dimension correntropy and it was able to build the convex cost function to calculate the output weights for the ELM. The experiments on the simulated data and real-world application were conducted to estimate the accuracy of the probability distribution of the signal and robustness on predicting the samples. The simulation results with the Sinc function proved that the proposed method can generate the multi-kernel correntropy with high accuracy on describing the probability distribution of the signals and fast converge on the evolution process. This leads to high robustness of the proposed method compared with the other methods. The performance comparisons on the benchmark datasets show that the proposed method can achieve higher accuracy and more stability than the other methods. Finally, the CTR prediction experiments show the proposed method can achieve high accuracy within acceptable time consumption on real world applications. Although the proposed algorithm has predictive advantages, there are still several limitations on the study. One limitation is the proposed method is only applicable for an ELM with one hidden layer, which requires extensions on multi-layer networks. The other limitation is that the proposed method only provides an offline training model. Therefore, how to update the online prediction model becomes another interesting topic for future research. The codes and data of the research are available at https://github.com/mwj1997/ECC-ELM.

**Author Contributions:** Conceptualization, W.M.; Data curation, L.D.; Funding acquisition, Z.L.; Investigation, J.H.; Methodology, W.M. and Y.S.; Project administration, Z.L.; Resources, J.H.; Software, W.M.; Supervision, Z.L.; Validation, Y.S.; Writing—original draft, W.M.; Writing—review & editing, L.D.

**Funding:** This work was supported by the National Natural Science Foundation of China under Grants No. U1830133 (NSAF) and No. 61271035.

**Conflicts of Interest:** The authors declare no conflict of interest.
