**6. Experiments**

## *6.1. The Simulation of the Sinc Function with Sas noises*

In this section, the simulation experiments using the Sinc function with random noises are presented. They compare between serval state-of-art algorithms with the proposed method, which are the R-ELM, the RCC-ELM, the MMCC-ELM and our method. The training and test samples were randomly assigned according to the Sinc function and random noises were added with respect to alpha-stable distribution. This is represented as follows:

$$\mathbf{y} = \mathfrak{a}\mathbf{S}\mathfrak{m}\mathfrak{c}(\mathbf{x}) + \mathfrak{p} \tag{40}$$

where α is the scale of the function which is set to 8.0 and Sinc(x) is the Sinc function. The Sinc function is represented as follows:

$$\text{Sinc}(\mathbf{x}) = \begin{cases} \quad \sin\left(\mathbf{x}\right) / \mathbf{x} & \mathbf{x} \neq \mathbf{0} \\ \quad 1 & \mathbf{x} = \mathbf{0} \end{cases} \tag{41}$$

Moreover, ρ is the noise that satisfies the following characteristic function [69]:

$$\rho = \begin{cases} \exp\left(-\delta^a |\theta|^a (1 - j\beta \text{sign}(\theta) \tan\left(\frac{\pi a}{2}\right))\right) + j\mu \theta & a \neq 1 \\\ \exp\left(-\delta^1 |\theta|^1 (1 - j\beta (\pi/2) \text{sign}(\theta) \log\left(\frac{\pi a}{2}\right))\right) & a = 1 \end{cases} \tag{42}$$

The parameters α, β, γ and μ are real and characterize the distribution of the random variable X. Here, the alpha-stable probability distribution function is denoted as S( <sup>α</sup>,β,γ,μ). In these experiments, the four parameters were assigned to three di fferent conditions to provide three types of noises. The assignment of the parameters in each sample is presented in Table 2.


**Table 2.** The assignments of the parameters in each sample.

Each sample contained 200 data, with half of the data being used for training and another half for testing. To ge<sup>t</sup> a proper estimation of the performances of each method, the experiments were operated with the best optimization of parameters. This is presented in Table 3.


**Table 3.** The assignment of the parameters for each algorithms.

Each experiment was conducted 30 times and the averages were taken. The comparison of the accuracies of these algorithms is presented in Table 4. Compared with other algorithms, the R-ELM and ECC-ELM achieve lower mean square errors due to the advantages of the correntropy. The performance of R-ELM is relatively poor due to the e ffect of noises. The performance of MMCC-ELM also improved by the correntropy. However, since the fixed dimension of the correntropy, the accuracy can be badly influenced by unnecessary assignments on the second order of the bandwidth. Furthermore, it is clear that the proposed algorithm achieves the lowest training MSE, which means that it is the most accurate method for simulation of the Sinc function.


**Table 4.** The comparison of the accuracies of the four algorithms.

To further analyze the predictive abilities of these four algorithms, Figure 3 depicts the di fferences between the actual function and the predicted function for each algorithm. It is clear that all the algorithms achieve relatively good prediction on the Sinc function. However, the prediction results of the ELM have been badly influenced by the noises in all three samples. Additionally, the MMCC-ELM performance is poor on sample 2 and sample 3, which is probably due to the assignments with high dimension parameters. The RCC-ELM and ECC-ELM provide good predictions, which are almost identical to the actual functions in all three samples. The ECCELM has the closet predicted function with the Sinc function, which also proves that the method has high reliability against noise.

**Figure 3.** *Cont*.

**Figure 3.** The performance comparison of each algorithm (**a**) comparison with sample 1; (**b**) comparison with sample 2 and (**c**) comparison with sample 3.

Furthermore, an experiment on sample 1 was conducted to compare the cost function for the output weights with the MMCC-ELM and ECC-ELM since they share similar cost functions. The results are shown in Figure 4, which show that the cost function of ECC-ELM is quite lower than the cost of MMCC-ELM. Additionally, the costs of the ECC-ELM become stable for less than 25 iterations for all three examples than MMCC-ELM. This shows the improvements on training the model with ECC-ELM taking the cooperating evolution technique. Since both algorithms finish the generation of the model when the cost function becomes stable, it can be concluded that the proposed model has faster convergence on training the prediction model.

**Figure 4.** The comparison on the cost function values of the extreme learning machine by maximum mixture correntropy criterion (MMCC-ELM) and ECC-ELM (**a**) comparison with sample 1; (**b**) comparison with sample 2; (**c**) comparison with sample 3.

Figure 5 illustrates the effects of the evolutionary process on the optimization of the kernel bandwidth and influence coefficients. From Figure 5, it can be seen that the cost function for the kernel bandwidth quickly drops during the evolution process. Moreover, Ef continuously decreases during the process, which means that the particle swarm become stable and the best solution occurs. Figure 6 compares the actual pdf function and the estimated pdf function. It can be seen that the algorithm achieves a comparatively accurate estimation of the distribution of the errors.

**Figure 5.** The dynamic changes of the evolution factor (Ef) and costs during the cooperative evolution.

**Figure 6.** Comparison between the estimated pdf and actual pdf.

## *6.2. The Performance Comparison on Benchmark datasets*

To further assess the proposed algorithm, the performance of the ECC-ELM and other methods were compared using the data set from the UCI machine learning repository [70], awesome public dataset [71] and the United Nations development program [72], which are listed in Table 5. The assignments of the parameters are shown in Table 6, all of which refer to the best performance of each algorithm. Each experiment was conducted 30 times and the average performance was reported.

The performance is compared in Table 7, which shows that the proposed algorithm is able to achieve better prediction accuracies than other methods. Additionally, the performance of the proposed method is relatively stable compared with other correntropy-based extreme learning machines.

Figure 7 compares the actual output value and the predicted value for the Servo data set. It is clear that the predicted values are basically identical to the actual output values, and it has not been influenced by the outliers in the data.

To illustrate the evolutionary processes for optimizing the bandwidth, Figure 8 depicts the distributions of the particles and the evolution of the optimal solutions. It can be seen that the distribution of the particles dynamically changes based on the state of the PSO process. The optimal solution is adjusted and stabilizes during the process, which allows the optimal solution of the bandwidth assignments to generate a more accurate model.


**Table 5.** The information on the data sets.

**Figure 7.** The comparison between the actual values and the predicted values under the data set, Servo.

**Figure 8.** The evolutionary process of the particles.




#### *6.3. The Performance Estimations for Forecasting the CTR of Optical Couplers*

Finally, to estimate the performance of a real application, the proposed method has been used to predict the current transfer ratio for optical couplers. This is one type of transmission device for electric signals and optical signals with wide applications to the isolation transfer of signals, A/D transmission, D/A transmission, digital communications and high-pressure control. For optical couplers, the CTR is an essential factor for estimating the operating status of optical couplers. In this section, the proposed method was used to give the predictions of CTR for the optical couplers to predict the health condition of the devices.

For the experiments, the degenerating signals of four optical couplers were recorded and transformed into the samples historical CTR value as input vectors and the CTR value of the next time as the expected output. The training data was the samples that were generated from the optical couplers' records over the first ten years and the testing data were the samples that were generated from the last ten years.

Figure 9 depicts the evolutionary process of the PSO procedure. It shows that the Ef value quickly decreases during the evolutionary process and stabilizes within 17 iterations, resulting in the optimal solution that is provided by the swarm.

Finally, the predicted results of the four optical couplers are shown in Figure 10. It is clear that the generated ELM network accurately predicts the CTR value of each optical coupler and is robust with the noises of the signals. Therefore, the proposed method is able to achieve good performance for the optical couplers.

**Figure 9.** Dynamic changes on Ef and costs during the cooperative evolution.

**Figure 10.** *Cont*.

**Figure 10.** The comparison with the predicted current transfer ratio (CTR) and actual CTR.

Table 8 presents the numerical results of the CTR prediction, which compares the actual CTR and the predicted CTR. It is clear that the proposed method can very accurately provide the prediction on the state of Optical Couplers (OCs). Additionally, the time consumption is presented in Table 8 which shows that the proposed method is able to obtain high accuracy on the prediction of the future CTR of the OC and the predicting time is quite low within 5 ms. Therefore, the proposed method can achieve high performance on real applications.

**Table 8.** The performance of the predicted model that is generated using the ECCELM.

