# Self-Organized Fuzzy Neural Network Nonlinear System Modeling Method Based on Clustering Algorithm
## 4 Simulation Analysis
### Mackey-Glass Time-Series Forecasting  
In the first benchmark test, we examine SOFNN-CA using chaotic Mackey-Glass time-series prediction [36]. This time series is generated by the differential delay equation:  
\[x(t+1)=(1-a)x(t)+\frac{bx(t-\tau)}{1+x^{10}(t-\tau)} \tag{58}\]  
where \(a=0.1\), \(b=0.2\), \(\tau=17\), and \(x(0)=1.2\) in the initial condition, according to the description in the literature [25], and the prediction model can be described as follows:  
\[x(t+\hat{H})=\overline{f}(x(t),x(t-6),x(t-12),x(t-18)) \tag{59}\]  
The 500 data pairs from \(t=1\) to 500 are selected as the training data set, and the remaining data pairs within the interval [501, 1000] are used as the test data set. In this experiment, we take k-means algorithm to generate six clusters, on which self-organized learning is performed. The parameters of the network are obtained by an empirical method, and the growth threshold ma_addth is set to 0.6, the width adjustment thresholds e_min and den_th are set to 0.2 and 10, respectively, and the merging threshold S_th is 7. The simulation results are shown in Figures 7 and 8. The comparison algorithms use the same training and test samples listed in Table 1, and the results are the means of multiple tests.  
Figure 7 shows the change of neurons as the number of iterations increases. The number of neurons first increases to 11 and then gradually decreases to 5, resulting from a self-organizing mechanism that learns the features of new samples and then merges similar neurons. From the curve of training the RMSE in Figure 8a, it can be seen that there are some peaks in the RMSE of SOFNN-CA, which are caused by merging neurons. Figure 8b shows that the method is effective. Furthermore, Table 1 lists the performance comparison details of SOFNN-CA and its competitors. It can be seen that the proposed SOFNN-CA obtained the best tested RMSE (0.0058) and the best tested APE (0.0055). Due to the neuronal merging mechanism, SOFNN-CA has the most compact network structure, and its performance is better than SOFNN-ALA, SOFNN-AGA, and SOFNN-ACA. Owing to the inherent drawbacks of the adopted EBP-based parameter optimization algorithm, although SOFNN-AGA, SOFNN-ACA, and FNN-EBP can achieve better test results, the training time is longer because more iterations are required to achieve the required accuracy. SOFNN-GA has the longest training time using the genetic algorithm. The results in Table 1 show that the SOFNN-CA proposed in this paper can obtain high prediction performance with a compact network structure.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Method** & **Rule Neurons** & **Testing RMSE** & **Testing APE** & **CPU Times (s)** \\ \hline SOFNN-CA & 5 & 0.0058 & 0.0055 & 6.62 \\ SOFNN-ALA [20] & 6 & 0.0066 & 0.0058 & 7.24 \\ SOFNN-AGA [10] & 6 & 0.0119 & 0.0076 & 21.20 \\ SOFNN-ACA [14] & 7 & 0.0201 & 0.0076 & 27.33 \\ SOFNN-GA [19] & 7 & 0.0132 & 0.0094 & 168.35 \\ FNN-EBP [37] & 8 & 0.0142 & 0.0131 & 37.05 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison results of different methods for predicting Mackey–Glass Time-Series.  
Figure 8: (**a**) Training RMSE chart; (**b**) test results chart.  
Figure 7: Neuron number change graph.