*4.3. Different Order*

In this section, we fixed the maximum AR and MA orders at 5 (AR (1–5) and MA (0–5)). We performed 100 Monte Carlo simulations on these 30 models and used the AIC, BIC, and BPNN to estimate the order. In Section 4.1, we found that the coefficient affects the estimate of the order. In order to make the results more representative, we randomized the coefficients of each model and met the conditions of causality and invertibility. In Section 4.2, we already knew that the length of the time series would affect the estimation of the order, so we set the length of the series to 1000 for better performance of all the methods. The setting of coefficient randomization can better study the effect of BPNN order estimation under different models. The length of the time series can make the accuracy of the AIC and the BIC higher, so as to better compare with the estimation results of the BPNN.

Figure 9 shows the order estimation results of different models. In Figure 9, the accuracy of BPNN is generally above 90%. The accuracy of the BIC is above 70% when the order is small, but it does not work well under higher-order models. Although the accuracy of the AIC is below 30%, the result is relatively stable. Combined with the present conditions, the order estimation of BPNN can have a prominent performance under random coefficient and different model orders. Another point worth noting is that when the order of the model is higher, only the BPNN can obtain satisfactory estimation results. From this example, it can be found that BPNN still has an excellent estimation effect, even though the ARMA model has mathematical symmetry.

**Figure 9.** Comparison of the correct number of estimation methods for each order under different orders.

Hossain et al., 2020, simulated physiological systems through ARMA and BPNN. As with their findings, the BPNN always shows an accuracy rate of more than 90% under different conditions. However, we found that AIC and BIC accuracy were low in our study, which may be due to a different coefficient selection. This may be because we did not have too much human intervention in the choice of coefficients.
