*4.2. Different Length*

In Section 4.1, we discussed the effect of each judgment criterion under different coefficients. Next, we study the effect of the length of the time series on the accuracy. We selected a set of representative coefficients from the three models in Section 4.1, where ARMA (1, 2) selected the 13th group, ARMA (2, 3) selected the 21st group, and ARMA (4, 2) selected the 9th group. Then, we simulated 100 different responses with time series lengths of 200, 400, 600, 800, and 1000. The expressions of the three models are shown in Equation (11).

$$\begin{aligned} \text{ARMA}(1,2): &X\_{l} - Z\_{l} + 0.5733X\_{l-1} + 0.1680Z\_{l-1} + 0.4097Z\_{l-2} \\ \text{ARMA}(2,3): &X\_{l} - Z\_{l} + 0.3272X\_{l-1} + 0.5129X\_{l-2} + 0.1052Z\_{l-1} + 0.1403Z\_{l-2} + 0.6325Z\_{l-3} \\ \text{ARMA}(4,2): &X\_{l} - Z\_{l} + 0.1401X\_{l-1} + 0.1314X\_{l-2} + 0.1436X\_{l-3} + 0.5957X\_{l-4} + 0.3971Z\_{l-1} + 0.2189Z\_{l-2} \end{aligned} \tag{11}$$

The model order estimation results are presented in Figure 8. From Figure 8, we can find that the BPNN can provide accurate order estimates for each length of the signal in models of different orders and the accuracy rate is above 90%. The accuracy of AIC and BIC has a certain upward trend with the increase of sequence length, while the estimated result of BIC can reach 90% under certain models. In Figure 8a,b, the judgment effect of the BIC and the BPNN is basically the same in the case of a long sequence. From Figure 8c, we can clearly find that the judgment result of the BPNN is much more accurate than the AIC and the BIC under the high-order model. In Figure 8c, the accuracy of AIC and BIC is below 40%. Similar to Section 4.2, the BIC performs better under low-order models than high-order models. In addition, we find that the accuracy of the AIC is low, but the effect is relatively stable under different models.

**Figure 8.** Comparison of the correct number of estimation methods for each order under different sequence length: (**a**) ARMA (1, 2), (**b**) ARMA (2, 3), (**c**) ARMA (4, 2).
