*4.1. Different Coefficients*

In the process of simulating data conforming to the ARMA model, the coefficients are restricted by many conditions. According to the definition of the ARMA model, the coefficients need to meet the requirements of causality and invertibility. In [17], the judgments of causality and invertibility are given by Equations (9) and (10).

Causality is equivalent to the condition

$$\phi(z) = 1 - \phi\_1 z - \dots - \phi\_p z^p \neq 0 \text{ for all } |z| \le 1 \tag{9}$$

Invertibility is equivalent to the condition

$$\theta(z) = 1 + \theta\_1 z + \dots + \theta\_q z^q \neq 0 \text{ for all } |z| \le 1 \tag{10}$$

where *φ*(·) and *<sup>θ</sup>*(·) are the *pth* and *qth*-degree polynomials. The complex *z* is used here since the zeros of a polynomial of degree *p* > 1 or *q* > 1 may be either real or complex. The region is defined by the set of complex *z* such that |*z*| = 1 is referred to as the unit circle. From Equations (9) and (10), the conditions of causality and invertibility are satisfied when the roots of *φ*(*z*) = 0 and *<sup>θ</sup>*(*z*) = 0 are outside the unit circle. In the calculation, we found that it is also necessary to consider whether the selected coefficients can effectively reflect the characteristics of the model in addition to causality and invertibility. In order to improve the sensitivity of the model, we set the minimum absolute value of the coefficient to 0.1. This avoids the fact that the coefficients are too small to make the polynomial difficult to identify.

In this section, we have simulated three ARMA models, each of which used 25 sets of different coefficients to simulate time series. Since the symmetrical ARMA model was less prone to result judgment difficulties, we used the asymmetrical ARMA model here. The three models were ARMA (1, 2), ARMA (2, 3), and ARMA (4, 2). Table 1 is the model coefficients we obtained through the random method, and Figure 6 is the verification of causality and invertibility.

In Figure 6, the roots of *φ*(*z*) = 0 and *<sup>θ</sup>*(*z*) = 0 for each model are shown in a different color. All roots are outside the unit circle, which shows that the coefficients meet the requirements of causality and invertibility. For the coefficients in Table 1, we simulated 30 different realizations of the system's response with time series lengths of 400. From these overdetermined ARMA model orders, the goal was to determine the correct ARMA model order using BPNN and compare its results with AIC and BIC. Another purpose was to study whether the effects of different order determination methods under random parameters were consistent. Model identification using the AIC and BIC was performed using functions in MATLAB R2017a.

Figure 7 is a stacked area diagram of the order estimation results. Figure 7 shows that the order estimation accuracy of BPNN is above 90%, AIC is below 10%, and the results of BIC are unstable. Comparing Figure 7a,b, the judgment results of BPNN and AIC are relatively stable, while the judgment efficiency of BIC criteria is significantly reduced and affected by the change of coefficients. For Figure 7c, the judgment effect of the BIC is basically the same as that of the AIC, and there is no obvious change in the BPNN. For the same model, the correct rate of BPNN is the highest, and the AIC is the lowest. The BIC is somewhere in between, but it is more sensitive to changes in model coefficients. For different ARMA models, the order estimation results of BPNN under different coefficients are relatively stable and accurate. In addition, the accuracy of the BIC is significantly reduced when the ARMA order is higher. For example, in ARMA (4, 2), its judgment effect is almost the same as that of the AIC. The influence of the change of order on different judgment criteria is analyzed in detail in Section 4.3. On the whole, the judgment result of BPNN has obvious advantages.


**Table 1.** ARMA coefficient.



**Figure 6.** The verification of causality and invertibility.

**Figure 7.** Comparison of the correct number of estimation methods for each order under different coefficients: (**a**) ARMA (1, 2), (**b**) ARMA (2, 3), (**c**) ARMA (4, 2).
