*2.1. ARMA (p, q) Model*

The autoregressive–moving-average is an important simulation method of stationary time series. Before discussing the order judgment of the time series, we need to analyze the structure of the ARMA model. {*Xt*} is an ARMA (p, q) process if {*Xt*} is stationary and if for every *t*,

$$X\_t - \phi\_1 X\_{t-1} - \dots - \phi\_p X\_{t-p} = Z\_t + \theta\_1 Z\_{t-1} + \dots + \theta\_q Z\_{t-q} \tag{1}$$

where {*Zt*} ∼ *WN*(0, *σ*<sup>2</sup>) and the polynomials (1 − *φ*1*<sup>z</sup>* − ··· − *φpz<sup>p</sup>*) and (1 − *θ*1*<sup>z</sup>* −···− *<sup>θ</sup>pz<sup>p</sup>*) have no common factors [17]. The *p* of the left-hand side of Equation (1) represents the order of the autoregressive (AR) process. Similarly, the *q* of the right-hand side is the order of the moving-average (MA) process. When they are equal, the ARMA model has mathematical symmetry.

For a mathematical model, it is an important requirement for the effectiveness to be able to fully represent information of the sequence, which is also true for the ARMA model. In Equation (1), *Zt* is actually a sequence of white noise. Its characteristic is that the value of *Zt* does not affect the trend of *Xt*. At the same time, when the parameters of Equation (1) and *Xt* are known, the value of *Zt* can also be determined within a certain range. The characteristics of *Zt* provide a basis for us to determine the order of the ARMA model through the neural network.
