3.2.1. ARMA Review of ARMA Model

A stochastic process *Xt* is called an autoregressive moving-average process (ARMA) with order *p* and *q*, namely ARMA (*p*, *q*) [48,49], if the process is stationary and satisfies a linear stochastic difference equation of the form,

$$X\_t = \phi\_1 X\_{t-1} + \dots + \phi\_p X\_{t-p} + e\_t + \theta\_1 e\_{t-1} + \dots + \theta\_q e\_{t-q} \tag{53}$$

where *et* is white gaussian noise (WGN), i.e., *et* ∼ *WN*(0, *<sup>σ</sup>*<sup>2</sup>), parameters *φ*1, *φ*2, ... , *φp* and *θ*1, *θ*2, ... , *θq* are coefficients of AR(*p*) and MA(*q*) models, and the polynomials are as follows,

$$\phi(z) = 1 - \phi\_1 z - \phi\_2 z - \dots - \phi\_p z^p \tag{54}$$

$$\theta(z) = 1 + \theta\_1 z + \theta\_2 z + \dots + \theta\_p z^p \tag{55}$$
