## 2 Background: Data and Autoregressive Models
### AutoRegressive Integrated Moving Average (ARIMA) Models  
The ARIMA model is a generalization of the ARMA model (AutoRegressive Moving Average model), suitable for handling non-stationary time series. As the classical ARMA model takes for granted the stationarity of the time series it is asked to analyze, the management of inherently non-stationary time series requires their transformation into a static data series by eliminating seasonality and trends, through a finite-point differentiation [3]. As mentioned earlier, a stationary time series can be thought of as a combination of signal and noise. The ARIMA model handles the time signal, after first separating it from the noise, and outputs its prediction for a subsequent time point [11]. As indicated by the method's acronym, its structural components are the following [13]:  
* AR: Autoregression. A regression model that uses the dependence relationship between an observation and a number of lagged observations (model parameter p).
* I: Integration. Calculating the differences between observations at different time points (model parameter d), aiming to make the time series stationary.
* MA: Moving Average. This approach considers the dependence that may exist between observations and the error terms created when a moving average model is used on observations that have a time lag (model parameter q). The AR model of order p, AR(p), can be written as a linear process as follows: \[x_{t}=c+\sum_{i=1}^{p}\phi_{i}x_{t-i}+\epsilon_{t},\] (7) where \(x_{t}\) the static variable, \(c\) a constant, the terms \(\phi_{i}\) are autocorrelation coefficients at the time delay steps \(1,2,\dots,p\) and \(\epsilon_{t}\) are the samples of the Gaussian white noise series, with zero mean and \(\sigma^{2}\) variance.  
A simple moving average model of order q, MA(q), can be given as:  
\[x_{t}=\mu+\sum_{i=0}^{q}\theta_{i}\epsilon_{t-i}, \tag{8}\]  
where \(\mu\), is the expected value of \(x_{t}\) (it mostly equals to 0), \(\theta_{i}\) the weights applied to the current and past values of the stochastic term of the time series, and \(\theta_{0}=1\). We consider \(\epsilon_{t}\) to be a Gaussian white noise series, with zero mean and \(\sigma_{\epsilon}^{2}\) variance.  
By combining these two models, autoregression and moving average, we create the ARMA model of class (p,q):  
\[x_{t}=c+\sum_{i=1}^{p}\phi_{i}x_{t-i}+\epsilon_{t}+\sum_{i=0}^{q}\theta_{i} \epsilon_{t-i}, \tag{9}\]  
where \(\phi_{i}\
eq 0,\theta_{i}\
eq 0,\sigma_{\epsilon}^{2}>0\). The parameters p, q constitute the order of the AR and MA models respectively.  
The general form of an ARIMA model is written as ARIMA(p, d, q), including the integration term that guarantees the stationarity of the time series [13], and it can be expressed as:  
\[\
abla^{d}x_{t}=c+\sum_{i=1}^{p}\phi_{i}\
abla^{d}x_{t-i}+\sum_{i=0}^{d}\theta_{ i}\epsilon_{t-i}, \tag{10}\]  
where \(\
abla^{d}\) is a differential factor, introducing a difference of order \(d\), aiming to remove the nonstationarity of the time series \(x_{t}\)[14].  
Table 1 displays the basic parameter combinations for the nonseasonal ARIMA models.