## 2 Forecasting Models
### ARIMA Models  
ARIMA is one of the most versatile linear models for forecasting seasonal time series. It has enjoyed great success in both academic research and industrial applications during the last three decades. The class of ARIMA models is broad. It can represent many different types of stochastic seasonal and nonseasonal time series such as pure autoregressive (AR), pure moving average (MA), and mixed AR and MA processes. The theory of ARIMA models has been developedby many researchers and its wide application was due to the work by [PERSON] et al. [8] who developed a systematic and practical model building method. Through an iterative three-step model building process--model identification, parameter estimation and model diagnosis--the Box-Jenkins methodology has been proven to be an effective practical time series modeling approach. The multiplicative seasonal ARIMA model, denoted as ARIMA\((p,d,q)\times(P,D,Q)_{m}\), has the following form [18]:  
\[\phi_{p}(B)\Phi_{P}(B^{m})(1-B)^{d}(1-B^{m})^{D}y_{t}=c+\theta_{q}(B)\Theta_{Q} (B^{m})\epsilon_{t}, \tag{14}\]  
where  
\[\phi_{p}(B) =1-\phi_{1}B-\cdots-\phi_{p}B^{p}, \Phi_{P}(B^{m}) =1-\Phi_{1}B^{m}-\cdots-\Phi_{P}B^{Pm},\] \[\theta_{q}(B) =1+\theta_{1}B+\cdots+\theta_{q}B^{q}, \Theta_{Q}(B^{m}) =1+\Theta_{1}B^{m}+\cdots+\Theta_{Q}B^{Qm},\]  
and \(m\) is the seasonal frequency, \(B\) is the backward shift operator, \(d\) is the degree of ordinary differencing, and \(D\) is the degree of seasonal differencing, \(\phi_{p}(B)\) and \(\theta_{q}(B)\) are the regular autoregressive and moving average polynomials of orders \(p\) and \(q\), respectively, \(\Phi_{P}(B^{m})\) and \(\Theta_{Q}(B^{m})\) are the seasonal autoregressive and moving average polynomials of orders \(P\) and \(Q\), respectively, \(c=\mu(1-\phi_{1}-\cdots-\phi_{p})(1-\Phi_{1}-\cdots-\Phi_{P})\), where \(\mu\) is the mean of \((1-B)^{d}(1-B^{m})^{D}y_{t}\) process and \(\varepsilon_{t}\) is a zero mean Gaussian white noise process with variance \(\sigma^{2}\). The roots of the polynomials \(\phi_{p}(B),\Phi_{P}(B^{m}),\theta_{q}(B)\) and \(\Theta_{Q}(B^{m})\) should lie outside a unit circle to ensure causality and invertibility [19]. For \(d+D\geq 2\), \(c=0\) is usually assumed because a quadratic or a higher order trend in the forecast function is particularly dangerous. The ARIMA models are useful in describing stationary time series. Although many time series are nonstationary, they can be reduced to stationary time series by taking proper degrees of differencing (regular and seasonal) and making mathematical transformations [20]. The main task in ARIMA forecasting is selecting an appropriate model order, which are the values of \(p,q,P,Q,d\) and \(D\). Usually, the following steps are used to manually identify a tentative model [20]: (1) plot the time series, identify any unusual observations and choose the proper variance-stabilizing transformation. A series with nonconstant variance often needs a logarithm transformation. Often, more generally to stabilize the variance, a Box-Cox transformation may be applied; (2) compute and examine the sample ACF (autocorrelation function) and the sample PACF (partial autocorrelation function) of the transformed data (if a transformation was necessary) or of the original data to further confirm a necessary degree of differencing (\(d\) and \(D\)) so that the differenced series is stationary. Because variance-stabilizing transformations such as the Box-Cox transformations require positive values and differencing may create some negative values, variance-stabilizing transformations should always be applied before taking differences. If the data have a strong seasonal pattern, it is recommend that seasonal differencing be done first because sometimes the resulting series will be stationary and there will be no need for a further first difference.