# Forecasting Irregular Seasonal Power Consumption.
## 2 Materials and Methods
### Forecasting Methods
#### 2.2.2 ARIMA Models  
ARIMA models were introduced by [PERSON] and [PERSON] [50] to model non-stationary series and allow predictions to be made. A description and in-depth analysis can be found in [51] and in the book by [PERSON] and [PERSON] [52]. Seasonal ARIMA models are usually denoted by \(\text{ARIMA}(p,d,q)x(P,D,Q)_{S}\). \(S\) indicates the length of the seasonal pattern under consideration. The compact representation of the ARIMA model is usually, as shown in (3), a function of autoregressive polynomials and polynomials of moving means, and of the difference operators.  
\[\begin{split}\phi_{P}(B)\Phi_{P}\big{(}B^{S}\big{)}\
abla^{d} \
abla_{S}^{D}\Big{(}& y_{t}^{(\lambda)}-c\Big{)}=\theta_{q}(B) \Theta_{Q}\big{(}B^{S}\big{)}\varepsilon_{t},\\ &\{\varepsilon_{t}\}\sim\text{N}\big{(}0,\sigma^{2}\big{)}.\end{split} \tag{3}\]  
\(\{y_{t},\ t=0,\pm 1,\pm 2,\ldots\}\) are the observed data of the univariate series. If the variability in the data grows with time, it is necessary to transform the data to stabilize the variance. The Box-Cox power transformation family is a general class of variance-stabilizing trans  
Figure 6: Function fitting neural network schema.  
formations. The Box-Cox transformation of \(y_{t}\) with power parameter \(\lambda\) to the transformed data \(y_{t}^{(\lambda)}\) is defined by (4).  
\[y_{t}^{(\lambda)}=\left\{\begin{array}{cc}\frac{y_{t}^{1}-1}{\lambda};&\text{ if }\lambda\
eq 0,\\ ln\,y_{t};&\text{if }\lambda=0\end{array}\right.. \tag{4}\]  
The power parameter \(\lambda\) is estimated by the maximum-likelihood method. The polynomials \(\phi_{p}(B)=1-\phi_{1}B-\phi_{2}B^{2}-\cdots-\phi_{p}B^{p}\) and \(\theta_{q}(B)=1-\theta_{1}B-\theta_{2}B^{2}-\cdots-\theta_{q}B^{q}\) represent the regular or non-seasonal autoregressive and the moving averages components, respectively, and the polynomials \(\Phi_{p}(B^{5})=1-\Phi_{1}B^{5}-\Phi_{2}B^{25}-\cdots-\Phi_{p}B^{PS}\) and \(\Theta_{\text{Q}}(B^{5})=1-\Theta_{1}B^{5}-\Theta_{2}B^{25}-\cdots-\Theta_{ \text{Q}}B^{QS}\) represent the seasonal autoregressive and the moving averages components, respectively, with \(B\) as the lag operator. \(\
abla\) is the is the backward difference operator, \([By_{t}=y_{t-1};\,B^{5}y_{t}=y_{t-S};\,\
abla=(1-B);\,\
abla^{d}=(1-B)^{d};\, \
abla_{\text{S}}^{D}=\left(1-B^{S}\right)^{D}]\). \(d\) and \(D\) are the number of differencings required to make the time series stationary (\(d\), \(D\leq 2\)). \(\{\epsilon_{t}\}\) is a Gaussian white noise process, \([\{\epsilon_{t}\}\sim N\big{(}0,\sigma^{2}\big{)}]\). \(c\) is the model constant.  
The orders of the polynomials \(\{p,\,d,\,P,\,Q\}\) are selected using the Akaike's Information Criterion (AIC, AICc) or [PERSON]'s or the Bayesian Information Criterion (_SIC_ or _BIC_). The model coefficients \(\big{\{}\,\Phi_{1},\Phi_{2},\ldots,\phi_{p};\,\theta_{1},\theta_{2},\ldots, \theta_{q};\,\Phi_{1},\Phi_{2},\ldots,\Phi_{p};\Theta_{1},\Theta_{2},\ldots, \Theta_{\text{Q}}\big{\}}\) and \(\sigma^{2}\) are estimated by the maximum likelihood method.  
ARIMA models can present more than one seasonality, as indicated in (5). To do this, the models are expressed as ARIMA \((p,d,q)x(P_{1},D_{1},Q_{1})_{S_{1}}x(P_{2},D_{2},Q_{2})_{S_{2}}\) where \(S_{1}\) and \(S_{2}\) indicate the two seasonalities to which they refer.  
\[\phi_{p}(B)\Phi_{\text{P}_{1}}\left(B^{S_{1}}\right)\Omega_{P_{2}}\left(B^{S_ {2}}\right)\
abla^{d}\
abla_{\text{S}_{1}}^{D_{1}}\
abla_{\text{S}_{2}}^{D_{2 }}\left(y_{t}^{(\lambda)}-\text{c}\right)=\theta_{q}\,\left(B)\Theta_{\text{Q }_{1}}\left(B^{S_{1}}\right)\Psi_{\text{Q}_{2}}\left(B^{S_{2}}\right)\epsilon_ {t}. \tag{5}\]  
The polynomials \(\Omega_{P_{2}}\left(B^{S_{2}}\right)\) and \(\Psi_{\text{Q}_{2}}\left(B^{S_{2}}\right)\) represent the second seasonal autoregressive and the moving averages components, respectively.