# Forecasting Irregular Seasonal Power Consumption.
## 2 Materials and Methods
### Forecasting Methods
#### 2.2.3 Multiple Seasonal Holt-Winters Models  
Exponential smoothing uses information from the past through weighted averages to make predictions. The weight decreases as newer values are entered into the time series, giving more importance to newer data over older. A smoothing parameter determines this weight. The introduction of these models dates back to the 1960s with the work of [PERSON] [53] and [PERSON] [54]. [PERSON] [20] presented the Holt-Winters models, in which exponential smoothing techniques are performed on the three components of the series: level (\(l_{t}\)), trend (\(b_{t}\)) and seasonality (\(s_{t}\)). The model includes a series of structured equations, called smoothing equations, the information from which is compiled by a forecast equation to provide forecasts. The equations can be combined with additive or multiplicative trends and seasonality.  
[PERSON] and [PERSON] [55] introduced a damping factor for the trend, and their model outperforms the previous models when the trend shows high variations [56]. [PERSON] broke down seasonality into two or three nested components so that the models can capture the series that present more than one seasonality, such as series for short-term demand [22; 23]. [PERSON] also included in the model an adjustment using the one-step-ahead error as proposed by [PERSON] [57]. This adjustment adds an AR(1) model for the residuals, obtaining the parameter at the same time as the smoothing parameters are obtained. In the same way, [PERSON] and [PERSON] [24] generalized the model including the way the initial values are obtained, to \(n\) seasonalities. The nHWT models are shown in Equations (6)-(9).  
\[l_{t}=\alpha\left(\frac{y_{t}}{\prod s_{t-s_{t}}^{(i)}}\right)+(1-\alpha)(l_{t -1}+\phi b_{t-1}), \tag{6}\]\[b_{t}=\gamma(l_{t}-l_{t-1})+(1-\gamma)\varrho b_{t-1}, \tag{7}\]  
\[s_{t}^{(i)}=\delta^{(i)}\left(\frac{y_{t}}{l_{t}\prod_{j\
eq i}s_{t-s_{j}}^{(j)}} \right)+\left(1-\delta^{(i)}\right)s_{t-s_{t}}^{(i)}, \tag{8}\]  
\[\hat{y}_{t+k}=\left(l_{t}+\sum_{j=1}^{k}\varrho^{j}b_{t}\right)\prod_{i}s_{t-s_ {i}+k}^{(i)}+\varphi_{AR}^{k}\varepsilon_{t}. \tag{9}\]  
The smoothing equations include the smoothing parameters \(\alpha\), \(\gamma\) and \(\delta^{(i)}\) for smoothing the level, the trend and the different seasonal indices \((i)\) of length \(s_{i}\). The equation \(\hat{y}_{t+k}\) provides the \(k\)-future prediction values from the observed values of the series \(y_{t}\). Here, \(\varepsilon_{t}\) is the one-step-ahead error, and the parameter \(\varphi_{AR}\) is the parameter for the AR(1) adjustment. The damping parameter for the trend is denoted by \(\varrho\)[58].  
The equations of the model are recursive, and therefore they need initial values so that they can fit the model. Several methodologies for initialization have been documented [56; 57]. To be able to use the models, it is necessary to estimate the smoothing parameters by minimizing the error using non-linear algorithms [59; 60]. The Nelder-Mead [61; 62] simplex method has been used, which minimizes the root of mean squared error (RMSE).