# Forecasting Irregular Seasonal Power Consumption.
## 2 Materials and Methods
### Forecasting Methods
#### 2.2.5 Multiple Seasonal Holt-Winters Models with Discrete Interval Moving Seasonalities  
nHWT models are robust to variations in the series, but sometimes special situations occur in which it is interesting to take these anomalies into account. One of the clearest examples is the influence of the calendar effect on electricity demand [65]. These anomalous and specific situations can sometimes be modeled as a discrete seasonality, if they follow a repetitive pattern. Despite being seasonal, since they are discrete, they have the particular quality that they are not located at fixed moments in the time series; therefore, they are not linked to a deterministic appearance, as would be the case for regular seasonality. These seasonalities are called discrete interval moving seasonality (DIMS). [PERSON] et al. [25] include the use of discrete seasonality in their model, so that the model seen in (6)-(9) now results in (21)-(25), which is named nHWT-DIMS:  
\[l_{t}=\alpha\left(\frac{y_{t}}{\prod s_{t-s_{i}}^{(i)}\prod D_{t_{t}^{*}-s_{t}^ {*}}^{(m)}}\right)+(1-\alpha)\ (l_{t-1}+\ \varrho b_{t-1}), \tag{21}\]  
\[b_{t}=\gamma(l_{t}-l_{t-1})+(1-\gamma)\varrho b_{t-1}, \tag{22}\]\[s_{t}^{(i)}=\delta^{(i)}\left(\,\frac{y_{t}}{l_{t}\prod_{j\
eq i}s_{t-s_{j}}^{(i)} \prod_{m}D_{t_{k}^{*}-s_{m}^{*}}^{(m)}}\,\right)\,+\,\left(1-\,\delta^{(i)} \right)s_{t-s_{i}^{\prime}}^{(i)} \tag{23}\]  
\[D_{t_{k}^{*}}^{(h)}=\delta_{D}^{(h)}\left(\,\frac{y_{t}}{l_{t}\prod_{j}s_{t-s_ {j}}^{(i)}\prod_{m\
eq h}D_{t_{k}^{*}-s_{m}^{*}}^{(m)}}\,\right)\,+\,\left(1- \delta_{D}^{(h)}\right)D_{t_{k}^{*}-s_{k}^{*}}^{(h)} \tag{24}\]  
\[\hat{y}_{t+k}=\left(l_{t}+\sum_{v=1}^{k}\varrho^{v}b_{t}\right)\prod_{i}s_{t-s _{i}+}^{(i)}\prod_{j}D_{t_{k}^{*}-s_{k}^{*}+k}^{(h)}+\varphi_{AR}^{k}\varepsilon _{t}. \tag{25}\]  
Here the term \(D_{t_{k}^{*}}^{(h)}\) is included, which represents the discrete seasonal indices, for each DIMS \((h)\) considered up to \(n_{DIMS}\). DIMS are only defined in the time intervals in which the special event takes place. These time intervals are designated using \(t_{h}^{*}\) for each DIMS \((h)\). This nomenclature is chosen in order to distinguish this from the continuous time interval \(t\). The great difference between this model and other methods of modeling special situations is that the effect produced by the anomaly in the series is modeled as an internal part of the model, as one more seasonality, and is smoothed with each new appearance, unlike the use of models with dummy variables and/or modifications of the original series. In the nHWT models, the seasonality equation shows a fixed recurrence for each seasonal pattern (\(s_{i}\)) being considered. With DIMS, this is not possible, since the occurrences of special events are not subjected to a deterministic pattern in the series. Therefore, the use of the variable \(s_{h}^{*}\) indicates, for each DIMS and each occurrence, which is the recurrence to consider. One possible situation with special events is the simultaneous occurrence of two events. In such a case, the forecaster should consider the option of using only one of the DIMS that occur at that time, or using both, if the effects produced by the two special events add up. An important aspect to consider is the initialization of the DIMS. A DIMS may have few occurrences in the time series and, therefore, its seasonal indexes must be calculated in such a way that it converges rapidly to the desired effect. The initialization method consists in first obtaining the initial values of the level, the trend, and the seasonal indices for the regular seasonality. Subsequently, a decomposition of the series is carried out using trend and multiple seasonality. It is common to use the multiple STL method (Seasonal-Trend decomposition procedure using Loess [66], where Loess is a method to estimate linear relationships). From the decomposition, the series can be reconstructed without including the irregular part, which is where the information necessary to obtain the desired indices is found.