# Forecasting Irregular Seasonal Power Consumption.
## 3 Results
### Application of Discrete Seasonality Models (nHWT-DIMS)
\begin{table}
\begin{tabular}{c c c c c} \hline
**DIMS** & **Nr.** & **Time Starts** & **Time Ends** & **Recursivity** \\ \hline DIMS a & 1 & 14 th November at 05:42 am & 14 th November at 08:24 am & \\  & 2 & 14 th at 04:00 pm & 14 th at 06:42 pm & 618 min. \\  & 3 & 15 th at 00:18 am & 15 th at 03:00 am & 498 min \\  & 4 & 15 th at 07:30 am & 15 th at 10:12 am & 432 min \\  & 5 & 15 th at 06:42 pm & 15 th at 09:24 pm & 672 min \\  & 6 & 16 th at 07:24 pm & 16 th at 10:06 pm & 1482 min \\  & 7 & 17 th at 10:42 am & 17 th at 01:24 pm & 918 min \\  & 8 & 18 th at 06:06 am & 18 th at 08:48 am & 1164 min \\  & 9 & 19 th at 02:30 am & 19 th at 05:12 am & 1224 min \\  & 10 & 19 th at 08:48 pm & 19 th at 23:30 pm & 1098 min \\  & 11 & 21 th at 06:06 pm & 21 th at 20:48 pm & 2718 min \\ DIMS b & 1 & 16 th November at 07:06 am & 16 th November at 09:48 am & \\  & 2 & 20 th at 05:06 am & 20 th at 07:48 am & 5640 min \\  & 3 & 20 th at 05:24 pm & 20 th at 08:06 pm & 738 min \\  & 4 & 21 th at 02:36 am & 21 th at 05:18 am & 552 min \\  & 5 & 22 th at 02:24 am & 22 th at 05:06 am & 1428 min \\ \hline \end{tabular}
\end{table} 
Table 4: Location in the time series of the discrete seasonalities (DIMS) and their recursivity. The column Nr. indicates the order of appearance of the corresponding DIMS. ’Time starts’ and ’Time ends’ reflect the moving interval in which DIMS is defined, and ’Recursivity’ shows the length of time since the previous appearance. Figure 12: Discrete interval moving seasonality (DIMS) locations and recursion design. Two additional seasonal patterns are located, the first mostly appears in (**a**) while the second pattern appears in (**b**). Vertical dashed lines delimitate the DIMS period length. The appearance number of each DIMS is numbered at the bottom of the figure in red. Lines with arrows represent the recursivity of the DIMS, with full lines for the DIMS in (**a**) and dashed lines for the DIMS in (**b**). The time span of the previous appearance is shown in minutes over the line. \[s_{t}^{(10)}=\delta^{(10)}\left(\frac{y_{t}}{l_{t}D_{t_{a}-s_{a}^{*}}^{(a)}D_{t_{b} ^{*}-s_{b}^{*}}^{(b)}}\right)+\left(1-\ \delta^{(10)}\right)s_{t-10}^{(10)}, \tag{29}\]  
\[D_{t_{a}^{*}}^{(a)}=\delta_{D}^{(a)}\left(\frac{y_{t}}{l_{s}^{(10)}D_{t_{a}^{*} -s_{b}^{*}}^{(b)}}\right)+\ \left(1-\delta_{D}^{(a)}\right)D_{t_{a}^{*}-s_{a}^{*}}^{(a)}, \tag{30}\]  
\[D_{t_{b}^{*}}^{(b)}=\delta_{D}^{(b)}\left(\frac{y_{t}}{l_{s}^{(10)}D_{t_{a}^{*} -s_{a}^{*}}^{(a)}}\right)+\ \left(1-\delta_{D}^{(b)}\right)D_{t_{b}^{*}-s_{b}^{*}}^{(b)}, \tag{31}\]  
\[\hat{y}_{t}(k)=(l_{t}+kb_{t})s_{t-s_{0}+k}^{(10)}D_{t_{a}^{*}-s_{a}^{*}+k}^{(a )}D_{t_{b}^{*}-s_{b}^{*}+k}^{(b)}+\vartheta_{AR}^{k}\varepsilon_{t}. \tag{32}\]  
Here, \(s_{t}^{(10)}\) is the seasonal equation for the regular seasonality of ten-time units with smoothing parameter \(\delta^{(10)}\). \(D_{t_{a}^{*}}^{(a)}\) and \(D_{t_{b}^{*}}^{(b)}\) are the DIMS as described in Table 4 with smoothing parameters \(\delta_{D}^{(a)}\) and \(\delta_{D}^{(b)}\) defined only in time \(t_{a}^{*}\) and \(t_{b}^{*}\). The recursivity \(s_{a}^{*}\) and \(s_{b}^{*}\) is defined in Table 4.  
To use the model, the procedure described in [25] is carried out. Initially, the initial values for the level are obtained as the moving average of the first period of one hour; for the trend as the slope between the first and second cycle of one hour; and for the seasonal indices, the weighting of the series in the first cycle on the moving average. Subsequently, the seasonal indices of the DIMS are obtained. The time series is decomposed into its trend, seasonality, and irregular components using STL decomposition with the period length of one hour. From these components, the series is rebuilt, but without the irregular component being included. The seasonal indices are obtained by weighting the original series over the reconstructed one. Once the initial values of the model have been determined, the model is fitted by minimizing the RMSE, and the smoothing parameters are obtained.