# Wavelet Time-Scale Modeling of Brand Sales and Prices
## 5 Results and Discussion
The perturbation is confirmed in the detail components \(D_{1}\) to \(D_{6}\). According to \(D_{1},D_{2}\) a certain pseudo-periodicity seems to take place, with some differences in the maximum prices. This may be explained by the fact that at short time scales (such as weeks in the classical treatments of the time factor), prices are influenced by competitors on the one hand and the global behavior of the market (national as well as international) on the other. One essential factor to be considered is the COVID-19 pandemic, which covered a great part of the period of study and which certainly resulted in decreased sales so that, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Brand** & **Level 1** & **Level 2** & **Level 3** & **Level 4** & **Level 5** & **Level 6** \\ \hline \multicolumn{7}{c}{The brand prices} \\ \(B_{1}\) & \(-0.0002\) & \(0.0007\) & \(0.0016\) & \(0.0147\) & \(0.0147\) & \(0.0912\) \\ \(B_{2}\) & \(0.0000\) & \(0.0000\) & \(-0.0001\) & \(-0.0054\) & \(-0.0054\) & \(-0.0041\) \\ \(B_{3}\) & \(-0.0000\) & \(-0.0004\) & \(0.0001\) & \(0.0042\) & \(0.0042\) & \(0.0656\) \\ \(B_{4}\) & \(-0.0000\) & \(-0.0001\) & \(-0.0008\) & \(0.0001\) & \(0.0001\) & \(-0.0106\) \\ \(B_{5}\) & \(-0.0000\) & \(0.0002\) & \(-0.0007\) & \(-0.0045\) & \(-0.0045\) & \(-0.0061\) \\ \(B_{6}\) & \(0.0000\) & \(0.0002\) & \(-0.0008\) & \(0.0054\) & \(0.0054\) & \(-0.0395\) \\ \multicolumn{7}{c}{The brand sales} \\ \(B_{1}\) & \(-0.0073\) & \(0.0196\) & \(-0.0206\) & \(0.0079\) & \(0.0079\) & \(0.0034\) \\ \(B_{2}\) & \(-0.0645\) & \(-0.0049\) & \(0.0065\) & \(0.2211\) & \(0.2211\) & \(0.7129\) \\ \(B_{3}\) & \(0.0012\) & \(-0.4976\) & \(-0.9052\) & \(36.9157\) & \(36.9157\) & \(20.2465\) \\ \(B_{4}\) & \(-0.2039\) & \(0.2151\) & \(-0.0401\) & \(0.3270\) & \(0.3270\) & \(1.0747\) \\ \(B_{5}\) & \(-0.2954\) & \(0.2021\) & \(-1.5023\) & \(4.1040\) & \(4.1040\) & \(21.1156\) \\ \(B_{6}\) & \(-0.0060\) & \(0.0060\) & \(-0.0210\) & \(0.4643\) & \(0.4643\) & \(0.5981\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mean wavelet coefficients for brand prices and sales at different levels. to compensate for the quantities in stock and thus to recover losses, the company had to increase prices, as in many other cases. This fact is easily shown in the \(B_{1}\) sales graphs in Figure 2, where we clearly observe an upward trend in sales whenever prices decreased and a downward trend in sales whenever prices increased. In addition, both prices and sales are volatile at long time scales, according to the detail components \(D_{3}\) to \(D_{6}\). We may thus conclude for this brand that short time scales are more comprehensive and easy to understand for consumers, investors, and managers. In Figure 3, there is a slight variation around the mean (median), with no exact periodicity but instead a slight pseudo-periodic structure. This behavior is repeated in  
Figure 1: The wavelet decomposition of the Jarir brand \(B_{1}\) prices at level 6. Figure 2: The wavelet decomposition of the Jarir brand \(B_{1}\) sales at level 6.  
the sales illustrated in Figure 4. However, some anomalies appear, essentially at all time scales. This behavior can be naturally understood from the fact that Almarai is a completely national company, based on completely national products. Therefore, compared to \(B_{1}\), for example, its prices and volumes are not strongly affected by importation. In addition, its products are the oldest, the best in the market (according to the majority of consumers), and are consumed daily. All these factors allow price stability in the market, despite a slight growth in the volume of production. In addition, prices and sales seem to reflect the same behavior at short and long time scales. These fluctuations are clearly illustrated by the detail components \(D_{1}\) to \(D_{6}\). Nevertheless, slight fluctuations remain around the mean for all the time scales. Figure 4: The wavelet decomposition of the Almarai brand \(B_{2}\) sales at level 6.