# Wavelet Time-Scale Modeling of Brand Sales and Prices
## 5 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Brand** & **Mean** & **Median** & **Min** & **Max** & **Std** & **Skewness** & **Kurtosis** & **Jarque–Bera (\(h,p\))** \\ \hline \multicolumn{10}{c}{The brand prices} \\ \(B_{1}\) & 161.42 & 159.2 & 105.44 & 225 & 26.48 & 0.21 & 2.37 & (1,\(10^{-3}\)) \\ \(B_{2}\) & 52.91 & 53.3 & 36.95 & 63.7 & 4.07 & \(-0.08\) & 3.15 & (0,0.34) \\ \(B_{3}\) & 100.25 & 99.75 & 67.10 & 139.2 & 16.35 & 0.20 & 2.52 & (1,\(10^{-3}\)) \\ \(B_{4}\) & 15.88 & 12.66 & 8.15 & 40.35 & 8.12 & 1.76 & 4.64 & (1,\(10^{-3}\)) \\ \(B_{5}\) & 20 & 20.34 & 14.46 & 29.4 & 3.55 & 0.39 & 2.35 & (1,\(10^{-3}\)) \\ \(B_{6}\) & 35.52 & 31.35 & 14.88 & 69.75 & 16.44 & 0.65 & 2.01 & (1,\(10^{-3}\)) \\ \multicolumn{10}{c}{The brands sales} \\ \(B_{1}\) & 153.63 & 109.31 & 7.56 & 4020 & 258.03 & 10.70 & 140.71 & (1,\(10^{-3}\)) \\ \(B_{2}\) & 564.28 & 418.03 & 34.70 & 12,140 & 689.47 & 9.25 & 129.83 & (1,\(10^{-3}\)) \\ \(B_{3}\) & 890.12 & 512.49 & 33.30 & 1.21310 & 3967.75 & 27.64 & 831.52 & (1,\(10^{-3}\)) \\ \(B_{4}\) & 809.01 & 223.43 & 8.24 & 22,200 & 1935.15 & 6.08 & 51.00 & (1,\(10^{-3}\)) \\ \(B_{5}\) & 2577.69 & 1720 & 111.39 & 37,490 & 3161.13 & 4.89 & 40.35 & (1,\(10^{-3}\)) \\ \(B_{6}\) & 243.89 & 64.04 & 0.01 & 5250 & 570.35 & 4.70 & 29.97 & (1,\(10^{-3}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Descriptive statistics for brands \(B_{i}\), \(1\leq i\leq 6\), prices and sales. Next, we conduct a wavelet multi-scale analysis for each of the variables \(B_{i}\), \(0=1,2,\ldots,6\) at level 6, based on the Daubechies mother wavelet Db8 (see [30]). Table 3 shows the wavelet coefficients for each of the brand prices and sales. Table 3 presents a global overview of the movements of brands (prices and sales) according to their mean wavelet coefficients estimated for different time horizons. We notice an overall increase, for example, in the prices of brand \(B_{1}\), accompanied by a decrease in sales. This behavior also occurs for brands \(B_{3}\), \(B_{4}\), \(B_{5}\), and \(B_{6}\), in contrast to \(B_{2}\) where the global behavior is fairly stable, apart from some extreme values, which appear mainly in the high levels for all the brands. This allows us to conclude that applying the wavelet method to these marketing time series effectively starts to indicate the influence of the time-scale behavior hidden in these series, and that a global description, although with wavelets, is not sufficient to extract the real and hidden structure of the series. Therefore, a deep time-scale study is necessary. To easily show the time-scale variations (fluctuations, increases, decreases) of these variables and to further understand their behavior according to the time scale, we reproduce the prices and sales for the brands \(B_{i}\), \(1\leq i\leq 6\), graphically. This yields, among other interpretations, an easy-to-read description of these variables according to the time scale. Figures 1, 3, 5, 7, 9, and 11 illustrate the wavelet decomposition of the prices for each brand at the level of decomposition \(J=6\). In addition, in Figures 2, 4, 6, 8, 10, and 12, we provide the wavelet decomposition due to the brand sales at the same decomposition level of \(J=6\). These graphs illustrate the variables with their trends and dynamics or fluctuations. On the \(y\)-axis, the variable \(A_{6}\) stands for the wavelet approximation at level 6, while the variable \(D_{i}\), \(1\leq i\leq 6\), stands for the detail component at level \(i\). The strong fitting between each variable and its approximation can clearly be seen. We now investigate these graphs case by case and deduce the effect of the time scale on the corresponding brand movement. Figure 1, illustrating the wavelet decomposition of \(B_{1}\) prices, shows a somewhat increasing behavior, clearly illustrated by the level 6 trend \(A_{6}\), reminiscent of some perturbations at medium-to-long scales. However, the trend shows no cycles (periodic behavior) for the \(B_{1}\) prices.