## 6 Discussion of Results  
Before going on commenting and discussing our results in the framework of quality of life sense, we quickly recall some facts about the accuracy and convergence of our wavelet decomposition. This may be in fact a theoretical task which is already guaranteed in wavelet theory. We know that in wavelet theory, the wavelet approximation of the statistical series at a level \(J\) converges in the \(L^{2}\)-sense (variance or standard deviation measures in statistics). Mathematically speaking, we know that the projection on the multi-resolution space \(V_{I}\) guarantees an approximation of the order \(2^{-sJ}\), and thus, as we increase the order \(J\), we keep a best convergence and accuracy. \(s=NLogN\) is the regularity of the wavelet used, where \(N\) is the length of the wavelet filter. In our case, Db8 is regular of order \(s=8 Log8=16.63\). So, the order of convergence is approximately \(2^{-32 Log8}\), which is an accurate order. Let us now comment on, interpret, and discuss our empirical results. Figure 1 illustrates the behavior of the variable \(V_{1}\), which represents the Gross Domestic Product. Notice that such a variable represents a global shape showing an increase in time. However, its global shape does not reflect the real and hidden behavior of the variable according to the time scale. Indeed, at low scales, \(j=1\), the component \(D_{1}\) shows quietly the same global behavior, an increasing variation according to time. However, by increasing the level or the scale, the real hidden behavior starts to appear, explaining some pseudo-periodicity and/or a fluctuation in the variation of the factor \(V_{1}\). By means of quality of life vocabulary, this means that the situation is not stable, and thus, the future quality of life situation should be carefully linked to the Gross Domestic Product. However, this is unstable according to the time scale. Related to the case study, especially the period chosen, this is a natural consequence of many critical phenomena that happened in this period. The first is the starting steps in implementing the 2030 vision plans. As usual, with the absence of a real future view, precise forecasting, and prevision of this plan, the statistics as well as the real behavior of investors and citizens is always careful. The COVID-19 pandemic has been also dispersed in all the word, leading to a perturbation and sometimes a permanent stop of many important activities such as oil exportation, foreign investments, etc. These facts affect the local or the national market, especially local or domestic production: for  
Figure 9: The plots of the \(HDI\) index, real and time scale model (5). example, by increasing prices of consummations, taxes, etc, which negatively affect the view of people regarding their lives, especially their economic situation. The variable \(V_{2}\) representing the Gross National Income is tending more toward a decreasing shape, with more stability at medium horizons. As the time scale increases, this income shows more instability with always a decreasing magnitude. This may be due as evoked previously to many causes, especially COVID-19, as well as the low movement of exportation. Recall also that the embargo against Qatar country where the exportation of a big amount of especially food industry products has been stopped for a long time, as estimated by the level \(j=3\). Another cause is the restoring legitimacy of the war in Yemen, which has largely affected the GCC continent in general. These phenomena have led to a large portion of money being directed to and reserved for the army activities and needs rather than being allocated to investments that bring more interest to the treasury. The variable \(V_{3}\) relative to Gross Savings is somehow increasing in low horizons, with a small perturbation in medium levels, and more instabilities at higher horizons.