# Wavelet Time-Scale Modeling of Brand Sales and Prices
## 2 Literature Review  
Generally, marketing researchers and/or modelers search for models taking into account one or more variables solely as functions of time. Next, the model is analyzed regarding its efficiency for interpolating (approximating) and extrapolating (predicting) these variables in time. Compared to other fields such as financial and economic applications, the use of mathematical models, especially wavelet-based models, is still not well developed in marketing. In [15], for example, the authors mention the difficulty of time series training for traditional research models, the lack of software adapted to marketing modeling, the lack of, and sometimes the poor quality of, time-series data, and the lack of a substantive area for marketing, allowing time-series-based models to be readily adopted. The problem of time scales in marketing has been raised in several studies. In [16], it is mentioned that marketing phenomena depend on frequencies in time relative to decisions and interactions in the market. In classical investigations, the most commonly used time intervals are weekly, monthly, quarterly, and annual cycles. Each period has its special and/or specific characteristics. Weekly cycles, for example, are for reductions in prices, and quarterly periods are generally related to regular price adjustments and competition. Moreover, many phenomena such as pandemics (COVID-19), wars, climate change, financial crises, and socio-political movements may induce severe perturbations in the usual considerations, especially time periods, leading to diversification in the use of time scales. This diversification in how to choose the time scale, according to what factors, what characteristics, and what aim, has led researchers in the marketing field to think about including the time scale strongly in the models. In [17], the authors applied certain types of short-, medium-, and long-term horizons to investigate the effect of the time scale. However, their choice was not adapted to the wavelet method. In marketing, according to [16; 18], the time scale is mostly applied at discrete, equally spaced intervals, with weekly periods dominating, although this hypothesis has been criticized. In [18], the author applied a wavelet-based method for forecasting brand sales. The problem of multicollinearity has been investigated, resulting in correlated vectors of coefficients which have been applied to provide the most accurate forecasting and the best dimension reduction. In [19], for example, it is mentioned that marketing models still present aggregations, even using weekly intervals. This allows us to conclude that more adequate tools for time/frequency scales should be applied. Many methods have been used in this context, such as Fourier analysis and spectral analysis [20; 21]. However, these tools have raised many problems such as the non-stationary behavior of the marketing data, the sizes of samples, etc. (see also [22; 23; 24; 25; 26; 27]). Wavelet analysis was introduced into marketing models to overcome limitations such as the non-localization aspect of spectral analysis and its concentration in the frequency domain. This is achieved by frequency decomposition of the statistical time series into components that are well localized in time/frequency. [PERSON] mentioned a very important characteristic of wavelet theory in [16; 18; 28]. It permits investigation of the causal links between time-series cycles in marketing according to time scale. The modelers differentiate the marketing-driver influences in long-duration cycles of sales. Wavelets are also good tools regarding estimation and prediction accuracy and the handling of non-stationary time series. The application of wavelets in marketing resides in covering/uncovering the frequency activity characteristics of marketing models. Wavelet crystals generated from the inverse of the wavelet transform are applied to localize the variation in economic variables over different horizons.