# Wavelet Time-Scale Modeling of Brand Sales and Prices
## 1 Introduction  
The aim of this work is to investigate a wavelet-based mathematical model for brand sales and prices. Wavelet theory has gained great interest in the fields of economics and finance since the last decades of the last century. Recall that almost all financial indices usually show volatile behavior as well as high fluctuations. As a result, their investigation using classical methods remains insufficient. Wavelets have indeed been shown to be useful in such investigations. Marketing, in its simplest sense, includes the institutions, the activities, and also the processes applied to deliver, create, and exchange goods, and to communicate offerings and needs to customers, partners, and clients, or more broadly to society [1]. While marketing consists of various activities, the principles of marketing rely on four elements, i.e., product, price, promotion, and place (distribution). Each one of these elements depends on qualitative and quantitative data. Quantitative data show the power of marketing in collecting, analyzing, optimizing, and executing, based on relevant numerical data. Product choices and portfolios, price hierarchies, numbers of outlets and stores, and, importantly, promotion budgets, especially for branding, brand management, and brand equity are also considered. Brands and the activity of branding are well represented in the literature, and brand equity refers to the value added to products [2]. Firms can benefit from a strong brand in various ways: it improves the product value and provides a chance to extendthe brand to new products, expand into new markets, and obtain a high level of customer loyalty and tolerance. A well-known brand acquires a price value [3]. Thus, brand sales- and equity-related data include things such as choice data, the brand design, the brand idea, time of purchases, the number of stores, the choice of price, and the competitors in the market. The state of the market itself affects the persistence and the success of the brand investment, strongly and simultaneously. Without sophisticated mathematical models that take into account all the factors and that permit a good understanding of the time and space evolution of the brand distribution and its evolution in the market, the investment in the brand may fail. In marketing, a mathematical model may include many factors such as the number of sales in specified periods such as weeks, months, or years, market shares per specified period, price evolution, the history of the market and its situation, competing brands, etc. The main characteristic of a mathematical model is its ability to deal with many factors simultaneously, according to time and space. It gathers the data into a time series form, permitting ordering over time. This has a very important consequence, as it yields information about the future state of the brand. The model uses a time series, as the best descriptor to extract, deduce, and/or accurately describe the properties of the empirical data. In the present paper, we aim to investigate the multidimensional modeling of time series obtained from brand sales and the evolution of prices. The model is of a general form, including many variables such as sales, prices, and advertising simultaneously. Our principal idea is based on the effect of the time factor on the brand evolution. In [4; 5], the authors mention that one of the limitations of conventional approaches in economic or financial studies and models is the lack of dependence of the models on the time scale, taking into account both the frequency and time domains. These facts are of great importance for the economic/econometric rationale. In many scientific fields, wavelets have provided efficient reformulations of many difficult problems such as the analysis and the modeling of time series, by transforming data from time domains to levels or horizons, according to the frequency. Their main advantage is their ability to be localized in the time/frequency domain.