# Wavelet Time-Scale Modeling of Brand Sales and Prices
## 3 Wavelets Toolkit  
As the literature on wavelets has grown dramatically since their discovery in the last century, we will review wavelet tools briefly in this part of the paper by focusing essentially on time series analysis and/or modeling. Wavelet analysis allows the representation of time series into forms relative to the time and frequency information, known as time-frequency decomposition. It involves decomposing a series into different frequency components at a scale-adapted resolution and thus allows the data to be observed and analyzed at different scales. The starting point of wavelet analysis is a mother wavelet \(\psi\left(\in L^{2}(\mathbb{R})\right)\), satisfying some appropriate assumptions). Such a mother wavelet yields the so-called wavelet basis \(\left(\psi_{j,k}=2^{-j/2}\psi(2^{j}-k),\ j,k\in\mathbb{Z}\right)\), where \(j\) is known as the scale, frequency, level, or horizon, and \(k\) is the position or translation parameter. The mother wavelet \(\psi\) gives rise to a second analyzing function called the father wavelet (or also the scaling function) \(\varphi\). For a fixed \(j\in\mathbb{Z}\), we introduce two types of functional spaces:  
* The \(j\)-level detail space \(W_{j}=span(\psi_{j,k},\,k)\);
* The \(j\)-level approximation space \(V_{j}=span(\varphi_{j,k},\,k)\). These spaces satisfy several useful properties such as  
\[L^{2}(\mathbb{R})=\bigoplus_{j}^{\perp}W_{j}=\overline{\bigcup_{j}V_{j}}\ \text{ and }\ V_{j+1}=W_{j}\oplus^{\perp}V_{j},\]  
where \(\varphi_{j,k}\) are defined in the same way as \(\psi_{j,k}\). The sequence \((V_{j})_{j}\) constitutes the multi-resolution analysis (multi-scale analysis) on \(\mathbb{R}\). Details on these concepts may be found in [7; 29; 30]. Using the above properties of wavelets and wavelet functional spaces, we deduce that any time series \(X(t)\) may be represented as  
\[X(t)=\sum_{j}D_{j}(t)=\sum_{j\leq I}D_{j}(t)+\sum_{j\geq I+1}D_{j}(t)=A_{J}(t) +\sum_{j\geq I+1}D_{j}(t), \tag{1}\]  
for all \(J\in\mathbb{Z}\), where the \(D_{j}(t)\) is the projection on \(W_{j}\), known as the detail component at level \(j\),  
\[D_{j}(t)=\sum_{k}d_{j,k}\psi_{j,k}(t), \tag{2}\]  
and where  
\[A_{J}(t)=\sum_{k}a_{j,k}\varphi_{J,k}(t)=\sum_{j\leq I}D_{j}(t) \tag{3}\]  
is the projection on \(V_{j}\), known also as the approximation of \(X(t)\) at level \(J\). This describes the global shape or the trend of \(X(t)\). The coefficients  
\[d_{j,k}=<X(t),\psi_{j,k}>=\int_{\mathbb{R}}X(t)\psi_{j,k}(t)dt, \tag{4}\]  
are known as the detail or wavelet coefficients of the series \(X(t)\) at level \(j\) and position \(k\), and the coefficients  
\[a_{j,k}=<X(t),\varphi_{j,k}>=\int_{\mathbb{R}}X(t)\varphi_{j,k}(t)dt \tag{5}\]  
are known as the approximation coefficients at level \(j\) and position \(k\). In the case of time series such as brand sales and prices, we use a discrete convolution to evaluate these coefficients:  
\[d_{j,k}=\sum_{n}X(n)\psi_{j,k}(n)\ \text{ and }\ a_{j,k}=\sum_{n}X(n)\varphi_{j,k}(n). \tag{6}\]  
In the context of marketing, the series \(X(t)\) may represent the prices, volumes, competitive prices, or promotion prices, as will be explained later for the marketing model (8) in Section 4. As a result, the wavelet series or the decomposition of \(X(t)\) is composed of a first part due to the global shape (the trend) \(A_{J}\), together with the details or the higher-frequency components \(D_{j}\). For empirical use, we obviously focus on a fixed maximal level \(J\) at which we truncate the decomposition:  
\[X_{J}(t)=A_{J_{0}}(t)+\sum_{J_{0}<j\leq J}D_{j}(t), \tag{7}\]  
where \(J_{0}<J\). In fact, there is no rule for fixing the levels \(J_{0}\) and \(J\). However, for \(J_{0}\), we always have some flexibility, as it does not affect the total decomposition (we usually take \(J_{0}-0\)). The maximal level \(J\) is selected relative to the desired error estimates. We know, in fact, that \(\|X_{J}-X\|_{2}\to 0\) as \(J\to\infty\).