## 3 Materials and Methods
### Data Preprocessing
#### 3.1.4 Feature Contribution Description  
To better understand the factors influencing sales predictions, this study decomposes the contributions of different dimensions into distinct models. Each dimension reflects specific aspects of the data and their interactions with the target variable. The contribution calculations are categorized as follows:  
* Time Dimension Contribution The contribution of the time dimension is modeled as: \[F_{\text{time}}(S_{f},H_{f})=w_{1}S_{f}^{2}+w_{2}H_{f}\cdot\sin(\frac{2\pi}{1 2}t)+w_{3}S_{f}\cdot H_{f}\] (1)  
The weights \(w_{1},w_{2},w_{3}\) control the contributions of nonlinear seasonal effects, periodic influences, and their interactions, respectively. This formulation captures seasonal variations, holiday influences, and their interactions. The term \(S_{f}^{2}\) accounts for nonlinear seasonal effects, while \(H_{f}\cdot\sin(\frac{2\pi}{12}t)\) introduces periodicity reflecting the calendar cycle.  
\begin{table}
\begin{tabular}{l l l} \hline
**Feature Name** & **Symbol** & **Feature Importance** \\ \hline Seasonal feature & \(S_{f}\) & 0.42 \\ Holiday feature & \(H_{f}\) & 0.38 \\ Latitude and longitude & \(L_{f}\) & 0.12 \\ Population density & \(P_{f}\) & 0.20 \\ Consumer confidence index & \(C_{f}\) & 0.24 \\ Consumer expenditure & \(E_{f}\) & 0.17 \\ Government spending & \(G_{f}\) & 0.21 \\ Packaging type & \(K_{f}\) & 0.02 \\ Advertising frequency & \(A_{f}\) & 0.31 \\ Store promotions & \(M_{f}\) & 0.41 \\ Supply-chain delay & \(D_{f}\) & 0.16 \\ Rainfall days & \(R_{f}\) & 0.08 \\ Temperature average & \(T_{f}\) & 0.22 \\ \hline \end{tabular}
\end{table}
Table 3: “Key Features and Their Importance Selected by RF-RFE”.  
* Geographical Dimension Contribution The geographical dimension contribution is defined as: \[F_{\text{geo}}(L_{f},P_{f})=w_{4}\log(1+P_{f})+w_{5}L_{f}\cdot\cos\left(\frac{2 \pi}{365}t\right)\] (2) The weights \(w_{4}\) and \(w_{5}\) control the contributions of population density effects and periodic regional geographical influences, respectively. Here, \(\log(1+P_{f})\) models the diminishing marginal impact of population density, while the term \(L_{f}\cdot\cos(\frac{2\pi}{365}t)\) reflects the periodic influence of regional geographical factors over time [24].
* Economic Dimension Contribution The contribution of the economic dimension is expressed as: \[F_{\text{econ}}(C_{f},E_{f},G_{f})=w_{6}C_{f}^{2}+w_{7}E_{f}+w_{8}G_{f}\cdot C _{f}\] (3) The weights \(w_{6},w_{7}\), and \(w_{8}\) control the contributions of consumer confidence, per capita consumption expenditure, and the interaction between government spending and consumer confidence, respectively. This equation captures key economic drivers, including consumer confidence (\(C_{f}\)), per capita consumption expenditure (\(E_{f}\)), and government spending in GDP (\(G_{f}\)). The interaction term \(G_{f}\cdot C_{f}\) reflects the combined effect of government policies and consumer sentiment.
* Product and Promotion Contribution The contribution of product dimension, advertising, and promotional activities is formulated as: \[F_{\text{product}}(K_{f},A_{f},M_{f},D_{f})=w_{9}\frac{A_{f}}{1+A_{f}}+w_{10}M _{f}\cdot\log(1+D_{f})+w_{11}K_{f}\] (4) The weights \(w_{9},w_{10}\), and \(w_{11}\) control the contributions of diminishing advertising returns, the delayed impact of promotions, and the influence of packaging type, respectively. This model accounts for the diminishing returns of advertising (\(\frac{A_{f}}{1+A_{f}}\)), the delayed impact of promotions through \(M_{f}\cdot\log(1+D_{f})\), and the influence of packaging type (\(K_{f}\)) on the target variable [21].
* Weather Dimension Contribution The weather dimension contribution is given by: \[F_{\text{weather}}(R_{f},T_{f})=w_{12}R_{f}\cdot T_{f}\] (5) The weight \(w_{12}\) regulates the combined impact of rainfall (\(R_{f}\)) and temperature (\(T_{f}\)) on the target variable. This formulation highlights the interaction between rainfall and temperature, capturing their joint influence on the target variable [25].