## 3 Materials and Method
### Data Preprocessing and Augmentation  
In the framework of DRL, data preprocessing and augmentation constitute critical steps to ensure model stability and accuracy in complex market environments. In practical applications, economic behavior data often suffer from missing values and outliers. Furthermore, due to delayed market feedback, handling lagged data and enhancing data diversity have emerged as core challenges in preprocessing. To address these issues, several effective preprocessing and augmentation techniques have been proposed to improve learning efficiency and model robustness. Initially, the imputation of missing values and treatment of outliers are essential preprocessing procedures. In economic behavior datasets--particularly transaction logs and market feedback--missing or anomalous data are frequently encountered. Missing values typically arise from incomplete user actions or data collection errors, while outliers refer to values that significantly deviate from expected ranges, such as sudden spikes in product sales or abrupt price fluctuations. Common imputation methods include mean imputation, interpolation, and prediction-based filling using other features. In this work, a \(k\)-nearest neighbors (KNN)-based imputation approach is adopted. The method estimates missing values by identifying and averaging the \(k\) most similar instances. Given a dataset \(D=(x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\), where \(x_{i}\) denotes the feature vectors and \(y_{i}\) represents the target variables, missing features \(x_{m}\) can be filled as follows:  
\[\hat{y}_{m}=\frac{1}{k}\sum i=1^{k}y_{i}, \tag{12}\]  
where \(k\) denotes the number of nearest neighbors, \(y_{i}\) are the target values of the \(k\) most similar data points, and \(\hat{y}_{m}\) is the imputed target. Outliers are identified and corrected using standard deviation thresholds or box plot-based statistical techniques, ensuring the rationality and consistency of the data. After addressing missing and anomalous values, the next step involves constructing state representations from the triplets of price, sales, and inventory. To enable DRL models to interpret economic behavior data, these variables must be transformed into state spaces amenable to model learning. While traditional pricing models often rely solely on price and sales data, inventory levels and competitive dynamics also critically influence pricing decisions. Accordingly, a multidimensional state representation is constructed by integrating the price (\(p_{t}\)), sales (\(q_{t}\)), and inventory (\(I_{t}\)), forming the state vector \(s_{t}=(p_{t},q_{t},I_{t})\). These state variables provide richer information about market dynamics and allow more informed optimization of pricing strategies. To further enhance the representational quality, weighted transformations of the price, sales, and inventory are introduced. The refined state representation is defined as  
\[s_{t}=\bigg{(}p_{t},\frac{q_{t}}{I_{t}+1},\log(I_{t}+1)\bigg{)}, \tag{13}\]where \(p_{t}\) represents the price at time \(t\), \(\frac{q_{t}}{t+1}\) denotes the sales-to-inventory ratio, and \(\log(I_{t}+1)\) captures the nonlinear influence of inventory levels. This formulation enables the model to better grasp the interdependencies among pricing, demand, and stock, thereby facilitating more precise pricing decisions. Subsequently, a price perturbation mechanism based on a normal distribution is introduced to simulate variations in market response. In real-world markets, price fluctuations often trigger nonlinear and delayed changes in sales. To emulate such feedback behavior, a stochastic price perturbation strategy is adopted. Let \(\Delta p_{t}\) denote the price perturbation at time \(t\), sampled from a normal distribution:  
\[\Delta p_{t}\sim\mathcal{N}(0,\sigma^{2}), \tag{14}\]  
where \(\sigma\) represents the standard deviation of the price perturbation.