## 3 Methodology
### Feature Selection Method
\(C_{i}\) indicates \(i\) cluster, \(x\) is a normalized data point, and \(\mu_{i}\) denotes the mean vector within the \(i\) cluster. \(d(x,\mu_{i})\) denotes the distance associated with data point \(x\) from center of mass \(\mu_{i}\), which involves the previously mentioned Pearson correlation, and is formulated as follows:  
\[d(x,y)=1-\gamma \tag{8}\]  
Next, the contour coefficient is chosen as the evaluation criterion, with \(s(x)\) denoting the contour coefficient of data point \(x\), and the overall contour coefficient is the mean of all data points \(s(x)\) for assessing the clustering quality. The specific formula is as follows:  
\[s(x)=\frac{b(x)-a(x)}{\max\{a(x),b(x)\}} \tag{9}\]  
where \(a(x)=\frac{1}{|C_{i}|-1}\sum_{y\in C_{i},x\
eq y}d(x,y)\) denotes the average distance of data point \(x\) from other points in the same cluster, and \(b(x)=\min_{C_{1}\
eq C_{i}}\frac{1}{|C_{i}|}\sum_{y\in C_{k}}d(x,y)\) denotes the average distance from data point \(x\) to the nearest other cluster. Using k \(-\) means \(++\) initialization, initialized centers of mass are selected through a probability distribution to maximize the center of mass spacing, where the first center of mass is chosen randomly and the probability of subsequent center of mass selection is inversely proportional to the distance squared with the following formula:  
\[P(x)=\frac{D(x)^{2}}{\sum_{x\in X}D(x)^{2}} \tag{10}\]  
where \(D(x)\) denotes the minimum distance from the current data point to the selected center of mass and \(X\) denotes all data points in the data set. Step 5. Upsampling  
Interpolation, as a meaningful input transformation, is based on the basic principle of using mathematical methods to reasonably infer the values of unknown points between known data points, thereby transforming the data into a form more suitable for model processing. We use the interpolation method to convert monthly data into daily data. Although the statistical characteristics of interpolated data differ from real daily data, its advantages far outweigh the possible disadvantages. The most important aspect is the adaptability of the data. Increasing the amount of data is beneficial for deep learning modelsand machine learning models to achieve better learning results, while also alleviating the problem of small samples and reducing the risk of overfitting. On the other hand, linear interpolation not only maintains the overall monthly trend but also introduces small fluctuations to simulate natural changes in actual scenarios. To enhance data diversity and yield more accurate prediction results, quadratic interpolation is applied to the data, transforming the monthly data into daily data. Assuming the time point of monthly data is the first day of each month, the value is \(y_{0},y_{1},\ldots,y_{n-1}\); the corresponding time coordinate is \(x_{0},x_{1},\ldots,x_{n-1}\); and \(x_{0}\) indicates the first day of the first month, \(x_{1}\) indicates the first day of the second month, and so forth. Based on all data points \((x_{0},y_{0})\),\((x_{1},y_{1})\),\(\ldots(x_{n-1},y_{n-1})\), divided into multiple overlapping three-month windows, with each window containing three consecutive points, the kth window is \((x_{k},y_{k})\),\((x_{k+1}y_{k+1})\),\((x_{k+2},y_{k+2})\); and \(k=0,1,\ldots,n-3\). Construct a quadratic polynomial\(-y(x)=ax^{2}+bx+c\)--the formula is as follows:  
\[\left\{ \begin{array}{l}a_{k}x_{k}^{2}+b_{k}x_{k}+c_{k}=y_{k}\\ a_{k}x_{k+1}^{2}+b_{k}x_{k+1}+c_{k}=y_{k+1}\\ a_{k}x_{k+2}^{2}+b_{k}x_{k+2}+c_{k}=y_{k+2}\end{array} \right. \tag{11}\]  
Solve for the coefficient  
\[\left\{ \begin{array}{l}a_{k}=\frac{(y_{k+2}-y_{k+1})(x_{k+1}-x_{k})-(y_{k+1} -y_{k})(x_{k+2}-x_{k+1})}{(x_{k+2}-x_{k+1})(x_{k+1}-x_{k})(x_{k+2}-x_{k})}\\ b_{k}=\frac{(y_{k+1}-y_{k})-a(x_{k+1}-x_{k}^{2})}{x_{k+1}-x_{k}}\\ c_{k}=y_{k}-a_{k}x_{k}^{2}-b_{k}x_{k}\end{array} \right.