## 2 Theory and Methodology
### MissForest Filling and Correction method  
The MissForest imputation method [29], based on the random forest algorithm, is introduced as a non-parametric approach for missing value imputation. Its key advantage lies in its low dependency on data distribution assumptions, making it suitable for handling mixed datasets containing both discrete and continuous variables. The core principle involves training a random forest predictive model using the complete portion of the dataset, then iteratively predicting and optimizing missing values until the imputed results converge. The hypothesis \(M=\{M_{1},M_{2},\cdots,M_{n}\}\) is a matrix of \(n\times p\) dimensions with missing values to be filled, \(M_{\text{s}}\) presents any variable containing missing values, Samples with missing values in the variable Ms are denoted as \(\langle\text{$\text{$s$}}\rangle_{\text{mis}}\subseteq\{1,2,\cdots,n\}\), while samples with observed values \(M_{\text{s}}\) are denoted as \(\langle\text{$\text{$i$}}\rangle_{\text{obs}}\subseteq\{1,2,\cdots,n\}\): \(\langle\text{$\text{$s$}}\rangle_{\text{obs}}\) represents the \(M_{\text{s}}\) observed value, \(\langle\text{$\text{$s$}}\rangle_{\text{mis}}\) represents the \(M_{\text{s}}\) missing value, \(\langle\text{$\text{$s$}}\rangle_{\text{obs}}\) represents \(M_{\text{s}}\) other observed values, and \(\langle\text{$\text{$s$}}\rangle_{\text{mis}}\) represents \(M_{\text{s}}\) other missing values. The implementation workflow of the algorithm is as follows:  
Preprocessing Phase: Perform initial imputation on missing values \(M\) using methods such as mean filling or linear interpolation.  
Variable Sorting: Sort variables in ascending order based on \(M_{\text{s}}\) their missing rate to generate the initial imputation matrix \(M_{\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$ $\text{$\text{$\text{$\text{$\text{$\text{$}}$}$}$}$}}}}}}$}}^{(s)}}^{(s)}\).  
Model Training: For each target variable \(M_{\text{s}}\), \(\langle\text{$\text{$s$}}\rangle_{\text{obs}}^{(s)}\) treat as input variables and \(\langle\text{$\text{$y$}}\rangle_{\text{mis}}^{(s)}\) as output variables, then train a random forest prediction model.  
Prediction and Imputation: Input \(\langle\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$ \text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$}}}$}$}$}$}}}}}$}}$}$} \rangle_{\text{mis}}\) into the trained model to obtain predicted missing values \(\langle\text{$\text{$\text{$y$}}\rangle_{\text{mis}}^{(s)}$}\), updating the imputation matrix as \(M_{\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{\text{$\text{$ \text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$}}}$}$}$}$}}}}}}$}} }^{(s)}}^{(s)}\).  
Iterative Optimization: Repeat steps 3-4 until convergence is met. The final imputed matrix is stored as \(M_{\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$ \text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$}}}$}$}$}$}}}}}}$}} }^{(m)}}^{(m)}\), while the previous version \(M_{\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{\text{$\text{$ \text{$\text{$\text{$\text{$\text{$\text{$$\text{$$\text{$\text{$ $}}}$}$}$}}}}}}}$}}$}}^{(m)}\) as \(M_{\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$\text{$ \text{\text{$\text{$\text{$\text{$$\text{$$\text{$$$\text{$$\text{$$ $}}$}$}$}}}}}}}$}}}$}}^{(m)}\) is retained.  
Figure 1: Algorithm flow chart.  
Convergence Criterion: Use the difference \(\Delta N\) between two consecutive iterations as the metric. Terminate the process when the difference \(\Delta N\) first increases, preserving the optimal imputation result.