## 2 Materials and Methods
### Imputation Methods  
#### 2.2.1 Ratio Method (R)  
The ratio method as proposed by [PERSON] and [PERSON] [37] relies on the assumption that the relationship between two variables can be expressed as a proportion. If two variables, \(X\) and \(Y\), are linearly related, with \(Y=k\times X\), where \(k\) is a constant, the ratio method estimates the missing value of \(Y\) by using the known value of \(X\) and the constant. The value of \(k\) can be derived from the observed data where both \(X\) and \(Y\) are present (Equation (1))  
\[k=\frac{\sum_{i=1}^{n}y_{i}}{\sum_{i=1}^{n}x_{i}} \tag{1}\]  
where  
\(n\)--period length;  
\(x_{i}\), \(y_{i}\)--paired values  
The ratio method offers several compelling advantages that make it a valuable approach to data imputation. One of its primary strengths is its ability to preserve the proportional relationships between variables, ensuring that the imputed values are consistent with the underlying data structure. This feature is particularly crucial in datasets where maintaining variable relationships is essential for accurate analysis. Additionally, the method's simplicity and computational efficiency make it accessible and practical for researchers, especially in resource-constrained environments or when dealing with large datasets. Unlike more complex techniques, the ratio method requires minimal computational resources while still delivering reliable results. Despite its advantages, the ratio method faces several limitations and challenges that can impact its effectiveness in certain contexts. One major limitation is its reliance on the assumption of a stable proportional relationship between variables, which may not hold true in all datasets, especially those with complex or nonlinear dynamics. This dependence on proportionality restricts its applicability to scenarios where such relationships are evi  
Figure 4: Frequency of gaps between years and months for different scenarios using uniform distribution. dent. Additionally, the method is highly sensitive to outliers, as extreme values can distort the calculated ratio and lead to inaccurate imputations. The ratio method is unsuitable for datasets where no strong relationships exist between the variables of interest, limiting its scope. Although traditionally used for gap imputation in hydrological data, its inability to account for noise makes it a suboptimal model for imputation purposes. In this study, we employ a self-imputation approach, raising the question of which data series should serve as the predictor. To address this, we created an adapted version of the ratio method, wherein pairwise correlations were calculated for each row and column (_A,B_) based on non-missing values. These correlation values were weighted by the proportion of usable value pairs (\(n_{A,B}\)) relative to the total length of the row or column (\(N\)), giving priority to scenarios with longer, gap-free data pairs (Equation (2)). \[w_{A,B}=\frac{n_{A,B}}{N}\qquad wCor(A,B)=w_{A,B}\cdot Cor(A,B) \tag{2}\]  
By considering the correlations between different column pairs, we tested for variable lag values measurable across months, while correlations between different row pairs allowed for the testing of lag values measurable across years. Thus, for imputing missing values in any given row or column, the predictor role is assigned to the row or column with the highest weighted correlation value (\(r^{*}\) or \(l^{*}\)), determined by the amount of usable (non-missing) data (Equations (3)-(5)).