## 2 Materials and Methods
### Linear Regression Model with Noise  
The expansion rate downstream of an earth dam can be modeled using a linear regression approach that incorporates deterministic and stochastic components. The model is expressed in Equation (2). The equation suggests that the expansion rate increases proportionally with the amount of rainfall, with some additional random variation (noise) to account for other influencing factors in the expansion rate downstream of an earth dam. These parameters include both rainfall data and soil properties. \[\text{Expansion Rate}=0.052\times\text{Rainfall Event}+\text{Noise}_{ sand}+\text{Noise}_{silt}+\text{Noise}_{clay}+\text{Noise}_{\text{organic matter}}\] (2) where \[\text{Noise}_{sand}=N\big{(}\mu_{sand},\sigma_{sand}^{2}\big{)}\] \[\text{Noise}_{silt}=N\big{(}\mu_{silt},\sigma_{silt}^{2}\big{)}\] \[\text{Noise}_{clay}=N\Big{(}\mu_{clay},\sigma_{clay}^{2}\big{)}\] \[\text{Noise}_{\text{organic matter}}=N\Big{(}\mu_{\text{organic matter}},\sigma_{\text{organic matter}}^{2}\Big{)}\]  
The coefficient 0.052 represents the proportionality between rainfall and expansion rate. This value was derived from a statistical analysis of historical data on rainfall events and corresponding expansion rates observed at the earth dam site. Specifically, we conducted a linear regression analysis on a dataset comprising 132 observations from the earth damsfrom 2010 to 2020. The coefficient of 0.052 was found to be statistically significant (\(p<0.001\)) and represents the average increase in expansion rate (in appropriate units) for each unit increase in rainfall. It is important to note that while this coefficient provides a general relationship between rainfall and expansion rate, its applicability may vary depending on specific site conditions. Engineers should consider local factors and potentially adjust this coefficient based on site-specific data. The noise captures the unobserved heterogeneity in soil types and other factors influencing the expansion rate. This stochastic component accounts for variations in soil properties, such as clay content and other geotechnical characteristics that may affect the soil's response to rainfall. By modeling these influences as random noise, we acknowledge their impact on the expansion rate while maintaining model parsimony. The choice of a normal distribution for the noise (Noise \(\sim\) N(\(\mu\), \(\sigma^{2}\))) is a standard assumption in linear regression analysis. It implies that the deviations from the linear relationship between rainfall and expansion rate are symmetrically distributed around zero, with more minor deviations being more common than larger ones. This assumption allows for statistical inference, such as hypothesis testing and confidence interval estimation, to assess the significance of the rainfall effect on the expansion rate. It is important to note that while this model provides a simple and intuitive representation of the relationship between rainfall, geotechnical properties' influence, and expansion rate, it assumes linearity and homoscedasticity (constant variance of errors). In practice, these assumptions should be validated through residual analysis. If violated, more complex models, such as nonlinear regression or heteroscedastic-robust techniques, may be warranted to better capture the underlying soil dynamics. Adjusted R-Square  
In linear regression analysis, the coefficient of determination (R\({}^{2}\)) is a widely used metric to assess model fit, quantifying the proportion of variance in the dependent variable explained by the independent variables. However, R\({}^{2}\) has a notable limitation: it invariably increases with adding predictors, even when these do not substantially improve the model's explanatory power. This phenomenon, known as overfitting, can lead to inflated R\({}^{2}\) values that misrepresent the model's true predictive capability. To mitigate this issue, statisticians employ the adjusted R-squared (R\({}^{2}\)a), a modified version of R\({}^{2}\) that penalizes model complexity [12].