Performing montecarlo simulation of regression coefficients

Steps:

1. Random Sampling: Randomly select 100 pairs of discharge (Q) and suspended sediment concentration (C) from the dataset. Convert these values to their logarithmic forms: log(Q) and log(C).
2. Compute the Slope (b) of the Regression Line:
   • Calculate the covariance between log(Q) and log(C).
   • Calculate the variance of log(Q).
   • Compute the slope as: $$b = \frac{\text{Cov}(\log Q, \log C)}{\text{Var}(\log Q)}$$
3. Compute the Intercept (logA) of the Regression Line:
   • Compute the mean of log(C) and log(Q).
   • Calculate the intercept using the equation: $$\log A = \text{Mean}(\log C) - (b \times \text{Mean}(\log Q))$$
4. Compute the Coefficient of Determination ($R^2$):
   • Compute the Pearson correlation coefficient $r$ between log(Q) and log(C): $$r = \frac{\text{Cov}(\log Q, \log C)}{\sigma_{\log Q} \cdot \sigma_{\log C}}$$
   • Compute $R^2$ as the square of the Pearson correlation coefficient: $$R^2 = r^2$$
5. Repeat Steps 1–4, 5000 Times:
   • Store the computed slope (b), intercept ($\log A$), and coefficient of determination ($R^2$) for each iteration.
6. Plot the Relationship:
   • Convert the stored intercept values back to the original scale: $$A = 10^{\log A}$$
   • Plot b (slope) vs. A (10$^\text{intercept}$) to analyze their relationship.

This version ensures mathematical correctness and clarity. Let me know if you'd like further refinements!