I'm currently working with some power law data of the form:
$Y_i = \beta \times X_i ^{-\gamma} $
Where $Y_i$ are my measurements at point $X_i$. The uncertainty on $X_i$ is vanishingly small and can be neglected. However, my $Y_i$ measurements are quite noisy.
I'm trying to understand what the best way of capturing the total uncertainty in my measurements is. Currently, I'm using the curve_fit module from scipy.optimize and extracting the uncertainty associated with the fitting procedure like so:
params, covariance = curve_fit(power_law_func, x, y)
std_fit = np.sqrt(np.diag(covariance))
Which returns a standard deviation for $\beta$ and $\gamma$. I'm also assessing uncertainty associated with the sampling of the data using a bootstrap approach, where I bootstrap the curve fit procedure $N$ times and get a standard deviation on $\beta$ and $\gamma$ from the N estimates.
Is it reasonable to assume that the standard deviations obtained from the curve fit and from the bootstrapping procedures are independent? And can I combine them through a simple addition? Or should I only be using one of these as my uncertainty for the parameters?
