If a growth function is doubling every x day it can be written as
$f(y) = f_0 * 2^{t/x}$
I am looking at a 7 day moving window of a measurement $y$ having exponential growth $f(y)=e^{\alpha t +\beta}$, fitting a linear curve to $ln(f(y))$ and taking the slope as an estimated $\hat{\alpha}$ of that 7 day window.
Knowing the estimated $\alpha$, what is the correct transformation to get an estimated doubling rate of this function $\hat{x}$?
When comparing to the curve it seems that $\hat{x}= \frac{ln(2)}{\hat{\alpha}}$ seems to be the correct formula, but I am not sure if this is true. How do I transform $\hat{\alpha}$ correctly to the doubling rate?
Thanks!
To double you go from $e^{\alpha t+\beta}$ to $e^{\alpha (t+x)+\beta}$, so we need $e^{\alpha x} = 2$. Taking logs, and rearranging gives $x=ln(2) / \alpha$ as you observed.