I'm a biology masters student working on a simple practical problem about some of the genetic tools we use. I'm very much a math novice, so please feel free to correct my notation/etc!
I'm working with some error propagation. I have a measurement of interest, which I'll call Z. It looks kinda like this:
$$ Z = f(x) = E - A \cdot \ln\left(B \cdot \frac{C - x}{x - D}\right)$$ Where ${A, B, C, D, E}$ are all treated as constants.
I'm interested in how an error in $x$ propagates into an error in $Z$.
I defined that error in $Z$ as:
$$Error_{Z}= h(g(x,y), f(x)) = \bigg[E - A\cdot\ln\left(B\cdot \frac{C - g(x,y)}{y-g(x,y)}\right)\bigg] - f(x)$$
Where $g(x, y)$ takes some $x$ and a level of relative error $y$ and returns an $x \pm (xy)$. Or, more specifically:
$$ x \pm Error_x = g(x, y) = x(1 \pm y)$$
So there are two different values of $Error_x$-- the $(1+y)$ version and the $(1-y)$ version (those just come from empirical measurements of how much $x$ varies). I'll plot the propagation of both of those on a $Error_Z$ by $Z$ graph. They'll look something like this:
(Where the purple is the |+error| and the black is the |-error|. In this case, I take the absolute value because I'm only interested in the absolute deviation of $Z$ from its true value--the direction is biologically irrelevant. I made this graph using a bunch of generated x's between C and D and a constant $y$ of 0.025. As I change $y$, the plot--and its minimum--change.)
I'm interested in the $Z$ where $Error_Z$ is minimum (the minimum of the graph above, where the black and purple curves hit).
I've noticed that, as $y \to \infty$ (or, practically, as $y \to 1$, at values around .5/50% error) that $Z$ is $E - A \cdot \ln(B)$.
That makes sense since the derivative of Z with respect to x $\frac{\partial Z}{\partial x}$ is at it's minimum at a Z of $E - A \cdot \ln(B)$--it's just that, since the absolute amount of $Error_x$ increases as $x$ increases (and it's that absolute amount that I'd plug into the derivative equation), that ideal doesn't work.
Anyways, does anyone know how to find the point $Z$ where $Error_Z$ is minimum, given some relative error in $x$ (defined as $y \in {\rm \Bbb R}$ and $y > 0$)?
Please let me know if you have any questions, or if there's anyway I can make this more clear.