I have a relatively straightforward problem. I have a value that gets multiplied by a bunch of correction factors to give a final answer.
$Final = Base * CF_1 * CF_2 * CF_3 * CF_4$
I am trying to re-create (approximate) this calculation with data from a different source. That will obviously mean that the values won't be perfect. Assuming I have the original values and the values from this separate source, I would like to somehow attribute how much error in the Final value can be attributed to each source (CF1, CF2, etc...). I will use a subscript T for the true values and A for the approximations.
The way I am currently doing this is by subbing in one approximation and then finding the difference. I do this for every term to find the "error" that can be attributed to that term assuming everything else was perfect.
$Error_{CF1} = Base_T * CF_{1,A} * CF_{2,T} * CF_{3,T} * CF_{4,T} - Final_T$
The problem is that all of my individual errors don't add up to the overall error. I believe this is because it is a non-linear problem with the product, but I can't figure out how to do this sort of error attribution and I have no idea what sort of terms to use to lookup the correct answer.
I know how to propagate uncertainties in this sort of situation but I don't think that applies since there isn't really a set uncertainty, the correction factors change for each sample point and that difference isn't constant.
https://www.geol.lsu.edu/jlorenzo/geophysics/uncertainties/Uncertaintiespart2.html
Any help would be appreciated.
If $z = xy$, are the true values while the approximate values are $z + \Delta z, x + \Delta x, y + \Delta y$, then $$\begin{align}z + \Delta z &= (x+\Delta x)(y+\Delta y) = xy + y\Delta x + x\Delta y + \Delta x\Delta y\\ \Delta z &= y\Delta x + x\Delta y + \Delta x\Delta y\end{align}$$
If $\Delta x, \Delta y$ are significantly smaller than $x, y$ themselves, the contribution from $\Delta x\Delta y$ is insignificant and can be dropped. Then we can also divide both sides by $z = xy$ to get $$\frac{\Delta z}z = \frac{\Delta x}x + \frac{\Delta y}y$$ That is, the relative error in a product is (to first order approximation) the sum of the relative errors in the multiplicands.
Multiplying back thru by $z$: $$\Delta z = \frac zx\Delta x + \frac zy\Delta y$$ I.e., the portion of the error in $z$ that comes from the error in $x$ is $\frac zx\Delta x$, and similarly for $y$.
This will hold true no matter how many factors are multiplied to get $z$: The contribution to the error from any of the factors $x$ will be $\frac zx\Delta x$, to first order approximation. Higher order contributions (those coming from the dropped $\Delta x\Delta y$ terms) still mix the contributions to error from multiple variables, but as long as the relative error is much smaller than $1$, these mixed contributions are small enough to be ignored.