Error propagation with asymmetric uncertainties

I red the Wikipedia page on error propagation. I get into some trouble when I want to calculate the error propagation in a specific case. I have real numbers $A$ and $B$ with related upper ($\Delta A_u$, $\Delta B_u$) and lower ($\Delta A_l$, $\Delta B_l$) uncertainties. I have two functions calculated as follows: $$F = \sqrt{A^2+B^2}$$ and $$G = \tan^{-1}\frac{B}{A}$$ What are the related upper and lower errors of $F$ and $G$?

I found general formulae for Gaussian error distributions ($\Delta A_u = \Delta A_l$ and $\Delta B_u = \Delta B_l$), for instance the first formula of the first answer on this other question. Is there a way to generalize that formula for arbitrary distributions with asymmetric errors as in my example?

• Why are your uncertainties asymmetric? Asymmetry implies that they are not altogether 'uncertain'. – sammy gerbil Jul 5 '17 at 19:36

Here are some thoughts about this question - but not a definitive answer: I don't believe a definitive answer exists.

In general, error propagation is founded on the assumption that the distribution of the errors is Gaussian, and that the error is small compared to the value of the quantity. In that case, a simple propagation of errors is possible.

For example, assuming that the error is small, you should be able to take the derivative of your function with respect to each of the variables - then you can use that to determine the error in the result.

For example, your case of

$$F =\sqrt{A^2+B^2}$$

Derivative with respect to A, B:

$$\frac{\partial F}{\partial A} = \frac{2A}{\sqrt{A^2+B^2}}\\ \frac{\partial F}{\partial B} = \frac{2B}{\sqrt{A^2+B^2}}$$

If the error in A is $\Delta A$, and the error in B is $\Delta B$, then the total expected error is the sum of squares of errors:

$$\Delta F = \frac{\sqrt{(2A\Delta A)^2 + (2B\Delta B)^2}}{\sqrt{A^2+B^2}}$$

However, the moment you state that your distribution is not symmetrical, the situation changes. If you have a sufficient number of variables with "small but non Gaussian" error distributions, the central limit theorem tells us that the result will nonetheless be Gaussian distributed: in that case you can compute the standard deviation of the (non Gaussian) individual distributions, and use those as a surrogate in your error propagation calculation. But if you have a small number of variables (like in your example), AND the distribution is not Gaussian, then there is no method I'm aware of to solve the question analytically. It can, however, be addressed with a simple Monte Carlo simulation.

In a Monte Carlo simulation, you sample the distributions of your input variables, and transform them according to the formula; you can then plot the resulting output distribution, and compute its shape etc.

The upper and lower limits of the output can in principle be computed by setting the input variables to their extreme values (this is sometimes done for "worst case analysis"); but it is rare that that gives you any really useful insights, since error distributions most often describe something stochastic rather than deterministic (which means that an upper limit is almost never "hard"). And as I said - the moment you have more than a small number of variables with similar weights, the output distribution will start to look Gaussian.

I would use the total derivative. For example, let's take a fct $F(x,y)$, which depends on two variables. Then the total derivative is $$dF = \frac{\partial F}{\partial x} dx + \frac{\partial F}{\partial y} dy$$ as descriptor here.