For a given function, for example f(x) = x * y I know the uncertainties of the variables x and y. For the sake of this question, we can assume x +/- 0.1 and y +/- 0.3.

I want to calculate the uncertainty of f(x).

I have seen that the formula to calculate the error propagation of a multiplication operation uses the relative error of the variables, as it is shown in the following image:

Formula of Error Propagation in Multiplication

The problem is the value of the variable x belongs to an interval [0 - 255] and y is a fixed value. The relative error of y it is easy to calculate but I do not know the approach to calculate the relative error of the variable x to be used in the formula.

Is there a way to calculate the uncertainty of f() without the relative error of x? Can I only use the uncertainties like in the addition formulas of the error propagation?

I have never worked with this kind of analysis, all that I know was self taught, so any detail given to me will be very appreciated.


1 Answer 1


Unfortunately the best answer here depends a bit on what exactly you're doing. One straightforward answer without a bunch of subtle assumptions is to use interval arithmetic. Assuming everything's positive, the interval arithmetic formulation of $xy$ is to multiply the two smallest possibilities and the two largest possibilities. So if you had $(x_m,x_M),(y_m,y_M)$ then your interval would become $(x_m y_m,x_M y_M)$. This is just because multiplication of positive numbers is monotone.

A serious defect of the interval arithmetic approach is that in real life, random errors tend to cancel one another out, but in interval arithmetic everything gets worse and worse as you do more operations with uncertain quantities. A system that avoids this is the one usually taught in physics. The mathematical basis for this is assuming that every quantity is Gaussian with a standard deviation much smaller than its mean, and each one is independent of all the others. You then deal with nonlinear functions like multiplication through a linearization trick. The mathematical basis for this linearization trick is exactly this assumption about the standard deviation vs. the mean of all your quantities.

The result of this system for multiplication is the rule in your link, which says that relative errors add "Euclidean style" i.e. you sum the squares of the relative errors and take the square root to get the final relative error. Even if you want to convert to absolute errors, the dependence on $x,y$ will not disappear:

$$xy \sqrt{\left ( \frac{\sigma_x}{x} \right )^2 + \left ( \frac{\sigma_y}{y} \right )^2}=\sqrt{y^2 \sigma_x^2 + x^2 \sigma_y^2}.$$

This shouldn't really be a surprise: the effect of $\sigma_x$ on the uncertainty of the whole quantity should be bigger if $y$ is bigger. (If that's not obvious to you, think about the case where $y$ is certain and only $x$ is uncertain.)


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