Uncertainty Propagation#
We follow the typical fundamental equations for the propagation of uncertainty for simple operations. We describe the methods for each operation as follows. The calculations below propagate uncertainty via variances, though we adapt the methods below for uncertainty propagation via standard deviations.
The variables with some provided uncertainty are \(A\) and \(B\) with their corresponding standard deviation uncertainty \(\sigma_A\) and \(\sigma\). We assume the covariance term for all of our operations are zero, so, for our purposes, \(\sigma_{AB} \triangleq 0\); however, for the sake of clarity, we leave this term in our equations below.
Addition and Subtraction#
The typical equations for the propagation of variance uncertainty for addition and subtraction are:
Multiplication and Division#
The typical equations for the propagation of variance uncertainty for multiplication and division are:
However, these formula are not very handy so we adapt it to remove possible division by zeros. This results in the following equations of the variance. The methodology is similar to Astropy.
For the case of multiplication without any variance (i.e. \(a\) and \(\sigma_a = 0\)), it simplifies to:
Exponentiation#
The typical equation for the propagation of variance uncertainty for exponentiation is:
Logarithms#
The typical equation for the propagation of variance uncertainty for exponentiation is:
Note, we assume that the logarithm base \(b\) is exact in this case, and does not have any associated uncertainty to its value.
Discrete Integration#
[[TODO]].
Weighted Mean#
A weighted mean \(f\) is defined by weights \(w_i\) and values \(A_i\). The weights need to be normalized, thus \(\sum w_i = 1\). The typical equation for the propagation of variance uncertainty for a weighted mean is:
Median#
[[TODO]].