Linear Error Propagation

Like the uncertainties and Measurements.jl libraries, Sigma is based on the theory of linear error propagation. Here we will provide a brief summary of that theory.

Assume a sequence of \( n \) variables \( \left(a_{i}\right)_{i=1}^{n} \) whose elements are defined as \( a_{i} = \bar{a}_{i} \pm \sigma_{a_{i}} \), where \( \bar{a}_{i} \) is the mean value of \( a_{i} \) and \( \sigma_{a_{i}} \) is the standard deviation of \( a_{i} \). \( F(A) \) is a function of \( A \), where \( A = \{a_{i} : i \in \mathbb{N}, 1 \leq i \leq n\} \). The uncertainty of \( F(A) \) can be determined as

\[ \normalsize \sigma_{F} = \sqrt{ \sum_{i=1}^{n} \left( \left( \left. \frac{\partial F}{\partial a_{i}} \right|_{a_{i}=\bar{a}_{i}} \sigma_{a_{i}} \right)^2 + 2 \sum_{j=i+1}^{n} \left( \left(\frac{\partial F}{\partial a_{i}}\right)_{a_{i}=\bar{a}_{i}} \left(\frac{\partial F}{\partial a_{j}}\right)_{a_{j}=\bar{a}_{j}} \sigma_{a_{i}a_{j}} \right) \right) } \]

where \( \sigma_{a_{i}a_{j}} \) is the covariance of \( a_{i} \) and \( a_{j} \). These covariances can be eliminated from the above equation if the uncertainties of the variables are independent from one another, which is a requirement imposed here. As such, the uncertainty of \( F(A) \) when the members of \( A \) are independent from one another is

\[ \normalsize \sigma_{F} = \sqrt{ \sum_{i=1}^{n} \left( \left. \frac{\partial F}{\partial a_{i}} \right|_{a_{i}=\bar{a}_{i}} \sigma_{a_{i}} \right)^2 } \]

Next, we consider a set \( B = \{x, y\} \) where \( x = a_{i} + a_{j} \) and \( y = a_{j} - a_{k} \), i.e. the elements of \( B \) are dependent on a subset of the independent variables in set \( A \). Given the function \( G(B) \), the uncertainty of \( G \) can be determined by application of the chain rule to relate the independent variables to \( G \) through their relationships with the dependent variables

\[ \normalsize \sigma_{G} = \sqrt{ \left( \left( \frac{\partial G}{\partial x} \frac{\partial x}{\partial a_{i}} \right)_{a_{i}=\bar{a}_{i}} \sigma_{a_{i}} \right)^2 + \left( \left( \frac{\partial G}{\partial x} \frac{\partial x}{\partial a_{j}} + \frac{\partial G}{\partial y} \frac{\partial y}{\partial a_{j}} \right)_{a_{j}=\bar{a}_{j}} \sigma_{a_{j}} \right)^2 + \left( \left( \frac{\partial G}{\partial y} \frac{\partial y}{\partial a_{k}} \right)_{a_{k}=\bar{a}_{k}} \sigma_{a_{k}} \right)^2 } \]