Basic Error Propagation¶

Let's have a function \(f\) of variables \(x_i\), which can be well linearised around some point \(f_0\):

\[ \begin{equation} f\approx f_0 + \sum_i \frac{\partial f}{\partial x_i} \end{equation} \]

then we can write the variance \(\sigma_f^2\) of \(f\) as follows:

\[ \begin{equation} \sigma_f^2 = J\Sigma J^T = \left(\begin{matrix} \frac{\partial f}{\partial x_1}&\frac{\partial f}{\partial x_2}&\cdots\end{matrix}\right)\cdot \left(\begin{matrix} \sigma_1^2&\sigma_{12}&\cdots\\ \sigma_{21}&\sigma_2^2&\cdots\\ \vdots& \vdots& \ddots \end{matrix}\right) \left(\begin{matrix} \frac{\partial f}{\partial x_1}\\\frac{\partial f}{\partial x_2}\\\vdots \end{matrix}\right) , \end{equation} \]

where \(J\) is the Jacobian of \(f\) and the \(\sigma_i\) are the uncertainties of variables \(x_i\), while \(\sigma_{ij}\) are the covariances between the variables. The covariance matrix \(\Sigma\) you can get, for instance, from a fitting function. We often assume that the variables are not correlated, i.e. covariance matrix has non-zero values only on the diagonal. In such a case, we get the famous formula:

\[ \begin{equation} \sigma_f^2=\sum_i \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_i^2 \end{equation} \]

However, sometimes we should not neglect the covariances. For details, see the more extensive Wikipedia article about propagation of uncertainty or your favourite statistics textbook.