1.2.5. Preventing “hidden” covariances

For the following, it is assumed now that the arithmetic expression for the output quantity \(y\) containing several expressions, e.g., \({\ a}_{1},\ a_{2},a_{3}\), each of which being functions of input quantities \(x_{i}\). Often, the uncertainties \({u(a}_{1}),\ \ u(a_{2}),{\ u(a}_{3})\) are calculated first from which then \(u(y\left( {\ a}_{1},\ a_{2},a_{3} \right))\) is derived. If, however, there are some of the input quantities \(x_{i}\), contained in more than one of the expressions \({\ a}_{1},\ a_{2},a_{3}\), then “hidden” or “overlooked” covariances exist between some of the \({\ a}_{1},\ a_{2},a_{3}\), which would have to be considered afterwards.

This problem does not occur during the uncertainty calculations within UncertRadio, because there the partial derivatives in its uncertainty propagation are always build from the equation of the output quantity. It is shown below, why this avoids the above problem.

The ansatz for the uncertainty propagation with partial derivatives which refer to the output quantity \(y = y({\ a}_{1}(\mathbf{x}),\ a_{2}(\mathbf{x}),a_{3}(\mathbf{x}))\), is formulated as follows with the vector \(\mathbf{x}\) of input quantities:

At first, the square within the sum is evaluated:

\[u^{2}(y) = \sum_{i = 1}^{ni}{\left( \frac{\partial y}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right)}\]

\[u^{2}(y) = \sum_{i = 1}^{ni}{\left( \sum_{j = 1}^{3}{\frac{\partial y}{\partial a_{j}}\frac{\partial a_{j}}{\partial x_{i}}} \right)^{2}u^{2}\left( x_{i} \right)}\]

\[\sum_{i = 1}^{ni}{\left( \frac{\partial y}{\partial a_{1}}\frac{\partial a_{1}}{\partial x_{i}} + \frac{\partial y}{\partial a_{2}}\frac{\partial a_{2}}{\partial x_{i}} + \frac{\partial y}{\partial a_{3}}\frac{\partial a_{3}}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right)}\]

Now, the summation over \(i\) is performed for each of the six terms, while at the same time the partial derivatives of \(y\) by \(a_{j}\) are factored out of the sums:

\begin{eqnarray} u^{2}(y) = \sum_{i = 1}^{n_i} & & \left( \frac{\partial y}{\partial a_{1}}\frac{\partial a_{1}}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right) + \\ && \left( \frac{\partial y}{\partial a_{2}}\frac{\partial a_{2}}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right) + \\ && \left( \frac{\partial y}{\partial a_{3}}\frac{\partial a_{3}}{\partial x_{i}} \right)^{2}u^{2}\left( x_{i} \right) + \\ && 2\left( \frac{\partial y}{\partial a_{1}}\frac{\partial a_{1}}{\partial x_{i}} \right)\left( \frac{\partial y}{\partial a_{2}}\frac{\partial a_{2}}{\partial x_{i}} \right)u^{2}\left( x_{i} \right) + \\ && 2\left( \frac{\partial y}{\partial a_{1}}\frac{\partial a_{1}}{\partial x_{i}} \right)\left( \frac{\partial y}{\partial a_{3}}\frac{\partial a_{3}}{\partial x_{i}} \right)u^{2}\left( x_{i} \right) + \\ && 2\left( \frac{\partial y}{\partial a_{2}}\frac{\partial a_{2}}{\partial x_{i}} \right)\left( \frac{\partial y}{\partial a_{3}}\frac{\partial a_{3}}{\partial x_{i}} \right)u^{2}\left( x_{i} \right) \end{eqnarray}

Now, each individual sum over \(i\) is representing a variance or a covariance of the expressions \({\ a}_{1},\ a_{2},a_{3}\):

\[\begin{split}u^{2}(y) = &\left( \frac{\partial y}{\partial a_{1}} \right)^{2}u^{2}\left( a_{1} \right) + \left( \frac{\partial y}{\partial a_{2}} \right)^{2}u^{2}\left( a_{2} \right) + \left( \frac{\partial y}{\partial a_{3}} \right)^{2}u^{2}\left( a_{3} \right) + \\ &2\frac{\partial y}{\partial a_{1}}\frac{\partial y}{\partial a_{2}} cov(a_{1},a_{2}) + 2\frac{\partial y}{\partial a_{1}}\frac{\partial y}{\partial a_{3}}cov(a_{1},a_{3}) + \\ & 2\frac{\partial y}{\partial a_{2}}\frac{\partial y}{\partial a_{3}}cov(a_{2},a_{3})\end{split}\]

Usually, a “hand-made” uncertainty propagation by first applying a decomposition of \(y\) into expressions or functions \(a_{j}\), only the first three terms in Eq. (6) are used, because covariances between the \(a_{j}\) often are not expected; this may explain the term “hidden” covariances.

The result of Eq. (6) is just the one which has to be expected when “hidden” covariances between the \(a_{j}\) are explicitly taken into account. This demonstrates that these covariances are considered by UncertRadio, automatically, only because it uses within its uncertainty evaluation according to Eq. (1), partial derivatives directly of the output quantity.