wip(thesis): work on preliminaries

src/thesis/Thesis_408230.tex

@@ -226,6 +226,15 @@
     in future applications of neural networks in their field
     \item that's why we consider only continuously differentiable activation functions
     such as parametric softplus and \textit{QuadLU}
+    \item for the sake of simplicity we do not consider the procedure to determine
+    the inputs' uncertainties but assume that all measures were taken to express it in
+    alignment with the GUM and its understanding
+    \item moreover we usually will conceive the uncertainty either given as an
+    expanded uncertainty with an appropriately large coverage factor or as stated
+    in~\cite[4.3.7]{jcgm_guide_2020}
+    \item for the latter cases, when the uncertainty is given in form of an
+    interval $[x_i - a, x_i + a]$ the associated variance $u^2(x_i)$ would be given
+    as $u^2(x_i) = a^2/3$
   \end{itemize}
 
 

src/thesis/preliminaries.tex

@@ -10,8 +10,8 @@ regarding those definitions.
 \section{Terms related to measurement uncertainty}\label{sec:terms-related-to
 -measurement-uncertainty}
 
-\begin{definition}
-  [uncertainty]\label{def:uncertainty}
+\begin{definition}[uncertainty]
+  \label{def:uncertainty}
   (or \enquote*{measurement uncertainty} or \enquote*{uncertainty of measurement})
   (non-negative) parameter, associated with the result of a measurement, that
   characterizes the dispersion of the values that could reasonably be attributed to
@@ -34,6 +34,10 @@ regarding those definitions.
   square root of a sum of terms, the terms being the variances or covariances of
   these other quantities weighted according to how the measurement result varies
   with changes in these quantities
+  \begin{equation}
+    \uc^2(y) = \sum_{i=1}^{N} \left( \frac{\partial f}{\partial x_i} \right)^2 u^2(x_i)
+    \label{eq:combined-standard-uncertainty}
+  \end{equation}
 \end{definition}
 
 \begin{definition}[Coverage factor]