uq.lyx

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\index Index
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\begin_body

\begin_layout Standard
\begin_inset FormulaMacro
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\begin_layout Standard
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\end_layout

\begin_layout Section
Classical propagation of uncertainties
\end_layout

\begin_layout Standard
In classical methods of uncertainty propagation, input quantities 
\begin_inset Formula $x\pm\Delta x$
\end_inset

 are taken from their best estimate 
\begin_inset Formula $x$
\end_inset

 together with a possible deviation 
\begin_inset Formula $\Delta x$
\end_inset

, usually drawn as error bars.
 In a single experiment, 
\begin_inset Formula $x$
\end_inset

 is the reading of the measured quantity and 
\begin_inset Formula $\Delta x$
\end_inset

 is set according to the accuracy of the measurement device, e.g.
 the last significant digit on a display if no other factors are known to
 introduce uncertainties.
 In a series of experiments, 
\begin_inset Formula $x$
\end_inset

 is usually set to the sample mean and 
\begin_inset Formula $\Delta x$
\end_inset

 to the sample standard deviation (or some multiple of it).
 To know how the uncertainty in 
\begin_inset Formula $x\pm\Delta x$
\end_inset

 propagates to a function 
\begin_inset Formula $y(x\pm\Delta x)$
\end_inset

 there are two straightforward ways.
\end_layout

\begin_layout Enumerate
Linear approximation
\begin_inset Newline newline
\end_inset

For sufficiently small 
\begin_inset Formula $\Delta x$
\end_inset

 and if the analytic form of 
\begin_inset Formula $y(x)$
\end_inset

 is known we can use a Taylor expansion
\begin_inset Formula 
\begin{equation}
y(x\pm\Delta x)\approx y(x)\pm y^{\prime}(x)\Delta x+\mathcal{O}(\,(\Delta x)^{2}\,).
\end{equation}

\end_inset

Thus we estimate 
\begin_inset Formula $y$
\end_inset

 and its uncertainty 
\begin_inset Formula $\Delta y$
\end_inset

 as
\begin_inset Formula 
\begin{align}
y & =y(x),\quad\Delta y=y^{\prime}(x)\Delta x.
\end{align}

\end_inset

Within this linear approximation the estimated 
\begin_inset Formula $y$
\end_inset

 stays centered within the error bars.
\end_layout

\begin_layout Enumerate
Direct evaluation
\begin_inset Newline newline
\end_inset

In the more general case one evaluates 
\begin_inset Formula $y(x)$
\end_inset

, 
\begin_inset Formula $y(x+\Delta x)$
\end_inset

 and 
\begin_inset Formula $y(x-\Delta x)$
\end_inset

 directly.
 This works also if no analytical expression for 
\begin_inset Formula $y(x)$
\end_inset

 exists, e.g.
 if it follows from a complicated numerical model.
 Here 
\begin_inset Formula $y(x)$
\end_inset

 does not necessarily lie centered between upper and lower error bar and
 we obtain two uncertainties
\begin_inset Formula 
\begin{align}
\Delta y^{+} & =y(x+\Delta x)-y(x),\\
\Delta y^{-} & =y(x)-y(x-\Delta x).
\end{align}

\end_inset

According to Taylor's theorem for differentiable functions 
\begin_inset Formula $y(x)$
\end_inset

 the difference 
\begin_inset Formula $\Delta y^{+}-\Delta y^{-}\approx y^{\prime\prime}(x)(\Delta x)^{2}$
\end_inset

 becomes ignorable for sufficiently small 
\begin_inset Formula $\Delta x$
\end_inset

, leading to results close to the linear approximation described above.
\end_layout

\begin_layout Section
Polynomial chaos expansion
\end_layout

\begin_layout Standard
In a more general sense we can model out input quantity as a random variable
 
\begin_inset Formula $X$
\end_inset

 according to some probability density function 
\begin_inset Formula $f_{X}(x)$
\end_inset

.
 In the pseudo-spectral approach we model
\begin_inset Formula 
\begin{equation}
f_{Y}(y)\approx\sum_{n}f_{n}\phi_{n}(x)
\end{equation}

\end_inset


\end_layout

\begin_layout Standard
Here 
\begin_inset Formula $\phi_{n}$
\end_inset

 are chosen as orthogonal polynomials (Hermite, Legendre, Jacobi, Laguerre).
 Their 
\series bold
weight function
\series default
 under which orthogonality holds corresponds to the 
\series bold
probability density function
\series default
 
\begin_inset Formula $f_{X}(x)$
\end_inset

 of the input variables.
 Coefficients 
\begin_inset Formula $f_{n}$
\end_inset

 are computed according to the optimum quadrature rule for the respective
 problem.
\end_layout

\end_body
\end_document