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Pierre-Francois Loos 2020-09-08 15:36:43 +02:00
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Manuscript/QUEST_WIREs.tex
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@ -282,14 +282,12 @@ where the probability $P(\mathcal{G})$ that the random variables are normally di
\begin{equation}
P(\mathcal{G}) = 1 - \chi^2_{\text{CDF}}(J,2)
\end{equation}
where $\chi^2_{\text{CDF}}(x,k)$ is the cumulative distribution function of the $\chi^2$-distribution with $k$ degrees of freedom.
where $\chi^2_{\text{CDF}}(x,k)$ is the cumulative distribution function (CDF) of the $\chi^2$-distribution with $k$ degrees of freedom.
As the number of samples is usually small, we use Student's $t$-distribution to estimate the statistical error.
The inverse of the cumulative distribution function of the $t$-distribution allows us to find how to scale the interval by a parameter
The inverse of the cumulative distribution function of the $t$-distribution, $t_{\text{CDF}}^{-1}$, allows us to find how to scale the interval by a parameter
\begin{equation}
%\beta = t_{\text{CDF}}^{-1} \left[
%\frac{1}{2} \left( 1 + \frac{P( \Delta E_{\text{FCI}} \in [ \Delta E \pm \sigma ] \; | \; \mathcal{G}) }{P(\mathcal{G})}\right), n \right]
\beta = t_{\text{CDF}}^{-1} \left[
\frac{1}{2} \left( 1 + \frac{0.6827}{P(\mathcal{G})}\right), n \right]
\beta = t_{\text{CDF}}^{-1} \qty[
\frac{1}{2} \qty( 1 + \frac{0.6827}{P(\mathcal{G})}), M ]
\end{equation}
such that $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p$.
Only the last $M>2$ computed energy differences are considered. $M$ is chosen such that $P(\mathcal{G})>0.8$ and such that the error bar is minimal.