mveril/QUESTDB
1
1
Fork 0
forked from PTEROSOR/QUESTDB
Code Issues Pull Requests Releases Wiki Activity

toto part

Browse Source
This commit is contained in:
Pierre-Francois Loos 2020-09-08 15:36:43 +02:00
parent e72c1b9751
commit ffbf841c10
1 changed files with 4 additions and 6 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

10
Manuscript/QUEST_WIREs.tex
View File

@ -282,14 +282,12 @@ where the probability $P(\mathcal{G})$ that the random variables are normally di
\begin{equation} \begin{equation}
P(\mathcal{G}) = 1 - \chi^2_{\text{CDF}}(J,2) P(\mathcal{G}) = 1 - \chi^2_{\text{CDF}}(J,2)
\end{equation} \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. 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} \begin{equation}
%\beta = t_{\text{CDF}}^{-1} \left[ \beta = t_{\text{CDF}}^{-1} \qty[
%\frac{1}{2} \left( 1 + \frac{P( \Delta E_{\text{FCI}} \in [ \Delta E \pm \sigma ] \; | \; \mathcal{G}) }{P(\mathcal{G})}\right), n \right] \frac{1}{2} \qty( 1 + \frac{0.6827}{P(\mathcal{G})}), M ]
\beta = t_{\text{CDF}}^{-1} \left[
\frac{1}{2} \left( 1 + \frac{0.6827}{P(\mathcal{G})}\right), n \right]
\end{equation} \end{equation}
such that $P( \Delta E_{\text{FCI}}^{(m)} \in [ \Delta E_{\text{CIPSI}}^{(m)} \pm \beta \sigma ] ) = p$. 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. 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.