Commit f2379bbd authored by Gregory Ashton's avatar Gregory Ashton
Browse files

Merge branch 'master' of gitlab.aei.uni-hannover.de:GregAshton/PyFstat

parents ea67dfa8 8886724c
- Add macros for the directed MC results
- Add macros for the all-sky MC results
- Adds macros for the examples values
- Investigate cause of all-sky losses again
- Run the transient search for a weaker signal
- Add results from Messenger et al for the transient - is there a fit?
- Weaken basyes factor definitions since they are not used, just put it in the context
\newcommand{\fdot}{\dot{f}}
\newcommand{\F}{{\mathcal{F}}}
\newcommand{\twoFtilde}{\widetilde{2\F}}
\newcommand{\A}{\boldsymbol{\mathcal{A}}}
\newcommand{\blambda}{\boldsymbol{\mathbf{\lambda}}}
\newcommand{\blambdaSignal}{\boldsymbol{\mathbf{\lambda}}^{\rm s}}
......
......@@ -31,6 +31,7 @@
\begin{tabular}[#1]{@{}c@{}}#2\end{tabular}}
\input{definitions.tex}
\input{macros}
% For editing purposes: remove before submition
\usepackage[normalem]{ulem} %% only added for 'strikeout' \sout
......@@ -881,7 +882,7 @@ chains explore the other `noise peaks' in the data.
\section{Monte Carlo studies}
In order to understand how well the MCMC follow-up method works, we will test
its ability to succesfully identify simulated signals in Gaussian. This will be
its ability to succesfully identify simulated signals in Gaussian noise. This will be
done in a Monte Carlo study, with independent random realisations of the
Guassian noise, amplitude, and Doppler parameters in suitable ranges. Such a
method is analagous to the studies performed in \citet{shaltev2013}, except
......@@ -894,7 +895,7 @@ the parameters of the signal in such a way to model the candidates generated
from directed and all-sky searches by drawing the signal parameters from
appropriate distributions. However, we do not draw $h_0$ randomly, but instead
run the MC study at a number of selected values chosen such that given the
fixed $\sqrt{S_n}=2\times10^{3}$, the signals are injected with a depth
fixed $\sqrt{S_n}=1\times10^{3}$, the signals are injected with a depth
$\mathcal{D} \in [100, 400]$. To simulate an isotropic distribution of
sources, we draw the remaining amplitude parameters for each signal uniformly
from $\phi \in [0, 2\pi]$, $\psi \in [-\pi/4, \pi/4]$, and $\cos\iota \in [-1,
......@@ -925,6 +926,24 @@ assumption as any search can (quite cheaply) increase the density of grid
points around any interesting candidates in order to better constrain their
uncertainty.
Before applying the directed follow-up to simulated signals in noise, we need
to characterise its behaviour in Gaussian noise alone. To do so, we simulate
$\DirectedMCNoiseN$ realisations of Gaussian noise and peform the follow-up
search on these. A histogram of the results is provided in
Figure~\ref{fig_hist_DirectedMCNoiseOnly}, the largest observed value was
found to be $\DirectedMCNoiseOnlyMaximum$. From this, we can set a threshold
for the detection statistic of $\twoFtilde_{\rm th} = 60$, an arbitrary
number chosen to be sufficiently larger than the maximum seen in noise and
consistent with the value chosen in \citet{shaltev2013}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{directed_noise_twoF_histogram}
\caption{Histogram of the recovered $\widetilde{2\F}$ values applying the
directed follow-up routine to $\DirectedMCNoiseN$ simulated Gaussian noise
realisations.}
\label{fig_hist_DirectedMCNoiseOnly}
\end{figure}
The behaviour of the follow-up is independent of the exact frequency and
spin-down values used (this is not true for an all-sky follow-up as discussed
in Section~\ref{sec_all_sky_follow_up}). As such, we can, without loss of
......@@ -937,8 +956,6 @@ from within this uncertainty box; this choice reflects the fact that the grid
is chosen such that the probability distribution of candidate signals is
uniform.
Having generated the data given the prescription above, we proceed to perform a
hierarchical MCMC follow-up. Given the data span and initial bounding box, we
compute the optimal heirarchical setup, the details of which are given in
......@@ -953,10 +970,9 @@ $\mathcal{R}=10$ and $\Nseg^0=20$.}
\input{directed_setup_run_setup}
\end{table}
This process yeilds a maximum detection statistic $\widetilde{2\F}^{\rm max}$.
This process yeilds a maximum detection statistic $\widetilde{2\F}_{\rm max}$.
The signal is considered `detected' if $\widetilde{2\F}^{\rm max} >
\widetilde{2\F}^{\rm th}$, where we set the threshold at $2\F^{\rm th}=60$,
corresponding to a p-value of \comment{Finish section}. In
\widetilde{2\F}_{\rm th}$, where we set the threshold at $2\F_{\rm th}=60$. In
Figure~\ref{fig_directed_MC_follow_up} we plot the the fraction of the MC
simulations which where recovered and compare this against the theoretical
maximum, given the threshold. The figure demonstrates that the recovery power
......@@ -995,6 +1011,25 @@ cause any significant bias. The frequency, spin-down, and amplitude parameters
are chosen in the same way as for the directed search
(Section~\ref{sec_directed_follow_up}).
Again, we first characterise the behaviour of the all-sky follow-up by applying
it to $\AllSkyMCNoiseN$ realisations of Gaussian noise. The resulting histogram
is provided in Figure~\ref{fig_hist_AllSkyMCNoiseOnly} and the largest $\twoFtilde$
value was found to be $\AllSkyMCNoiseOnlyMaximum$. This is larger than the
value found for the directed search, although both use the same number of
Gaussian noise trials, and therefore must result from the increased number of
search parameters. \comment{Ask Reinhard about Miroslavs statement on number of
templates}. As a result we will correspondinly increase our detection threshold
for the all-sky search to $\twoFtilde_{\rm tr} = 70$; again this is an arbitary
choise, but is consisent with the values chosen in \citet{shaltev2013}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.5\textwidth]{allsky_noise_twoF_histogram}
\caption{Histogram of the recovered $\widetilde{2\F}$ values applying the
all-sky follow-up routine to $\AllSkyMCNoiseN$ simulated Gaussian noise
realisations.}
\label{fig:}
\end{figure}
Producing \CHECK{1000} indepedant MC simulations we the perform a follow-up on
each using the setup given in Table~\ref{tab_allsky_MC_follow_up}. The
resulting recovery fraction as a function of the injected signal depth is given
......
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