diff --git a/Paper/Todo.txt b/Paper/Todo.txt new file mode 100644 index 0000000000000000000000000000000000000000..dbe2e6f0a96114cee717888a4517a3067dbc9121 --- /dev/null +++ b/Paper/Todo.txt @@ -0,0 +1,8 @@ +- 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 + diff --git a/Paper/definitions.tex b/Paper/definitions.tex index 233a66e55fe61797cf43f0482788a9ac978ce1b1..1d4d00eff6d6374fbbfae67ce40eece676c3968a 100644 --- a/Paper/definitions.tex +++ b/Paper/definitions.tex @@ -1,5 +1,6 @@ \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}} diff --git a/Paper/paper_cw_mcmc.tex b/Paper/paper_cw_mcmc.tex index cf8081ef101f51ffaf52ae206f8983b7542d9442..2dc7f005b9d34a00a5b09526a8dc1c6c0741224e 100644 --- a/Paper/paper_cw_mcmc.tex +++ b/Paper/paper_cw_mcmc.tex @@ -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