parent e6370ecf
 \begin{tabular}{c|cccccc} Stage & $\Nseg$ & $\Tcoh^{\rm days}$ &$\Nsteps$ & $\V$ & $\Vsky$ & $\Vpe$ \\ \hline 0 & 27 & 3.7 & 100 & 1.0 & 1.0 & 1.0 \\ 1 & 15 & 6.7 & 100 & 1.0 & 1.0 & 1.0 \\ 2 & 8 & 12.5 & 100 & 1.0 & 1.0 & 1.0 \\ 3 & 4 & 25.0 & 100 & 1.0 & 1.0 & 1.0 \\ 4 & 1 & 100.0 & 50,50 & 1.0 & 1.0 & 1.0 \\ \end{tabular}
 ... @@ -27,7 +27,7 @@ DeltaF1 = VF1 * np.sqrt(45/4.)/(np.pi*Tspan**2) ... @@ -27,7 +27,7 @@ DeltaF1 = VF1 * np.sqrt(45/4.)/(np.pi*Tspan**2) depth = 100 depth = 100 nsteps = 50 nsteps = 20 run_setup = [((nsteps, 0), 20, False), run_setup = [((nsteps, 0), 20, False), ((nsteps, 0), 7, False), ((nsteps, 0), 7, False), ((nsteps, 0), 2, False), ((nsteps, 0), 2, False), ... ...
 \begin{tabular}{c|cccccc} \begin{tabular}{c|cccccc} Stage & $\Nseg$ & $\Tcoh^{\rm days}$ &$\Nsteps$ & $\V$ & $\Vsky$ & $\Vpe$ \\ \hline Stage & $\Nseg$ & $\Tcoh^{\rm days}$ &$\Nsteps$ & $\V$ & $\Vsky$ & $\Vpe$ \\ \hline 0 & 20 & 5.0 & 100 & $2{\times}10^{2}$ & 10.0 & 10.0 \\ 0 & 20 & 5.0 & 50 & $2{\times}10^{2}$ & 10.0 & 10.0 \\ 1 & 11 & 9.1 & 100 & $2{\times}10^{3}$ & 40.0 & 50.0 \\ 1 & 11 & 9.1 & 50 & $2{\times}10^{3}$ & 40.0 & 50.0 \\ 2 & 6 & 16.7 & 100 & $2{\times}10^{4}$ & $1{\times}10^{2}$ & $2{\times}10^{2}$ \\ 2 & 6 & 16.7 & 50 & $2{\times}10^{4}$ & $1{\times}10^{2}$ & $2{\times}10^{2}$ \\ 3 & 3 & 33.3 & 100 & $1{\times}10^{5}$ & $2{\times}10^{2}$ & $6{\times}10^{2}$ \\ 3 & 3 & 33.3 & 50 & $1{\times}10^{5}$ & $2{\times}10^{2}$ & $6{\times}10^{2}$ \\ 4 & 1 & 100.0 & 100,100 & $8{\times}10^{5}$ & $3{\times}10^{2}$ & $2{\times}10^{3}$ \\ 4 & 1 & 100.0 & 50,50 & $8{\times}10^{5}$ & $3{\times}10^{2}$ & $2{\times}10^{3}$ \\ \end{tabular} \end{tabular}
 ... @@ -413,3 +413,18 @@ volume = {91}, ... @@ -413,3 +413,18 @@ volume = {91}, year = {2015} year = {2015} } } @article{wette2012, arxivId = {1111.5650}, author = {Wette, Karl}, doi = {10.1103/PhysRevD.85.042003}, eprint = {1111.5650}, file = {:home/greg/Dropbox/Papers/Wette{\_}2012.pdf:pdf}, isbn = {0556-2821}, issn = {15507998}, journal = {Physical Review D}, number = {4}, title = {{Estimating the sensitivity of wide-parameter-space searches for gravitational-wave pulsars}}, volume = {85}, year = {2012} }
 \begin{tabular}{c|cccc} Stage & $\Nseg$ & $\Tcoh^{\rm days}$ &$\Nsteps$ & $\Vpe$ \\ \hline 0 & 20 & 5.0 & 20 & 10.0 \\ 1 & 7 & 14.3 & 20 & $1{\times}10^{2}$ \\ 2 & 2 & 50.0 & 20 & $1{\times}10^{3}$ \\ 3 & 1 & 100.0 & 20,20 & $2{\times}10^{3}$ \\ \end{tabular}
 ... @@ -790,6 +790,13 @@ fraction of the volume at adjacent stages. That is we define ... @@ -790,6 +790,13 @@ fraction of the volume at adjacent stages. That is we define \end{equation} \end{equation} where $\mathcal{R} \ge 1$ as $\Tcoh^{i+1} > \Tcoh^{i}$. where $\mathcal{R} \ge 1$ as $\Tcoh^{i+1} > \Tcoh^{i}$. For the MCMC simulations to be succesful, this initial bounding box, given the segment setup which produced the candidate, must be small compared to the width of the signal (at that segment setup). If we start out follow-up with the same search setup used in the source search, i.e. we set $\Nseg^0$ equal to the number of segments used in the input search, then for the MCMC simulation to work, we require that $\V(\Nseg^{0}) \sim \mathcal{O}(100)$. Given a fixed prior on the Doppler parameters and a fixed span of data, the Given a fixed prior on the Doppler parameters and a fixed span of data, the metric volume $\V_{\Nstages}$ for $\Tcoh^{\Nstages} = \Tspan$ is fixed, or in metric volume $\V_{\Nstages}$ for $\Tcoh^{\Nstages} = \Tspan$ is fixed, or in terms of the number of segments, $\Nseg^{\Nstages} = 1$. terms of the number of segments, $\Nseg^{\Nstages} = 1$. ... @@ -824,12 +831,12 @@ in the noise. ... @@ -824,12 +831,12 @@ in the noise. First, we must define the setup for the run. Using $\mathcal{R}=10$ and First, we must define the setup for the run. Using $\mathcal{R}=10$ and $\V^{\rm min}=100$ our optimisation procedure is run and proposes the setup $\V^{\rm min}=100$ our optimisation procedure is run and proposes the setup layed out in Table~\ref{tab_signal_follow_up}. In addition, we show the layed out in Table~\ref{tab_follow_up_run_setup}. In addition, we show the number of steps taken at each stage. number of steps taken at each stage. \begin{table}[htb] \begin{table}[htb] \caption{The search setup used in Figure~\ref{fig_follow_up}, generated with \caption{The search setup used in Figure~\ref{fig_follow_up}, generated with $\mathcal{R}=10$ and $\V^{\rm min}=100$.} $\mathcal{R}=10$ and $\V^{\rm min}=100$.} \label{tab_follow_up} \label{tab_follow_up_run_setup} \input{follow_up_run_setup} \input{follow_up_run_setup} \end{table} \end{table} ... @@ -842,7 +849,7 @@ found to result in the MCMC simulations loosing' the peaks between stages, we ... @@ -842,7 +849,7 @@ found to result in the MCMC simulations loosing' the peaks between stages, we conservatively opt for 10 here, but values as large as 100 where also successful. conservatively opt for 10 here, but values as large as 100 where also successful. In Figure~\ref{fig_follow_up} we show the progress of the MCMC sampler during In Figure~\ref{fig_follow_up} we show the progress of the MCMC sampler during the follow-up. As expected from Table~\ref{tab_follow_up}, during the follow-up. As expected from Table~\ref{tab_follow_up_run_setup}, during the initial stage the signal peak is broad with respect to the size of the the initial stage the signal peak is broad with respect to the size of the prior volume, therefore the MCMC simulation quickly converges to it. Subsequently, prior volume, therefore the MCMC simulation quickly converges to it. Subsequently, each time the number of segments is reduced, the peak narrows and the samplers each time the number of segments is reduced, the peak narrows and the samplers ... @@ -855,9 +862,8 @@ Eventually, the walkers all converge to the true signal. ... @@ -855,9 +862,8 @@ Eventually, the walkers all converge to the true signal. \caption{In the top three panels we show the progress of the 500 parallel \caption{In the top three panels we show the progress of the 500 parallel walkers (see Figure~\ref{fig_MCMC_simple_example} for a description) during the walkers (see Figure~\ref{fig_MCMC_simple_example} for a description) during the MCMC simulation for each of the search parameters, frequency $f$, MCMC simulation for each of the search parameters, frequency $f$, right-ascension $\alpha$ and declination $\delta$. Each vertical dashed line right-ascension $\alpha$ and declination $\delta$. Each vertical dashed line indicates the start of a new stage of the search, the parameters for all stages indicates the start of a new stage of the search, the parameters for all stages are listed in Table~\ref{tab_follow_up_run_setup}.} are listed in Table~\ref{tab_follow_up}.} \label{fig_follow_up} \label{fig_follow_up} \end{figure} \end{figure} ... @@ -869,52 +875,89 @@ chains explore the other noise peaks' in the data. ... @@ -869,52 +875,89 @@ chains explore the other noise peaks' in the data. \section{Monte Carlo studies} \section{Monte Carlo studies} In order to understand how well the MCMC follow-up method works, we will now In order to understand how well the MCMC follow-up method works, we will test study the recovery fraction as a function of signal depth. This will be done in its ability to succesfully identify simulated signals in Gaussian. This will be a Monte Carlo study, with independent random realisations of the Guassian done in a Monte Carlo study, with independent random realisations of the noise, amplitude, and Doppler parameters in suitable ranges. Such a method is Guassian noise, amplitude, and Doppler parameters in suitable ranges. Such a analagous to the studies performed in \citet{shaltev2013}, except that we method is analagous to the studies performed in \citet{shaltev2013}, except present results as a function of the fixed signal depth, rather than the that we present results as a function of the fixed injected signal depth, squared SNR. rather than the squared SNR. In particular we will generate \comment{N} realisations of Gaussian noise data In particular we will generate a number of 100-day data sets containing lasting for 100 days, each with a simulated CW signal. We choose the parameters independent realisations of Gaussian noise and a simulated CW signal. We choose of the signal in such a way to model the candidates generated from directed and the parameters of the signal in such a way to model the candidates generated all-sky searches by drawing the signal parameters from appropriate from directed and all-sky searches by drawing the signal parameters from distributions. However, we do not draw $h_0$ randomly, but instead run the MC appropriate distributions. However, we do not draw $h_0$ randomly, but instead study at a number of selected values chosen such that given the fixed run the MC study at a number of selected values chosen such that given the $\sqrt{S_n}=2\times10^{3}$, the signals are injected with a depth $\mathcal{D} fixed$\sqrt{S_n}=2\times10^{3}$, the signals are injected with a depth \in [100, 400]$. $\mathcal{D} \in [100, 400]$. To simulate an isotropic distribution of sources, we draw the remaining amplitude parameters for each signal uniformly To simulate an isotropic distribution of sources, we draw the remaining from $\phi \in [0, 2\pi]$, $\psi \in [-\pi/4, \pi/4]$, and $\cos\iota \in [-1, amplitude parameters for each signal uniformly from$\phi \in [0, 2\pi]$,$\psi 1]$. \in [-\pi/4, \pi/4]$, and $\cos\iota \in [-1, 1]$. To provide a reference, we will compare the MC follow-up study against the expected maximum theoretical detection probability for an infinitly dense fully-coherent search of data containing isotropically-distributed signals as calculated by \citet{wette2012}. Note however that we will parameterise with respect to the signal depth (i.e. using Equation~(3.8) of \citet{wette2012} to relate the averaged-SNR to the depth). The probability is maximal in the sense that signals are lost' during the follow-up due simply to the fact that they are not sufficiently strong to rise above the noise. \subsection{Follow-up candidates from a directed search} \subsection{Follow-up candidates from a directed search} \label{sec_directed_follow_up} In a directed search, the sky location parameters $\alpha$ and $\delta$ are In a directed search, the sky location parameters $\alpha$ and $\delta$ are fixed - in our study we fix them on the location of the Crab pulsar, but this fixed - in our study we fix them on the location of the Crab pulsar, but this choice is arbitrary and holds no particular significance. The ouput of a choice is arbitrary and holds no particular significance. The ouput of a gridded directed search would provide the grid-point with the highest detection semi-coherent gridded directed search would provide the grid-point with the statistic, and some estimate of the iso-mismatch contours in which the highest detection statistic, and some box bounding the candidate given some candidate is expected to exist. To simulate this, we will define a frequency uncertainty. We will assume in this section that given the search setup and spindown of $f_0=30$~Hz and $\dot{f}_0=10^{-10}$Hz/s and a surrounding box $\Nseg^{0}$ of the input search, the bounding box is sufficiently small to $\Delta f$ and $\Delta \dot{f}$ which correspoinds to a fully-coherent ensure that $\V(\Nseg^0)\sim \mathcal{O}(100)$. This is not a limiting $\V=10^{4}$ with $\V_{f}=\V_{\dot{f}}$. Then, we pick a candidate we first pick assumption as any search can (quite cheaply) increase the density of grid a point randomly in the unit circle, then using the PE-phase-metric we convert points around any interesting candidates in order to better constrain their this into a random point in an isomismatch contour. In addition, we also select uncertainty. the amplitude parameters We then select a set of particular values of 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 generality define our Monte-Carlo follow-up in the following way. First, we select an arbitrary frequency and spindown of $f_0=30$~Hz and $\dot{f}_0=10^{-10}$Hz/s and a surrounding uncertainty box $\Delta f$ and $\Delta \dot{f}$ chosen such that the uncertainty in frequency and spindown are roughly equivalent in terms of mismatch. Then, we pick a candidate uniformly 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 Table~\ref{tab_directed_MC_follow_up}, this setup was used for all MC simulations. This table also lists the number of steps, which in this case we chose to be 10 - quite a small number of steps for a typical MCMC simulation. \begin{table}[htb] \begin{table}[htb] \caption{Run-setup for the directed follow-up Monte-Carlo study, generated with \caption{Run-setup for the directed follow-up Monte-Carlo study, generated with $\mathcal{R}=10$ and $\V^{\rm min}=100$} $\mathcal{R}=10$ and $\Nseg^0=20$.} \label{tab_directed_MC_follow_up} \label{tab_directed_MC_follow_up} \input{directed_setup_run_setup} \input{directed_setup_run_setup} \end{table} \end{table} 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 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 of the MCMC follow-up shows only a small margin of loss compared to the theoretical maximum. \comment{Probably best to run with more steps and see if this doesn't remove the loss} \begin{figure}[htb] \begin{figure}[htb] \centering \centering \includegraphics[width=0.45\textwidth]{directed_recovery} \includegraphics[width=0.45\textwidth]{directed_recovery} ... @@ -924,13 +967,39 @@ Table~\ref{tab_directed_MC_follow_up}.} ... @@ -924,13 +967,39 @@ Table~\ref{tab_directed_MC_follow_up}.} \label{fig_directed_MC_follow_up} \label{fig_directed_MC_follow_up} \end{figure} \end{figure} \subsection{Follow-up candidates from an all-sky search} \subsection{Follow-up candidates from an all-sky search} \label{sec_all_sky_follow_up} We now test the follow-up method when applied to candidates from an all-sky search which, by definition, have uncertain sky-position parameters $\alpha$ and $\delta$. Searching over these parameters, in addition to the frequency and spin-down not only increases the parameter space volume that needs to be searched, but also adds difficulty due to correlations between the sky-position and spin-down \comment{Does some of Karl's work discuss this?}. To replicate the condition of candidates from an all-sky search, we draw the candidate positions isotropically from the unit sphere (i.e. $\alpha$ uniform on $[0, 2\pi]$ and $\delta = \cos^{-1}(2u{-}1){- }\pi/2$ where $u$ is uniform on $[0, 1]$). We then place an uncertainty box containing the candidates with a width $\Delta\alpha=\Delta\delta=0.05$; this box is not centered on the candidate, but chosen in such a way that the location of the candidate has a uniform probability distrubution within the box. This neglects the non-uniform variation in $\delta$ over the sky-pacth, but given the small area should not 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}). 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 in Figure~\ref{fig_allsky_MC_follow_up}. \begin{table}[htb] \begin{table}[htb] \caption{Run-setup for the all-sky follow-up Monte-Carlo study, generated with \caption{Run-setup for the all-sky follow-up Monte-Carlo study, generated with $\mathcal{R}=10$ and $\V^{\rm min}=100$} $\mathcal{R}=10$ and $\Nseg^0=20$.} \label{tab_allsky_MC_follow_up} \label{tab_allsky_MC_follow_up} \input{AllSky_run_setup} \input{allsky_setup_run_setup} \end{table} \end{table} \begin{figure}[htb] \begin{figure}[htb] ... ...
 ... @@ -2365,8 +2365,8 @@ class MCMCFollowUpSearch(MCMCSemiCoherentSearch): ... @@ -2365,8 +2365,8 @@ class MCMCFollowUpSearch(MCMCSemiCoherentSearch): nsteps = rs nsteps = rs else: else: nsteps = '{},{}'.format(*rs) nsteps = '{},{}'.format(*rs) line = line.format(i, rs, Tcoh, nsteps, line = line.format(i, rs, '{:1.1f}'.format(Tcoh), texify_float(Vpe)) nsteps, texify_float(Vpe)) f.write(line) f.write(line) f.write(r'\end{tabular}' + '\n') f.write(r'\end{tabular}' + '\n') ... ...
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