From ce6f4793ff05b11c1ab3ab5878718c33eaff3ffe Mon Sep 17 00:00:00 2001 From: Gregory Ashton <gregory.ashton@ligo.org> Date: Thu, 23 Mar 2017 17:50:27 +0100 Subject: [PATCH] Improvements to glitch section of the paper --- Paper/Examples/single_glitch.py | 54 +++-- .../SingleGlitchMCNoiseOnly/generate_data.py | 2 +- Paper/definitions.tex | 3 +- Paper/macros.tex | 11 + Paper/paper_cw_mcmc.tex | 225 +++++++++++++----- Paper/single_glitch_F0F1_grid_2D.png | Bin 101031 -> 117892 bytes Paper/single_glitch_glitchSearch_corner.png | Bin 756734 -> 840289 bytes 7 files changed, 209 insertions(+), 86 deletions(-) diff --git a/Paper/Examples/single_glitch.py b/Paper/Examples/single_glitch.py index 0abb7c4..1da8f5d 100644 --- a/Paper/Examples/single_glitch.py +++ b/Paper/Examples/single_glitch.py @@ -1,6 +1,7 @@ import pyfstat import numpy as np import matplotlib.pyplot as plt +import matplotlib sqrtSX = 1e-22 tstart = 1000000000 @@ -26,7 +27,8 @@ DeltaF0 = VF0 * dF0 DeltaF1 = VF1 * dF1 # Next, taking the same signal parameters, we include a glitch half way through -dtglitch = duration/2.0 +R = 0.5 +dtglitch = R*duration delta_F0 = 0.25*DeltaF0 delta_F1 = -0.1*DeltaF1 @@ -35,6 +37,7 @@ glitch_data = pyfstat.Writer( F1=F1, F2=F2, duration=duration, Alpha=Alpha, Delta=Delta, h0=h0, sqrtSX=sqrtSX, dtglitch=dtglitch, delta_F0=delta_F0, delta_F1=delta_F1) glitch_data.make_data() +FCtwoFPM = glitch_data.predict_fstat() F0s = [F0-DeltaF0/2., F0+DeltaF0/2., 1*dF0] F1s = [F1-DeltaF1/2., F1+DeltaF1/2., 1*dF1] @@ -45,7 +48,12 @@ search = pyfstat.GridSearch( 'single_glitch_F0F1_grid', 'data', 'data/*single_glitch*sft', F0s, F1s, F2s, Alphas, Deltas, tref, tstart, tend) search.run() -search.plot_2D('F0', 'F1') +ax = search.plot_2D('F0', 'F1', save=False) +ax.xaxis.set_major_locator(matplotlib.ticker.MaxNLocator(6)) +plt.tight_layout() +plt.savefig('data/single_glitch_F0F1_grid_2D.png', dpi=500) +FCtwoF = search.get_max_twoF()['twoF'] + theta_prior = {'F0': {'type': 'unif', 'lower': F0-DeltaF0/2., 'upper': F0+DeltaF0/2}, @@ -55,21 +63,6 @@ theta_prior = {'F0': {'type': 'unif', 'lower': F0-DeltaF0/2., 'Alpha': Alpha, 'Delta': Delta } -ntemps = 3 -log10temperature_min = -0.05 -nwalkers = 100 -nsteps = [500, 500] - -mcmc = pyfstat.MCMCSearch( - 'single_glitch', 'data', sftfilepath='data/*_single_glitch*.sft', - theta_prior=theta_prior, tref=tref, minStartTime=tstart, maxStartTime=tend, - nsteps=nsteps, nwalkers=nwalkers, ntemps=ntemps, - log10temperature_min=log10temperature_min) - -mcmc.run() -mcmc.plot_corner(figsize=(3.2, 3.2)) -mcmc.print_summary() - theta_prior = {'F0': {'type': 'unif', 'lower': F0-DeltaF0/2., 'upper': F0+DeltaF0/2}, @@ -80,8 +73,8 @@ theta_prior = {'F0': {'type': 'unif', 'lower': F0-DeltaF0/2., 'Delta': Delta, 'tglitch': {'type': 'unif', 'lower': tstart+0.1*duration, 'upper': tend-0.1*duration}, - 'delta_F0': {'type': 'halfnorm', 'loc': 0, 'scale': DeltaF0}, - 'delta_F1': {'type': 'norm', 'loc': 0, 'scale': DeltaF1}, + 'delta_F0': {'type': 'halfnorm', 'loc': 0, 'scale': 1e-7*F0}, + 'delta_F1': {'type': 'norm', 'loc': 0, 'scale': abs(1e-3*F1)}, } ntemps = 3 log10temperature_min = -0.1 @@ -93,7 +86,30 @@ glitch_mcmc = pyfstat.MCMCGlitchSearch( tref=tref, minStartTime=tstart, maxStartTime=tend, nsteps=nsteps, nwalkers=nwalkers, ntemps=ntemps, log10temperature_min=log10temperature_min) +glitch_mcmc.write_prior_table() glitch_mcmc.run() glitch_mcmc.plot_corner(figsize=(6, 6)) glitch_mcmc.print_summary() +GlitchtwoF = glitch_mcmc.get_max_twoF()[1] + +from latex_macro_generator import write_to_macro +write_to_macro('SingleGlitchTspan', '{:1.1f}'.format(duration/86400), + '../macros.tex') +write_to_macro('SingleGlitchR', '{:1.1f}'.format(R), '../macros.tex') +write_to_macro('SingleGlitchDepth', + '{}'.format(pyfstat.texify_float(sqrtSX/h0)), '../macros.tex') +write_to_macro('SingleGlitchF0', '{}'.format(F0), '../macros.tex') +write_to_macro('SingleGlitchF1', '{}'.format(F1), '../macros.tex') +write_to_macro('SingleGlitchdeltaF0', + '{}'.format(pyfstat.texify_float(delta_F0)), '../macros.tex') +write_to_macro('SingleGlitchdeltaF1', + '{}'.format(pyfstat.texify_float(delta_F1)), '../macros.tex') +write_to_macro('SingleGlitchFCtwoFPM', '{:1.1f}'.format(FCtwoFPM), + '../macros.tex') +write_to_macro('SingleGlitchFCtwoF', '{:1.1f}'.format(FCtwoF), + '../macros.tex') +write_to_macro('SingleGlitch_FCMismatch', + '{:1.1f}'.format((FCtwoFPM-FCtwoF)/FCtwoFPM), '../macros.tex') +write_to_macro('SingleGlitchGlitchtwoF', '{:1.1f}'.format(GlitchtwoF), + '../macros.tex') diff --git a/Paper/SingleGlitchMCNoiseOnly/generate_data.py b/Paper/SingleGlitchMCNoiseOnly/generate_data.py index 7c2d8b4..cdedbdc 100644 --- a/Paper/SingleGlitchMCNoiseOnly/generate_data.py +++ b/Paper/SingleGlitchMCNoiseOnly/generate_data.py @@ -59,7 +59,7 @@ glitch_data.make_data() startTime = time.time() -theta_prior = {'F0': {'type': 'unif', 'lower': F0-DeltaF0/2., +theta_prior = {'F0': {'type': 'unif', 'lower': F0-DeltaF0/2., # PROBLEM 'upper': F0+DeltaF0/2}, 'F1': {'type': 'unif', 'lower': F1-DeltaF1/2., 'upper': F1+DeltaF1/2}, diff --git a/Paper/definitions.tex b/Paper/definitions.tex index 1d4d00e..c6d071a 100644 --- a/Paper/definitions.tex +++ b/Paper/definitions.tex @@ -1,5 +1,6 @@ \newcommand{\fdot}{\dot{f}} \newcommand{\F}{{\mathcal{F}}} +\newcommand{\twoFhat}{\widehat{2\F}} \newcommand{\twoFtilde}{\widetilde{2\F}} \newcommand{\A}{\boldsymbol{\mathcal{A}}} \newcommand{\blambda}{\boldsymbol{\mathbf{\lambda}}} @@ -7,7 +8,7 @@ \newcommand{\tglitch}{t_{\rm glitch}} \newcommand{\tstart}{t_{\rm start}} \newcommand{\tend}{t_{\rm end}} -\newcommand{\Nglitches}{N_{\rm glitches}} +\newcommand{\Nglitches}{N_{\rm g}} \newcommand{\Tspan}{T_{\rm span}} \newcommand{\Tcoh}{T_{\rm coh}} \newcommand{\tref}{t_{\rm ref}} diff --git a/Paper/macros.tex b/Paper/macros.tex index df6253a..6ec6b01 100644 --- a/Paper/macros.tex +++ b/Paper/macros.tex @@ -14,3 +14,14 @@ \def\BasicExamplenprod{50.0} \def\DirectedMCNoiseN{1.0{\times}10^{4}} \def\DirectedMCNoiseOnlyMaximum{52.4} +\def\SingleGlitchDepth{10.0} +\def\SingleGlitchFCMismatch{0.7} +\def\SingleGlitchFCtwoF{718.5} +\def\SingleGlitchFCtwoFPM{2645.7} +\def\SingleGlitchFone{-1e-10} +\def\SingleGlitchFzero{30.0} +\def\SingleGlitchGlitchtwoF{2622.1} +\def\SingleGlitchR{0.5} +\def\SingleGlitchTspan{100.0} +\def\SingleGlitchdeltaFone{$-2.9{\times}10^{-13}$} +\def\SingleGlitchdeltaFzero{$3.2{\times}10^{-6}$} diff --git a/Paper/paper_cw_mcmc.tex b/Paper/paper_cw_mcmc.tex index fa4045b..a7d2c12 100644 --- a/Paper/paper_cw_mcmc.tex +++ b/Paper/paper_cw_mcmc.tex @@ -835,7 +835,7 @@ of parallel tempering (see Sec.~\ref{sec_parallel_tempering}). This was intenion not used here to highlight the utility of the convergence statistic. -\section{Follow-up} +\section{Zoom follow-up} \label{sec_follow_up} Incoherent detection statistics trade significance (the height of the peak) for @@ -964,11 +964,12 @@ possible for this to occur during an ordinary MCMC simulation (with $\Tcoh$ fixed at $\Tspan$), it would take substantially longer to converge as the chains explore the other `noise peaks' in the data. -\section{Monte Carlo studies} +\subsection{Monte-Carlo studies} +\label{sec_MCS_zoom} In order to understand how well the MCMC follow-up method works, we will test 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 +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 that we present results as a function of the fixed injected signal depth, @@ -996,7 +997,7 @@ 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} +\subsubsection{Follow-up candidates from a directed search} \label{sec_directed_follow_up} In a directed search, the sky location parameters $\alpha$ and $\delta$ are @@ -1074,7 +1075,7 @@ Table~\ref{tab_directed_MC_follow_up}.} \end{figure} -\subsection{Follow-up candidates from an all-sky search} +\subsubsection{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 @@ -1139,23 +1140,22 @@ Table~\ref{tab_directed_MC_follow_up}.} \label{fig_allsky_MC_follow_up} \end{figure} -\section{Alternative waveform models} +\section{Alternative waveform models: transients} +\label{sec_transients} In a grided search, the template bank is constructed to ensure that a canonical CW signal (i.e. when it lasts much longer than the observation span and has a -phase evolution well-described by a Equation~\eqref{eqn_phi}) will be -recovered with a fixed maximum loss of detection statistic; this loss can be -described as the `template-bank mismatch'. In addition to this mismatch, CW -searches may experience a mismatch if the waveform model differs from the -matched-filter template. There are of course an unlimited number of ways this -may manifest given our ignorance of neutron stars, but from studying pulsars -three obvious mechanisms present themselves: transient, glitching, and noisy -waveforms. In the following sections we will discuss the first two of these, we -discussed the effect of random jitters in the phase evolution (noisy waveforms) -in \citet{ashton2015} and concluded it was unlikely to be of immediate concern. - -\subsection{Transients} -\label{sec_transients} +phase evolution well-described by a Equation~\eqref{eqn_phi}) will be recovered +with a fixed maximum loss of detection statistic; this loss can be described as +the `template-bank mismatch'. In addition to this mismatch, CW searches may +experience a mismatch if the waveform model differs from the matched-filter +template. There are of course an unlimited number of ways this may manifest +given our ignorance of neutron stars, but from studying pulsars three obvious +mechanisms present themselves: transient, glitching, and noisy waveforms. In +the this section we will discuss the first of these, then in +Section~\ref{sec_glitches} the second, we discussed the effect of random +jitters in the phase evolution (noisy waveforms) in \citet{ashton2015} and +concluded it was unlikely to be of immediate concern. The term \emph{transient-CWs} refers to periodic gravitational wave signals with a duration $\mathcal{O}(\textrm{hours-weeks})$ which have a @@ -1237,85 +1237,180 @@ a simulated transient signal and Gaussian noise.} -\subsection{Glitches} +\section{Alternative waveform models: Glitch-robust follow-up search} \label{sec_glitches} -Observations of radio pulsars show that occasionally neutron stars undergo -sudden glitch events during which the pulsation frequency suddenly -\emph{increases} quite distinct from the usual spin-down due to EM or GW -torques (see \citet{espinoza2011} for a review). These events are typically -Poisson distributed \citet{melatos2008}, but the exact mechanism is not yet -fully understood. However, it seems plausible that the glitch mechanism will -similarly effect any CW emission, i.e. CW signals will also undergo glitches. -In \citet{ashton2016} we demonstrated that if this is the case, CW sources in -directed and all-sky searches have a reasonable probability of undergoing one -or more glitches during typical observations spans and that these glitches may -be sufficiently large to disrupt the detection; this is particuarly true for -areas of parameter space with large spindowns. However, the effect is mitigated -by the use of incoherent searches. As such, we may expect all-sky and directed -searches to correctly identify glitching signals as candidates, but -subseuqnetly `loose' them during the follow-up process. As such, we will now +Observations of radio pulsars show that occasionally the regular spin-down of a +neutron stars due to electromagnetic and (potentially) gravitational torques is +interrupted by sudden glitch events during which the rotation frequency +suddenly \emph{increases} (see for example \citet{espinoza2011} for an +observational review). The glitch mechanism is not yet fully understood, but as +argued in \citet{ashton2017}, it seems plausable that the glitch mechanism, as +it effects the radio pulsars, will also effect gravitars. As such, wide +parameter space searches which look for unknown neutron stars spinning down +primarily by gravitational wave emission (i.e. gravitars), could be hampered by +the existence of glitches in the signals. This was investigated by +\citet{ashton2017} and it was found that for many recent and ongoing searches, +there exists a reasonable probability of one or more glitches occuring in the +signal during typical observations spans and that these glitches may be +sufficiently large such that the loss of detection statistic could be of order +unity. This is particuarly true for searches looking for signals with a large +spin-down and over long data span ($\gtrsim 100$~days). Semi-coherent searches +are less effected (in terms of the loss of detection statistic) such that, +provided the number of glitches in a signal during a search is less than say +3, the initial search should still find the signal. However, that signal may +look like a transient signal, or be `lost' during the follow up process. + +This work motivates the existence of a \emph{glitch-robust} follow-up search +which is capable of fitting for the glitch parameters. As such, we will now discuss how the $\F$-statistic optimisation routine discussed in this work can be used to search for glitching signals. -We will model a glitch as a sudden discontinuous change in the Doppler -parameters $\delta \blambda$ at some time $\tglitch$: that the change is only -in the frequency and spin-down Doppler parameters, i.e. $\delta\blambda = [0, -0, \delta f, \delta \fdot, \ldots]$. This is to be interpretted, as is done for -radio-pulsar glitches, as the instantaneous change in frequency and spindown at -the time of the glitch. Moreover, multiple glitches can be included (we will -define $\Nglitches$ as the number of glitches) by adding -an index to the glitch magnitude and glitch time. +\subsection{Glitch-robust detection statistic} +In our analysis, we consider a signal which undergoes $\Nglitches$ glitches. +We will model a glitch $\ell$ as a sudden discontinuous change in the frequency +and spin-down Doppler parameters $\delta \blambda^\ell = [0, 0, \delta f, +\delta \fdot, \ldots]$ at some time $\tglitch^\ell$. This is to be +interpretted, as is done for radio-pulsar glitches, as the instantaneous change +in frequency and spindown at the time of the glitch. For convienience, we +define $\delta\blambda^0 = [0, 0, 0, 0 \ldots]$, $\tglitch^0 = +t_\mathrm{start}$ (i.e. the start time of the observation), and +$\tglitch^{\Nglitches+1}=t_\mathrm{end}$ (i.e. the end time of the +observation). We refer to $\{\delta\blambda^{\ell}\}, \{\tglitch^\ell\}$ as +the set of all glitch parameters. An ideal glitch search would include these new parameters into the phase model itself and compute the $\F$-statistic fully-coherently i.e. rewriting Equation~\eqref{eqn_B_S_N} as \begin{multline} -\Bsn(x| \tglitch, \delta\blambda, \Pic, \blambda, I) \equiv \\ +\Bsn(x| \{\tglitch^\ell\}, \{\delta\blambda^\ell\}, \Pic, \blambda, I) \equiv \\ \int -\mathcal{L}(x ;\A, \blambda, \tglitch, \delta\blambda) +\mathcal{L}(x ;\A, \blambda, \{\tglitch^\ell\}, \{\delta\blambda^\ell\}) P(\A| \Hs, I) d\A \end{multline} -where $\tglitch, \delta\blambda$ modify the phase model. However, we propose the -following pragmatic approach: let us instead use an semi-coherent detection -statistic with the epoch between \emph{segments} as $\tglitch$, i.e. +where the glitch parameters subsequently modify the phase model. However, we +propose instead the following pragmatic approach: let us instead use a +semi-coherent detection statistic with the epoch between \emph{segments} as +$\tglitch^\ell$, i.e. \begin{multline} \Bsn(x| \tglitch, \delta\blambda, \Pic, \blambda, I) \equiv \\ -\int_{\tstart}^{\tglitch} -\mathcal{L}(x ;\A, \blambda) -P(\A| \Hs, I) d\A \\ + -\int_{\tglitch}^{\tend} -\mathcal{L}(x ;\A, \blambda{+}\delta\blambda) +\sum_{\ell=0}^{\Nglitches} +\int_{\tglitch^\ell}^{\tglitch^{\ell+1}} +\mathcal{L}(x ;\A, \blambda+\delta\blambda^\ell) P(\A| \Hs, I) d\A \end{multline} +or, written in terms of the $\mathcal{F}$-statistic, +\begin{multline} +\twoFhat(N_{\rm glitch}, \lambda, \{\tglitch^\ell\}, \{\delta\blambda^{\ell}\})= \\ +\sum_{\ell=0}^{N_{\rm glitch}} +\twoFtilde(\alpha, \delta, \blambda+\delta\blambda^\ell; +t_{\rm glitch}^\ell, t_{\rm glitch}^{\ell+1}) +\label{eqn_glitch_likelihood} +\end{multline} +\comment{Need to think about how to connect these two statements and if the +first one is required} This simplistic method leverages readily available and tested code with a loss -of sensitivity over the fully-coherent search by a factor -$\sqrt{\Nglitches+1}$. Such a loss is acceptable, given that the signals must -be sufficiently strong to have been identified by the initial all-sky or directed +of sensitivity over the fully-coherent search by a factor $\sqrt{\Nglitches+1}$ +\citep{wette2015}. Such a loss is acceptable, given that the signals must be +sufficiently strong to have been identified by the initial all-sky or directed search. +The general methodology for a glitch-robust statistic (i.e. +Eq.\eqref{eqn_glitch_likelihood}) could be implemented in any number of ways, +the most straightforward of course beging a gridded search over the glitch +parameters. However, for each glitch this introduces at least two new +parameters to search over. For reasonable priors on the unknown glitch +parameters, the likelihood peak occupies a small fraction of the total search +space and so a gridded search will spend the majority of the compute time in +exploring the empty regions of parameter space. However, provided that the +initial Doppler parameters $\blambda$ are well constrained, the fraction of the +search space occupied should not be so small as to make application of the MCMC +procedures described in this work inefficient. On the contrary, we find in +practise that the MCMC parameter estimation to be highly efficient. + +\subsection{Case study: directed glitching signals} + As an illustration of the use of this method, we simulate a signal in Gaussian -noise which undergoes a glitch. Firstly in Figure~\ref{fig_glitch_grid} we show -the fully-coherent $2\F$ value computed for a grid of points: this distinctly -shows two peaks, a feature indicative of a glitching signal, with a maximum -value of $\sim\CHECK{400}$. Using our incoherent glitch statistic, we plot -the resulting posterior distributions in Figure~\ref{fig_glitch_posteriors}. +noise which undergoes a single glitch; the properties of this signal and the +data are given in Table~\ref{tab_single_glitch}. + +\begin{table} +\caption{Values used to generate the simulated glitching signal} +\label{tab_single_glitch} +\begin{tabular}{c|c} +& \\ \hline +$T$ & \SingleGlitchTspan~days \\ +$\sqrt{X}/h0$ & \SingleGlitchDepth \\ +$f$ & \SingleGlitchFzero~Hz \\ +$\dot{f}$ & \SingleGlitchFone~Hz/s \\ +$\delta f$ & \SingleGlitchdeltaFzero~Hz \\ +$\delta \dot{f}$ & \SingleGlitchdeltaFone~Hz/s \\ +$\frac{\tglitch - \tstart}{T}$ & \SingleGlitchR +\end{tabular} +\end{table} + +In Figure~\ref{fig_glitch_grid} we show the fully-coherent $\twoFtilde$ value +computed on a grid over $f$ and $\dot{f}$ (i.e. with a non-glitching template). +The maximum computed $\twoFtilde$ value is found to be $\SingleGlitchFCtwoF$. +This can be compared to the expected $\twoFtilde$ for a perfectly matched +signal in the same data which is $\SingleGlitchFCtwoFPM$: the glitch results in +the loss of $\SingleGlitchFCMismatch$ of the detection statistic. The figure +distinctly shows two peaks, a feature indicative of a glitching signal. \begin{figure}[htb] \centering \includegraphics[width=0.5\textwidth]{single_glitch_F0F1_grid_2D} -\caption{} +\caption{The fully-coherent $\twoFtilde$ computed over a grid in $f$ and +$\dot{f}$ for the simulated glitching signal with the properties described in +Table~\ref{tab_single_glitch}.} \label{fig_glitch_grid} \end{figure} -\begin{figure}[htb] +To perform the glitch-robust search we define a prior as follows. Uniform in +$f$ and $\dot{f}$ (over the range shown in Figure~\ref{fig_glitch_grid}; for +$\tglitch$ we define a uniform prior over the entire duration except the first +and last 10\%; then for $\delta f$ we asign a prior of $\CHECK{\mathcal{N}(0, +10^{-7} f_0)}$ and for $\delta\dot{f}$ a prior of $\CHECK{|\mathcal{N}(0, +10^{-3}\dot{f})|}$ (i.e. a range of physical reasonable values preferring no +glitch, but allowing an arbitrarily large glitch). Running the MCMC simulation +with such a prior, we find the walkers converge to a single solution after +$\sim 1000$ steps. Taking the converged sample (after the burn-period), in +Figure~\ref{fig_glitch_posteriors} we plot the resulting posteriors. Finally, +the maximum recovered $\twoFhat$ was found to be $\SingleGlitchGlitchtwoF$. +Comparing this to the fully-coherent $\twoFtilde=\SingleGlitchFCtwoFPM$, we see +that we have recovered the signal with negligible mismatch. + +\begin{figure*}[htb] \centering -\includegraphics[width=0.5\textwidth]{single_glitch_glitchSearch_corner} -\caption{} +\includegraphics[width=0.8\textwidth]{single_glitch_glitchSearch_corner} +\caption{The posterior distributions for the Doppler and glitch parameters +applied to a simulated glitching signal. The simulation parameters are listed +in Table~\ref{tab_single_glitch}. The extent of these plots does not correspond +to the prior range.} \label{fig_glitch_posteriors} +\end{figure*} + +\subsection{Monte-Carlo studies} +\label{sec_MCS_glitch} + +The glitch-robust statistic is similar to a semi-coherent +$\mathcal{F}$-statistic, but with additional degrees of freedom due to the +glitch parameters. As such, to compute a $p$-value (or false alarm rate) for a +given $\twoFhat$ we must understand the distribution of $\twoFhat(\Nglitches)$ in +noise. We follow the same prescription laid out in Sec.~\ref{sec_MCS_zoom}, +namely we repeatedly run a single glitch search on independent 100~day data +span with Gaussian noise. Taking the maximum $\twoFhat$ from each search, we +plot the resulting histogram in Figure~\ref{fig_single_glitch_twoFhat_histgram}. + +\begin{figure}[htb] +\centering +\includegraphics[width=0.5\textwidth]{single_glitch_noise_twoF_histogram} +\caption{The distribution of $\twoFhat$ } +\label{fig_single_glitch_twoFhat_histgram} \end{figure} + \section{Conclusion} \label{sec_conclusion} diff --git a/Paper/single_glitch_F0F1_grid_2D.png b/Paper/single_glitch_F0F1_grid_2D.png index 81d06390abb97886f7c7090f74af7defd22f0905..d0b2eea7e47eb37fb01b09b1962df6eb403066c7 100644 GIT binary patch literal 117892 zcmeAS@N?(olHy`uVBq!ia0y~yU|Yh#!19QLje&vTj$cy(0|Ns~v6E*A2L}g74M$1` z0|NtRfk$L91B1^~5N0er_hBjng93x6i(^Q|oHuuU^JT(kefw!W>5PhMc=^P26Vn>z z#fj{Ex%^?e^E89plJ{Ll4mjw${uI9Pi%n_I>$wiP)$Ui_1*S7!X<GAi0ndpiX?JGa z)8L-rc+glVC)M}#v=b`EX$!?>7sZH+nj1Ht@%dNZz-TF9Q}}3)yIk5G*T`w93~<1k z8Cwc+IYhwV?u`jd5GDhI!{SlSXc$nC4jK$Uzqq*g#*G_49#rm)(>F6WPcAJry>|V2 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