@@ -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 $\mathcal{R}\ge1$ as $\Tcoh^{i+1} > \Tcoh^{i}$.

where $\mathcal{R}\ge1$ 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