1. J. Ellis, F. Jenet, & M. McLaughlin
Practical Methods for Continuous Gravitational Wave Detection Using Pulsar Timing Data
J. Ellis, F. Jenet, & M. McLaughlin

2. Overview
Aim: “Determine the most sensitive practical detection technique for continuous GW sources in PTA data”
Methods compared:
Matched filtering (various realisations)
Power spectral summing
Figures of merit:
Minimum detectable amplitude (relative to other methods)
Compute load
Important simplifying assumptions:
No frequency evolution
Perfect sinusoids (no eccentricity)
Data: white, regularly-sampled, “stationary”
Builds from previous work:

3. Data used
Up to 100 pulsars
Uniform sky distribution, random distance distribution (0.5-3kpc)
10 years, 250 TOAs , regular (~bimonthly) sampling
White residuals, ns RMS
“In real pulsar timing data, the residuals will be unevenly sampled and the noise may have various red components. In addition, the pulsar timing residuals will not be stationary, as a quadratic must be fit out of the data to account for the spindown of the pulsar. Specifically, our definition of the inner product in Eq. 16 no longer holds as we will need to include the covariance matrix of the data and incorporate a linear operator that takes into account this fitting. This […] will be addressed in future papers. However, here we will deal with the simple case to illustrate the efficacy of the studied search techniques on a data set of optimal quality.”
Usual caveats included which is to say “we’re simply comparing methods not doing any real data analysis”

4. Matched filtering process
Define a vector of search parameters:
Compute inner product of data (x) with template (r):
Define a (log-)likelihood function:
False alarm probability?
Maximise this likelihood function using various l realisations.
Binary orientation parameters, frequency, mass, distance, and a vector of N pulsar distances.
Likelihood function includes a hypothesis test, so is some relative likelihood of probability of signal vs. probability of having no signal
Determine threshold value over which this statistic must exceed to claim a detection

5. NGWtrials NDtrials^(Npulsars) 10400
“Coherence”
N templates needed
e.g. for NGWt = NDt = Np = 100
Notes
FULL
(earth+pulsar terms)
Fully coherent
NGWtrials NDtrials^(Npulsars)
10400
Pulsar distance added as search term. Basically impossible for low expected SNR signals
EARTH-ONLY
Pulsar term = noise source
NGWtrials
100
Lower SNR detection than FULL method. D and W strongly correlated.

6. Full filter
Earth term only

7. NGWtrials NDtrials^(Npulsars) 10400
“Coherence”
N templates needed
e.g. for NGWt = NDt = Np = 100
Notes
FULL
(earth+pulsar terms)
Fully coherent
NGWtrials NDtrials^(Npulsars)
10400
Pulsar distance added as search term. Basically impossible for low expected SNR signals
EARTH-ONLY
Pulsar term = noise source
NGWtrials
100
Lower SNR detection than FULL method. D and W strongly correlated.
PAIRWISE
(earth + sum-of-likelihood-stat-for-all-pulsar-pairs)
Semi-coherent
NGWtrials  NDtrials2  Npulsars(Npulsars-1)
~1010
Positional inaccuracies remain unless large Npulsars

8. Power Spectral Summing
Similar to Yardley et al. (2010)
Claim detection when this value > threshold based on false alarm rate
“Bad”: gives detection but no source parameters
“Good”: no reliance on templates
Npsr
Calculate power spectrum, sum the power weighted by dataset variance, look for maximum value over all f bins

9. Result
In the case that you already know about a SMBH and are just searching for *that one*.
Lower false-alarm probability assuming only one template, matching the data

10. Result
The “best we’ll ever do” might not be that good! (Note, longer Tobs not considered)
In the case that you already know about a SMBH and are just searching for *that one*.
Pairwise/Full: Similar results for current experiments!

11. lower false-alarm rate
Summary
Your analysis method choice will depend on:
Your compute cluster size
Your expected signal strength
Whether you need to locate the position really well
Whether you know anything about your target vs. blind sky search
How many pulsars you have
fewer templates =
lower false-alarm rate