Transcript houghlsc_nov_03 - LIGO Hanford Observatory

Searching for pulsars using the Hough transform
Badri Krishnan
AEI, Golm
(for the pulsar group)
LSC meeting, Hanford
November 2003
LIGO-G030580-00-Z
Outline
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The Hough transform
The frequency-time pattern
Hough map statistics
Frequentist upper limits
Code status
Conclusions
Need for a hierarchical scheme
• Want to perform an all sky search for unknown pulsars
Parameters involved : {a,d,f0,fi}
• If we want to search for one spin-down parameter over an
observation time of say 6 months and frequency band of a
few 100 Hz using the optimal search - would need a 1016
Flops computer
• Idea of a hierarchical search: perform a first sub-optimal
search to reduce parameter space volume and use optimal
method only for small number of candidates in parameter
space
Hierarchical Hough transform strategy
Pre-processing
raw data
GEO/LIGO
Divide the data set in N chunks
Template placing
Coherent search (a,d,fi)
Construct set of short
FT (tSFT)
in a frequency band
Incoherent search
Peak selection
in t-f plane
Set upper-limit
Hough transform
(a, d, f0, fi)
Candidates
selection
Candidates
selection
Incoherent Hough search:
Pipeline for S2
Pre-processing
raw data
GEO/LIGO
Divide the data set in N chunks
Construct set of short FT
(tSFT=1800s)
Incoherent search
Peak selection
in t-f plane
Hough transform
(a, d, f0, fi)
Want to perform an all sky
search over a frequency
range of ~ few 100 Hz
including one spin down
parameter
Candidates
selection
Set upper-limit
The Hough Transform
 Looks for patterns in frequency-time plane
Expected pattern depends on
{a,d,f0,fi}
c
f (t )  f 0 (t )  f 0 (t )
 nˆ
v (t )

Resolution in sky position ~ 10-2rad
Spindown resolution ~ df / Tcoh
~ 1.6 x 10-10 Hz/s
f
n
t
a
d
Basic pipeline for a single stage
Normalized power
r
Break up data into
N segments and combine
them incoherently
r0
Used running-median to estimate noise floor
Frequency
Thresholds chosen by optimizing false
dismissal for fixed false alarm rate
n  n0 
n
Candidates
For next stage
no
a
d
Noise only case
Sensitivity of the Hough search
 For coherent directed search with false alarm of 1% , false dismissal
rate of 10% :
S f 
h0  11.4 n GW
Tobs
 For incoherent Hough search with 1% false alarm and 10% false
dismissal rate for large N and looking at a single pixel on the celestial
sphere:
h0 
10.0
N 1/ 4
Sn  f GW 
S f 
10.0 N 1/ 4 n GW
Tcoh
Tobs
 Sub-optimal nature of Hough search leads to loss in sensitivity by a
factor of N 1/ 4
 Currently code is being used with ~ 1900 SFTs each 1800s long
 In future will also be run with input from F statistic code which allows
a smaller value of N and larger Tobs
Statistics of the number counts
 Distribution of power: c2 with 2 d.o.f. with non centrality parameter

 Detection probability :
~
| h ( f gw ) |2
Tcoh S n ( f gw )

   p ( r |  ) dr
ro
 This statistic changes between SFTs because of non-stationarities in
the noise and also due to amplitude modulation of signal. Thus
detection probability changes with time. Thus to set upper limits we
have to perform Monte Carlo simulations
 To obtain a number count value n, we must have selected n SFTs and
rejected (N-n) . The probability for this happening is
N
1
p(n | N , h, r 0 , n0 ) 
i1i2 ...in (1  in1 )...(1   N )

n!( N  n)! im 1
Reduces to binomial distribution when ’s are same
Example: Run Hough code in 1Hz frequency band
(263.5-264.5Hz), sky patch 0.5 rad on a side centered on
the South pole with 1887 SFTs each 1800sec long:
Observed Maximum number count:
n  443
Setting the upper limit.
Frequentist approach.
n*=395  =0.295
 h095%
0.95 
95 %
 pn|h 
N
9 5%
0
n  n*
The Monte-Carlo simulation
 Aim is to set an upper limit for every 1Hz band
 Run the Hough driver over the 1Hz band for a large sky-patch and
obtain the Hough maps, the distribution of the number counts, and files
with the detector velocity at the timestamps of the SFTs.
 To obtain upper limits we must obtain the number count distribution
with a signal injected and find the value of the amplitude h0 for which
the number count is at least n  with 95% probability.
 Generate signals with random values of sky position within the chosen
sky-patch, frequency within the 1Hz band, and pulsar parameters i,y,
f0 but with fixed value of the amplitude h0.. Inject these signals into
SFTs and find the Hough number count at the correct point in
parameter space.
 Repeat for a range of amplitudes and find the value of h0 which gives
the right confidence level for the observed value of n 
Current status of code
 Driver code ready: Stand alone code accepts SFTs as input and
produces Hough maps for a reasonably large sky patch and includes
spindown parameters.
 Code for Monte Carlo signal injection is also ready and preliminary
results are available.
To-do list
 Perform Monte-Carlo runs over a large frequency range and large
sky-patch for a range of signal amplitudes and find the upper limit :
by march lsc