1. Status of the Hough CW search code - Plans for S2 -
A.M. Sintes,
B. Krishnan, M.A. Papa
Universitat de les Illes Balears, Spain
Albert Einstein Institut, Germany
This talk is in the content of continuous gravitational waves emitted by pulsars.
This work is done in collaboration with Map, Badri
Proposal Started from Rome group, worked for about 1.5 years together on the basic aspects, actual code development and pipeline implementation independent – somewhat related to different deadlines for the projects.
LSC Meeting
Hannover, August 2003
LIGO-G Z

2. Outline The hierarchical Hough search. Pipeline for S2
Statistics of the Hough maps. Frequentist analysis
Code status
Results on simulated & H1 data

3. The concept of the Hough transform
The idea is to perform a search over the total observation time using an incoherent (sub-optimal) method:
We propose to search for evidence of a signal whose frequency is changing over time in precisely the pattern expected for some one of the parameter sets
The method used is the Hough transform
The method we propose is to perform, at some stage, a search over the total observation time using an incoherent sub-optimal method able to do a search efficiently over a large parameter space.
In particular we would search for evidence…
The method used here is a particular implementation of the Hough transform. The basic idea is the following:The input is a set of points in a time-frequency plane.The output: histogram in the parameter space (f0, f1) and sky location.
What we do, is for each point in the t-f plane we search all those pixels in parameter space that are consistent, and we enhance their number count by unity. After having transformed all the data points if a signal is present we would find a significant clustering in the pixel corresponding to the source parameters f
{a,d,f0,fi} t

4. Hierarchical Hough transform strategy
raw data GEO/LIGO
Pre-processing
Divide the data set in N chunks
Template placing
Coherent search (a,d,fi)
in a frequency band
Construct set of short FT (tSFT)
Incoherent search
Hough transform (a, d, f0, fi)
Peak selection
in t-f plane
Candidates
selection
Similar strategy is pursued by the Virgo Rome Group
Set upper-limit
Candidates
selection

5. The time-frequency pattern
SFT data
Demodulated data
For SFT data the frequency evolution of the signal depends on the spin-down parameters and on the Doppler modulation, thus on the frequency of the signal, the instantaneous relative velocity between source and detector, and the location of the source in the sky.
If we are using the output of a coherent search, I.e., data that has been demodulated with respect some parameters: a particular sky location and spin down. The frequency pattern will now related to the miss-match between the source parameters and the ones used in the coherent search. This will be different from the previous SFT case, although because of similarities in the governing equation, we can use the same strategy to analyze the two different kinds of input data, and also the same code we’d develop handles both kinds of data.
Time at the SSB for a given sky position

6. Incoherent Hough search: Pipeline for S2
Pre-processing
Divide the data set in N chunks
raw data GEO/LIGO
Construct set of short FT (tSFT<1800s)
Candidates
selection
Incoherent search
Hough transform (a, d, f0, fi)
Peak selection
in t-f plane
Develop a search code able to analyze a large sky area or full sky (on a coarse grid) in a given frequency band, with possible 1 spin-down parameter. The main purpose is to determine a detail implementation of the incoherent Hough search using real IFO data, set an strategy for setting an upper limit, and candidate selection (since this will be a module in a more sensitive hierarchical coherent-incoherent-coherent-… search procedure).
Issues to be addressed mainly concern automatic handling of noise features and robust PSD estimation.
Future plans include performing an incoherent search using F statistic as input data (but not for S2).
The (LAL) Hough routines are general and can already handle this case.
Set upper-limit

7. Peak selection in the t-f plane
Input data: Short Fourier Transforms (SFT) of time series (Time baseline: 1800 sec, calibrated)
For every SFT, select frequency bins i such exceeds some threshold r0
 time-frequency plane of zeros and ones
p(r|h, Sn) follows a 2 distribution with 2 degrees of freedom:
The false alarm and detection probabilities for a threshold r0 are

8. Hough statistics
After performing the HT using N SFTs, the probability that the pixel {a,d,f0,fi} has a number count n is given by a binomial distribution:
The Hough false alarm and false dismissal probabilities for a threshold n0
 Candidates selection

9. Frequentist Analysis 30% 90%
Perform the Hough transform for a set of points in parameter space l={a,d,f0,fi} S , given the data:
HT: S  N
l  n(l)
Determine the maximum number count n*
n* = max (n(l)): l  S
Determine the probability distribution p(n|h0) for a range of h0
30%
90%

10. Frequentist Analysis
Perform the Hough transform for a set of points in parameter space l={a,d,f0,fi} S , given the data:
HT: S  N
l  n(l)
Determine the maximum number count n*
n* = max (n(l)): l  S
Determine the probability distribution p(n|h0) for a range of h0
The 95% frequentist upper limit h095% is the value such that for repeated trials with a signal h0 h095%, we would obtain n  n* more than 95% of the time
Compute p(n|h0) via Monte Carlo signal injection, using l  S , and 0 [0,2],  [-/4,/4], cos [-1,1].

11. Code Status Stand-alone search code in final phase
Test and validation codes (under CVS at AEI)
Hough routines are part of the LAL library:
Future plans include performing an incoherent search using F statistic as input data (but not for S2).
The (LAL) Hough routines are general and can already handle this case.
We make use of an efficient implementation of a HT: stereographic projection is used to tile the sky (to avoid unnecessary distortions and facilitate calculations) .use of LUT. computational savings if spin-down search parameters are considered. Driver code automatically updates LUTs in the search band
Statistical analysis of the Hough maps. Optionally output maps, candidates, loud events, etc.

12. Code Status II
Auxiliary functions C-LAL compliant (under CVS at AEI to be incorporated into LAL or LALApps):
Read SFT data
Peak Selection (in white & color noise)
Statistical analysis of the Hough maps
Compute <v(t)>
Under development:
Implementation of an automatic handling of noise features. Robust PSD estimator.
Monte Carlo signal injection analysis code.
Condor submission job.
Input search parameter files

13. Validation code -Signal only case- (f0=500 Hz)
bla
South pole half a sky
As seen on the Stereographic projected plane
60 time stamps of 1800s, separated by a day Tobs~60 days

14. Validation code -Signal only case- (f0=500 Hz)

15. Signal only case II

16. Signal only case II

17. Results on simulated data
Statistics of the Hough maps
f0~300Hz
tobs=60days
N=1440 SFTs
tSFT=1800s
a=0.2231
<n> = Na= n=(Na(1-a))=

18. Noise only case

19. Number count probability distribution
1440 SFTs
1991 maps
58x58 pixels

20. Comparison with a binomial distribution

21. Set of upper limit. Frequentist approach.
95 %
n*=395  =0.295
 h095%

22. H1 Analysis – the SFTs 1887 Calibrated SFT’s H1
TSFT = 1800s, Band : Hz
26 SFT, 0.95 or 1.05

23. H1 Results 264-265 Hz Input: 1861 SFT 0.5x0.5 srad Output: 19811 maps
(51x51 sky locations)
N=1861 SFTs
a=0.2019
Mean number count=
<n> = Na= n=(Na(1-a))=

24. H1 Results 264-265 Hz “upper limit estimate”
95% n*
Loudest event n*=473
95 % upper limit
n*=473  =0.2715
h095% ~ 5x ?
Very preliminary
We have to do monte-carlos
Moreover, I do not trust how we got h0, we should repeat the exercise!