optimized hierarchical continuous-wave searches badri krishnan curt cutler, iraj gholami aei, golm...
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Optimized hierarchical continuous-wave searches
Badri KrishnanBadri KrishnanCurt Cutler, Iraj GholamiCurt Cutler, Iraj Gholami
AEI, GolmAEI, Golm
G040155-00-Z
LSC meeting, March 2004 ASIS session
Motivation
Full coherent searches for unknown pulsars Full coherent searches for unknown pulsars not computationally feasiblenot computationally feasible
Require incoherent, sub-optimal methodsRequire incoherent, sub-optimal methods Illustrative example: The stack slide searchIllustrative example: The stack slide search
5obsp TN
Example: Searching for young, fast pulsars over the whole sky and including two spin-down parameters for 10 days data requires a 1017 Flops computer
P.Brady and T.Creighton, PRD 61, 082001 (2000)
The Stack-Slide method Basic idea: Take the Fourier transform of each segment and track the Basic idea: Take the Fourier transform of each segment and track the
Doppler shift by adding power in the frequency domain (Stack and Doppler shift by adding power in the frequency domain (Stack and Slide)Slide)
First step : Break up data into N segmentsFirst step : Break up data into N segments
Calculate power spectrum or DeFT for each segmentCalculate power spectrum or DeFT for each segment
NTT obs
obsT
Frequency
Tim
eAdd power after frequency bins are shifted according to the time-frequency pattern
General hierarchical pipeline Break data into segments
Analyze eachsegment coherently
Combine segments incoherently
Select candidates
Analyze candidatescoherently using all available data
Acquire more data
Detection orupper limit
Incoherent stepcan be either stack-slide orHough
The search parameters
Ni : Number of stacks Ti : Time-baseline of each stack i : Mismatch in signal power Xi: Threshold on summed power
Variables for each incoherent stage Variables for final coherent stage
Tobs : Total observation time coh : Mismatch in signal power
Given : • Available computer power C0
• Data of a certain time duration Tobs
• Weakest signal strength we wish to detect h0
• Desired confidence level
Number of incoherent stages : n
We want to know:• Optimal values of the search parameters
Minimize computational cost subject to constraints
Two different search modes
Take fresh data in each stageTake fresh data in each stage
Re-use old data Re-use old data
time
Ist stage IInd stage IIIrd stage
Ist stage
IInd stage
IIIrd stage
False alarm rate for ith stage
Determines number of candidates for next stage to analyze
False dismissal rate for ith stage
Determines weakest signal that can be detected
StatisticsSummed power follows chi-square distribution with 2N d.o.f
N
k k0
X
X
dhP
dhP
0)|(
)0|(
False alarm and false dismissal rates :
Total false alarm rate always determined by final coherent stage
Choose thresholds for each stage by fixing false dismissal rate
),,,,(),,,,( 0)0()0(
0 hTNXXXhTN iiiiiiiiiii
Template countingParameter space metric calculated by Brady & Creighton
)(
2/
1111
)2(4
!3.01max
2/
2/
2/)1(min
2/max
2/)3(2max
222max
2max
0
min1
{0,1,2,3}s
NGss
sTfN
CBAs
fM
MckrGNMN
ss
s
sss
ss
s
s
s
k s
kk
sssp
63
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00243.0)(
0708.0)(
524.0)(1)(
NNG
NNG
NNGNG
35
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118.0
1046.0
014.0
NdayTC
dayTB
A
Number of points in coarse grid : Number of points in fine grid : ),,(
)1,,(
max
max
NTNN
NTNN
ppf
ppc
Computational costs
Computational cost of each stage is essentially cost of calculating FFTplus cost of summing the power
For each point in parameter space, number of floating point operations for first, intermediate and coherent stages:
)2(log6
)1(2log3
2
612log6
max2max)()(
)()()(
)1()()(
)()()(
max2max)1(
)()()()1()(
)1()1(
)1()1(
max2)1()1()1(
max)1(
obsobsnpf
nobs
cohpfobsncoh
iipc
i
iipfi
ipf
ipc
iiii
pcpfpc
TfTfNTNT
FC
TNNNN
TffN
NTNFC
NNNNTfNTNfC
F(i) = Number of candidates which survive the ith stage
Optimization strategy
• Basic strategy is to minimize Total computational cost subject to constraint that amount of analyzed data is lesser than available data
• False alarm rate is not really a constraint because false alarm rate is set by final coherent step
• Computationally limited searches can only see strong signals and when we do see them, it is usually easy to build up confidence • Want to analyze data in (roughly) real time• Function to be optimized is
obsa TT
n
coh
SThNFA
20)( exp
power nalComputatio )( obsa TTSf
S(x) = 1 if 0 < x < 1 and very large otherwise
Preliminary resultsExample of search criteria :
• Total observation time : 1 year• Signal strength we wish to detect : • Allowed false dismissal rate for each stage : 1%• Mismatch in coherent stage : 0.10• Don’t reuse old data• Max number of spindowns included : 3• All-sky search• Largest frequency searched over : fmax = 1000 Hz• Smallest spindown age : min = 40 yr
15/20 nobs STh
Optimization carried out by simulated annealing and amoeba method
Two stage search with coherent follow-up: Length of each stack : 0.37 days ; 0.25 days Number of stacks : 199 ; 956 Observation time : 73 days ; 240 days Power mismatch : 0.51 ; 0.23 Computational requirement : 8.8 x 1013 Flops
Three stage search with coherent follow-up: Length of each stack : 0.17 days ; 0.15 days ; 0.48 days Number of stacks : 511; 1028 ; 261 Observation time : 84 days ; 149 days ; 125 days Power mismatch : 0.48 ; 0.33 ; 0.02 Computational requirement : 7.8 x 1012 Flops
Single stage search with coherent follow-up: Length of each stack : 1.9 days Number of stacks : 39 Observation time : 74 days Power mismatch : 0.53 Computational requirement : 6.3 x 1016 Flops
Conclusions• Optimization scheme for hierarchical stack-slide search presented• Tells us what the search pipeline parameters must be• Expect similar results for hierarchical Hough search also• Does not consider cost of Monte Carlo simulations or memory issues• Shows that hierarchical schemes are absolutely essential for large parameter space blind searches
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