Optimized Search Optimized Search StrategiesStrategies

for for Continuous Gravitational Continuous Gravitational

WavesWavesIraj GholamiIraj Gholami

Curt Cutler, Badri KrishnanCurt Cutler, Badri Krishnan

Max-Planck-Institut fMax-Planck-Institut für Gravitationsphysikür Gravitationsphysik(Albert-Einstein-Institut) (Albert-Einstein-Institut)

Golm, GermanyGolm, Germany

GWDAW9 Friday, 17 December 2004

Annecy, France

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),},{,( )( nffParameters of Parameters of waveform:waveform:

Optimize ALL SKY searches

for UNKNOWNUNKNOWN PULSARSPULSARS

Due to detectors’ motion wrt Solar System Due to detectors’ motion wrt Solar System Barycenter;Barycenter;

=> => Amplitude and Phase modulationAmplitude and Phase modulation

Initial work was done by Brady and Creighton, who considered a two stage Hierarchical Search

P. Brady and T. Creighton, PRD 61, 082001 (2000)

Demodulation performed by F-Demodulation performed by F-StatisticStatistic

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Problem:Problem: Full coherent searches for unknown pulsars Full coherent searches for unknown pulsars

by using the present computational by using the present computational resources is not feasibleresources is not feasible

5obsp TN

Searching for young fast pulsars over the whole sky and including Searching for young fast pulsars over the whole sky and including two spin-down parameters just for 10 days data, requires a 10two spin-down parameters just for 10 days data, requires a 101717 Computer Flops.Computer Flops.

Why Hierarchical SearchWhy Hierarchical Search

Need an inexpensive sub-optimal Need an inexpensive sub-optimal techniques totechniques to discard uninteresting regions in discard uninteresting regions in parameter spaceparameter space

Optimal method:Optimal method: A full coherent A full coherent searchsearch

Example:Example:

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TheThe Stack-Slide methodStack-Slide method

Computing power spectrum for each segmentComputing power spectrum for each segment

N

TT obs

obsT

Break up data into shorter lengths (Break up data into shorter lengths (StacksStacks))

Phase correction in each stack using a mesh of correction points Phase correction in each stack using a mesh of correction points

sufficient to confine a putative signal to ~ 1 frequency bin in each stack.sufficient to confine a putative signal to ~ 1 frequency bin in each stack.

(One example of semi-coherent method)

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Frequency

Tim

e Shift the individual power spectra relative to each other (Shift the individual power spectra relative to each other (SlideSlide))

Add corrected power spectra in the frequency domainAdd corrected power spectra in the frequency domain

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Hierarchical Hierarchical SearchSearch Perform search in several (n) stages

At each stage consider only candidates that survived previous stage

Do finer search with more data near surviving candidates

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Two different methods of Two different methods of

incorporating new dataincorporating new data

Ist stage

IInd stage

IIIrd stage

1. 1. Re-use already-analyzed dataRe-use already-analyzed data

111 TNT

222 TNT

333 TNT

obsT

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2. Ignore previously analyzed data until final coherent 2. Ignore previously analyzed data until final coherent follow up (Fresh Mode)follow up (Fresh Mode)

time

Ist stage IInd stage IIIrd stage

111 TNT 222 TNT 333 TNT

obsT

For both above methods, we consider one final For both above methods, we consider one final coherent stage that search over entire data coherent stage that search over entire data coherently, but taking just those candidates coherently, but taking just those candidates could pass the last incoherent stage.could pass the last incoherent stage.

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The search parameters to The search parameters to optimizeoptimize

Ni: Number of stacks Ti: Time-baseline of each stack i: Mismatch in signal power

Variables for each incoherent stageVariables for each incoherent stage Variables for final coherent stageVariables for final coherent stage

NNcohcoh = 1 = 1 TTobs obs : Total observation time: Total observation time

coh coh : Mismatch in signal power: Mismatch in signal power

Given : • Amount of data, Tobs

• Weakest signal strength we wish to detect, h0

• Set false dismissal in each stage (= few %)

Number of incoherent stages : Number of incoherent stages : nn

We numerically solve for:• Optimal values of the search parameters

By minimizing Computational Cost subject to constraintsBy minimizing Computational Cost subject to constraints

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ResultsResults

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Example: All sky search for young fast Example: All sky search for young fast pulsarspulsars

(Taking Fresh data in each stage)(Taking Fresh data in each stage) Minimum spin-down age: Minimum spin-down age: 40 years40 years Maximum frequency searched over: Maximum frequency searched over: 1000 Hz1000 Hz False dismissal rates: False dismissal rates: 11stst stage stage = = 10%10%, , subsequent stages subsequent stages = =

1%1% Weakest detectable signal has Weakest detectable signal has 1-year SNR = 39.721-year SNR = 39.72 Total data available: Total data available: one yearone year

What is the Computational Cost to find such a pulsar?What is the Computational Cost to find such a pulsar?What is the best hierarchical strategy?What is the best hierarchical strategy?

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Optimized Computational Cost vs Optimized Computational Cost vs Number of StagesNumber of Stages

Based on taking Fresh data in each stageBased on taking Fresh data in each stage 1. 1. 1.320e+11.320e+15 (F)5 (F)

1.320e+11.320e+15 (O)5 (O)

2. 2. 4.245e+14.245e+13 (F)3 (F)

4.422e+14.422e+13 (O)3 (O)

3. 3. 9.764e+19.764e+12 (F)2 (F)

1.057e+11.057e+13 (O)3 (O)

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Stage (days) N Stage (days) N TTobsobs (days) (days) 1 2.630 1 2.630 0.76690.7669 6 6 15.77515.775 2 3.013 2 3.013 0.07380.0738 7 7 21.09121.091 3 32.838 3 32.838 0.81240.8124 10 10 328.384328.384

Computational Cost: (Computational Cost: (Number of OperationsNumber of Operations)) 1st1st 2nd 2nd 3rd3rd Coherent (stages) Coherent (stages) 1.828e+20 4.030e+19 1.384e+18 2.679e+16 1.828e+20 4.030e+19 1.384e+18 2.679e+16

Minimum Computational Power required: Minimum Computational Power required: 9.764e+12 9.764e+12 FlopsFlops

MaxT

Optimal search Optimal search ParametersParameters

Based on taking Fresh data in each stageBased on taking Fresh data in each stage

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Signal strength required for different spin-down ages Signal strength required for different spin-down ages

when we fix the Computational Cost to be 10when we fix the Computational Cost to be 101313 Flops. Flops.

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Computational Cost vs minimum spin-down Computational Cost vs minimum spin-down ageage

for fixed signal strengthfor fixed signal strength

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ConclusionsConclusions 3-stage hierarchical searches significantly better than 1 or 2 stages,3-stage hierarchical searches significantly better than 1 or 2 stages, but no point going beyond 3 stages.but no point going beyond 3 stages.

Have solved for the optimum search strategy, and foundHave solved for the optimum search strategy, and found the minimum computational cost for given sensitivity the minimum computational cost for given sensitivity

Have not considered cost of Monte Carlo simulations or memory Have not considered cost of Monte Carlo simulations or memory issuesissues

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Computational Cost vs DeltaT1 for fixed values of other Computational Cost vs DeltaT1 for fixed values of other parameters’ pointsparameters’ points

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Computational Cost vs N1 for fixed values of other parameters’ Computational Cost vs N1 for fixed values of other parameters’ pointspoints

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Computational Cost vs Mu1 for fixed values of other Computational Cost vs Mu1 for fixed values of other parameters’ pointsparameters’ points

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Computational Cost vs Mu2 for fixed values of other Computational Cost vs Mu2 for fixed values of other parameters’ pointsparameters’ points

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Computational Cost vs Mu3 for fixed values of other Computational Cost vs Mu3 for fixed values of other parameters’ pointsparameters’ points

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Differences between BC work Differences between BC work and the current workand the current work

They just considered 2 stages hierarchical searchThey just considered 2 stages hierarchical search They fixed the confidence level to be 99%, but if They fixed the confidence level to be 99%, but if

you reduce the number of candidates in the last you reduce the number of candidates in the last stage to the few one, therefore you can have a stage to the few one, therefore you can have a confidence level more than 99%.confidence level more than 99%.

They did not consider any final coherent stageThey did not consider any final coherent stage In current work we used the F-Statistic, but they In current work we used the F-Statistic, but they

ignored the polarization.ignored the polarization. They just considered re-use old data for each They just considered re-use old data for each

stagestage

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Full details of the search Full details of the search parametersparameters

For Fresh dataFor Fresh data ############ INPUT ######################## INPUT ############ Fresh modeFresh mode All sky searchAll sky search Minimum spindown age: 40.00 yearsMinimum spindown age: 40.00 years Maximum frequency searched over: 1000.00 HzMaximum frequency searched over: 1000.00 Hz False dismissal rates: 0.100 0.010 0.010 False dismissal rates: 0.100 0.010 0.010 Smallest signal that can cross thresholds: 5.00e-05Smallest signal that can cross thresholds: 5.00e-05 1.00-Year Signal to Noise Ratio (SNR): 39.721.00-Year Signal to Noise Ratio (SNR): 39.72 Total data available: 1.00 yearsTotal data available: 1.00 years Length of sfts if sft method used: 1800.00 secondsLength of sfts if sft method used: 1800.00 seconds ############ OUTPUT ####################### OUTPUT ########### False alarm rates: 6.047e+13 5.096e+08 1.000e+00 False alarm rates: 6.047e+13 5.096e+08 1.000e+00 False alarm prob.: 2.028e-06 3.167e-08 0.000e+00 False alarm prob.: 2.028e-06 3.167e-08 0.000e+00 Npf: 1.205e+11 3.205e+13 8.101e+20 Npf: 1.205e+11 3.205e+13 8.101e+20 Npc: 1.025e+08 1.100e+10 3.205e+13 Npc: 1.025e+08 1.100e+10 3.205e+13 NpfCoh: 1.597e+24NpfCoh: 1.597e+24 Thresholds: 2.631e+01 3.548e+01 4.609e+02 Thresholds: 2.631e+01 3.548e+01 4.609e+02 Computational Cost: Computational Cost: Incoherent Part: Incoherent Part: 1.344e+20 1.121e+18 1.323e+18 1.344e+20 1.121e+18 1.323e+18 Coherent Part: Coherent Part: 4.842e+19 3.918e+19 6.131e+16 4.842e+19 3.918e+19 6.131e+16 Total: Total: 1.828e+20 4.030e+19 1.384e+18 2.679e+16 1.828e+20 4.030e+19 1.384e+18 2.679e+16 Stage DeltaT (days) muMax N Tobs (days) SNR Stage DeltaT (days) muMax N Tobs (days) SNR 1 2.865 0.7669 5.507 15.7751 2.865 0.7669 5.507 15.775 8.26 8.26 2 2.865 0.0738 7.363 21.0912 2.865 0.0738 7.363 21.091 9.55 9.55 3 34.163 0.8124 9.612 328.3843 34.163 0.8124 9.612 328.384 37.66 37.66 Minimum computational power required: 9.764e+12 FlopsMinimum computational power required: 9.764e+12 Flops

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Full details of the search Full details of the search parametersparameters

For re-use old dataFor re-use old data ############ INPUT ######################## INPUT ############ Re-use old data modeRe-use old data mode All sky searchAll sky search Minimum spindown age: 40.00 yearsMinimum spindown age: 40.00 years Maximum frequency searched over: 1000.00 HzMaximum frequency searched over: 1000.00 Hz False dismissal rates: 0.100 0.010 0.010 False dismissal rates: 0.100 0.010 0.010 Smallest signal that can cross thresholds: 5.00e-05Smallest signal that can cross thresholds: 5.00e-05 1.00-Year Signal to Noise Ratio (SNR): 39.721.00-Year Signal to Noise Ratio (SNR): 39.72 Total data available: 1.00 yearsTotal data available: 1.00 years Length of sfts if sft method used: 1800.00 secondsLength of sfts if sft method used: 1800.00 seconds ############ OUTPUT ####################### OUTPUT ########### False alarm rates: 4.473e+13 3.344e+08 1.000e+00 False alarm rates: 4.473e+13 3.344e+08 1.000e+00 False alarm prob.: 1.410e-06 1.310e-14 0.000e+00 False alarm prob.: 1.410e-06 1.310e-14 0.000e+00 Npf: 1.284e+11 1.032e+14 2.278e+21 Npf: 1.284e+11 1.032e+14 2.278e+21 Npc: 1.033e+08 1.217e+10 1.032e+14 Npc: 1.033e+08 1.217e+10 1.032e+14 NpfCoh: 1.597e+24NpfCoh: 1.597e+24 Thresholds: 2.689e+01 5.565e+01 5.158e+02Thresholds: 2.689e+01 5.565e+01 5.158e+02 Computational Cost: Computational Cost: Incoherent Part: Incoherent Part: 1.461e+20 3.579e+18 8.418e+17 1.461e+20 3.579e+18 8.418e+17 Coherent Part: Coherent Part: 4.961e+19 4.102e+19 5.094e+16 4.961e+19 4.102e+19 5.094e+16 Total: Total: 1.957e+20 4.460e+19 8.927e+17 9.528e+15 1.957e+20 4.460e+19 8.927e+17 9.528e+15 Stage DeltaT (days) muMax N Tobs (days) SNR Stage DeltaT (days) muMax N Tobs (days) SNR 1 2.861 0.7618 5.605 16.0381 2.861 0.7618 5.605 16.038 8.32 8.32 2 2.861 0.0700 10.433 29.8522 2.861 0.0700 10.433 29.852 11.36 11.36 3 38.842 0.8074 9.404 365.2503 38.842 0.8074 9.404 365.250 39.72 39.72 Minimum computational power required: 1.057e+13 FlopsMinimum computational power required: 1.057e+13 Flops