Data analysis for science Lee Samuel Finn Center for Gravitational Wave Physics 28 October 2002 Grav. Wave Source Simulation & 1 Data Analysis Workshop
Overview • “Data analysis” goals • Distinguishing signal from noise: Examples • What this means for you 28 October 2002 Grav. Wave Source Simulation & 2 Data Analysis Workshop
Data Analysis v. Source Simulation • Source Simulation – Goal: Identify source science impressed on gravitational wave signal – Important question: how is source science encoded in radiation? • Data Analysis – Goal: • Distinguish between signal and noise • Discriminate to identify source science in signal – E.g., source parameters like ns/bh masses, or spins,population statistics, etc. • Interpretation: place observations in (astro)physical context – Important question: how to maximize contrast between signal, noise? 28 October 2002 Grav. Wave Source Simulation & 3 Data Analysis Workshop
What distinguishes? • Measure of distinction: • What is d? likelihood or sampling – Depends on noise, sought- distribution for signal – P(d| Θ , I) : prob of – We’ll return to this point! observing d given Θ , I • P(d| 0 , I) – d – quantity (not necessarily – Probability that observation h[t k ]) calculated from is of noise alone (no signal measurement at detector • P(d| Θ , I) – Θ - all the parameters that – Probability observation is of distinguish among signals noise + signal Θ • Amplitude, population, • Likelihood of d: “odds” etc. signal v. noise: – I - everything relevant about detector and noise – Λ = P(d| Θ , I)/P(d| 0 , I) Goal: Make probabilities P(d| Θ , I), P(d| 0 , I) as different as possible! 28 October 2002 Grav. Wave Source Simulation & 4 Data Analysis Workshop
The Detection Game • The Game: – Observe h[t k ] – Calculate d – Calculate P(d| 0 , I) – If P(d| 0 , I) < P 0 , buy tux, tickets to Stockholm • Choice P 0 : false alarms, false dismissal – False alarm prob: frequency with which noise alone (no signal) would give d such that P(d| 0 , I) < P 0 – False dismissal prob: frequency with which noise + signal Θ would give d such that P(d| 0 , I) > P 0 – Efficiency := 1 – (false dismissal prob) • Note: the more different P(d| 0 , I) , P(d| Θ , I), the smaller the false dismissal for a given false alarm 28 October 2002 Grav. Wave Source Simulation & 5 Data Analysis Workshop
Expressing the contrast: False alarm v. efficiency • Guessing: pick a random number between 1 and 100 1 – If less than N+1 then say detected • False alarm probability? efficiency – N/100 • Efficiency? – N/100 • Close to diagonal is close to random guessing • Better tests have greater lift off diagonal 0 1 False alarm – High efficiency for low false alarm probability 28 October 2002 Grav. Wave Source Simulation & 6 Data Analysis Workshop
Clearing the Clutter • Goal: make contrast P(d| Θ , I)/P(d|0,I) large – How? Can’t choose, change signal, noise – Only possibility: choose d! – Choice of d based on signal characteristics and their uncertainty (in nature or knowledge) • Examples: – Stochastic gravitational wave signal – Periodic signals – Gravitational waves from γ -ray bursts – Bursts: things that go “bump” in the night 28 October 2002 Grav. Wave Source Simulation & 7 Data Analysis Workshop
Stochastic gravitational wave signal • “Signal” is noise – How do we distinguish gw contribution to total “noise”? • What’s distinguishes signal, instrumental contributions? – Physically distinct detectors respond coherently to gravitational waves • Quantity that distinguishes – Cross correlation: ∫∫ ( ) h 2 f ( ) Q t 1 − t 2 ( ) d = dt 1 dt 2 h 1 t – Choose kernel Q to extremize contrast in d between signal present, absent cases • Key point: look for, choose measure that draws the greatest contrast between signal, noise 28 October 2002 Grav. Wave Source Simulation & 8 Data Analysis Workshop
(nearly) Periodic Signals • Signal • Identify a quantity that – s(t) = A sin [ Φ (t)+ φ 0 ] large for signal, small – Know Φ (t) accurately, for noise: unknown φ 0 , A ρ 2 = x 2 + y 2 • What distinguishes? T x = 1 – Noise not periodic with ∫ ( ) cos Φ ( t ) dt h t known phase T 0 – Signal has no power T except at frequencies y = 1 ∫ ( ) sin Φ ( t ) dt h t near d Φ /dt T 0 – Phase φ 0 not important Key point: phase must be known s.t. ∆Φ << π for all t 28 October 2002 Grav. Wave Source Simulation & 9 Data Analysis Workshop
The γ -ray Burst Story Hypernovae; • Key Facts: Black hole + collapsars; NS/BH, debris torus – Multiple, indistinguishable He/BH, WD/BH triggers mergers; AIC; … – Rapidly rotating (Jc/GM 2 ~1) BH – γ -ray production far from BH – Sources likely too distant γ -rays generated by (z~1) to detect individuals internal or external shocks – Gravitational wave Relativistic strength, time dependence fireball unknown 28 October 2002 Grav. Wave Source Simulation & 10 Data Analysis Workshop
What science might we learn? Hypernovae; • Progenitor mass, angular Black hole + collapsars; NS/BH, momentum debris torus He/BH, WD/BH – Expect radiated power to peak at mergers; AIC; … frequency related to black hole M, J • Differentiate among progenitors – Radiation originating from stellar collapse, binary coalescence have different gw intensity, spectra • Internal vs. external shocks γ -rays generated by – Elapsed time between gw, g-ray internal or external burst depends on whether shocks are internal or external shocks • Analysts goal: describe an analysis Relativistic that brings science into contrast fireball – Spectra, elapsed time between g, gw bursts 28 October 2002 Grav. Wave Source Simulation & 11 Data Analysis Workshop
The LIGO Lock-in Incident waves give h ij correlated detector output Integrated cross-correlated detector output: T H − t L − ′ ( ) s L t γ ( ) K t − ′ ( ) s H , s L ≡ dtd ′ t s H t γ t t ∫ ∫ 0 Collect catalog of <s,s> x off = n H , n L associated, not associated x on ≡ n H + h H , n L + h L with GRBs, Source ≡ x off + h H , n L + n H , h L + h H , h L population x off = µ off Collect & compare on- average burst, off-burst catalogs: 28 October 2002 Grav. Wave Source Simulation & 12 x on = µ off + h H , h L Data Analysis Workshop
Are distributions different? Incident waves give h ij correlated detector output • Accuracy of estimated means increases the more samples are available x off = µ off • Gather enough samples and any difference x on = µ off + h H , h L becomes distinguishable from zero • Ref: Finn et al. Phys. Rev. D. 60 (1999) 121101 28 October 2002 Grav. Wave Source Simulation & 13 Data Analysis Workshop
Discovery: Things that go bump in the night • How to discover? • What distinguishes signal, noise? – Signal time-limited – Signal (s), noise (n) uncorrelated: <sn>=0 • Important: <(n+s) 2 > greater than <n 2 > • Analysis method: look for anomalies – Where is detector output unusual? – Where do noise statistics change? 28 October 2002 Grav. Wave Source Simulation & 14 Data Analysis Workshop
Example: Power • How does power in a frequency band evolve? – |h(f)| 2 measured over short intervals of time • Spectrogram – Band-limited signals – Signals that exhibit interesting “time- frequency” behavior • Refs. – Anderson et al. Phys.Rev. D63 (2001) 042003 (gr- qc/0008026) – Sylvestre. Phys Rev. in press. (gr-qc/0210043) 28 October 2002 Grav. Wave Source Simulation & 15 Data Analysis Workshop
Example: Change-point analyses • Look for places where statistics change – Statistics? Mean, variance – Needn’t assume any particular mean, variance: look for changes • Refs. – Poisson statistics cf. Scargle Ap. J. v.504, p.405 – Time series: Finn & Stuver in progress 28 October 2002 Grav. Wave Source Simulation & 16 Data Analysis Workshop
What does this mean for you? • Source simulator’s job – Identify science reflected in the gravitational waves • The science is the signal! – Find wave description draws the science into sharpest focus • Frequency, bandwidth, duration, polarization, …? – Connect the source to an astrophysical context • Amplitude, rate, space density, etc. – Don’t forget uncertainties! • The data analyst’s job – Develop analyses that makes science stand-out • The science is the signal! – Provide astrophysical interpretation of observations 28 October 2002 Grav. Wave Source Simulation & 17 Data Analysis Workshop
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