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Analysis of Fermi LAT data. Day 2

Analysis of Fermi LAT data. Day 2

Introductory lecture on the analysis of Fermi LAT data. Day 2.

• Basic theory of likelihood analysis
• Likelihood fit → Characterize spectra of a source
• obtain spectral energy distribution (SED)
• build a light curve

These are the tutorials for the hands on, practical session on the analysis of *Fermi* Large Area Telescope (aka LAT) gamma-ray observations for the São Paulo School of Advanced Science on High Energy and Plasma Astrophysics in the CTA Era. The goal of this activity is to get you started on the analysis of Fermi LAT data while giving you a concrete overview of the steps involved.

Link for material: https://github.com/rsnemmen/Fermi-LAT-tutorial

Rodrigo Nemmen

May 27, 2017
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  1. Analysis of Fermi LAT data Analysis of Fermi LAT data

    Rodrigo Nemmen IAG USP May 24, 25 2017 High energy school @ IAG USP Likelihood tutorial
  2. Overview of activities Day 1 Day 2 Introduction, overview Obtaining

    and exploring LAT data Inspecting the data: count map Basics of modeling Likelihood analysis of a blazar Create a SED Produce a light curve R. Nemmen, Fermi LAT hands-on ✅
  3. Structure of this talk Basic theory of likelihood analysis Likelihood

    fit → Characterize spectra of a source obtain spectral energy distribution (SED) build a light curve see jupiter notebook hands-on R. Nemmen, Fermi LAT hands-on
  4. Model fitting: given this model, what parameters best fit my

    data? Model selection: given two potential models, which better describes my data? Fundamental questions of science statistics R. Nemmen, Fermi LAT hands-on
  5. Pi = 1 p 2⇡✏2 i e (yi yM )2

    2✏2 i Probability for a single measurement
  6. Pi = 1 p 2⇡✏2 i e (yi yM )2

    2✏2 i Probability for a single measurement L = Y i Pi = Y i 1 p 2⇡✏2 i e (yi yM )2 2✏2 i Likelihood:
  7. Pi = 1 p 2⇡✏2 i e (yi yM )2

    2✏2 i Probability for a single measurement L = Y i Pi = Y i 1 p 2⇡✏2 i e (yi yM )2 2✏2 i Likelihood:
  8. Pi = 1 p 2⇡✏2 i e (yi yM )2

    2✏2 i Probability for a single measurement L = Y i Pi = Y i 1 p 2⇡✏2 i e (yi yM )2 2✏2 i Likelihood: Log-likelihood: log L = 1 2 N X i=1 ( log(2⇡" 2 i ) + [yi yM (xi; ✓)] 2 " 2 i )
  9. Experiment counting events during a fixed interval: Poisson distribution n:

    number of detected events m: expected number (model)
  10. Experiment counting events during a fixed interval: Poisson distribution n:

    number of detected events m: expected number (model) P = mne m n! Probability of observing n events
  11. Experiment counting events during a fixed interval: Poisson distribution Standard

    deviation = p n n: number of detected events m: expected number (model) P = mne m n! Probability of observing n events
  12. Experiment counting events during a fixed interval: Poisson distribution P

    = mne m n! γ e- e+ γ e- e+ γ e- e+ Standard deviation = p n
  13. Experiment counting events during a fixed interval: Poisson distribution P

    = mne m n! γ e- e+ γ e- e+ γ e- e+ Detected: n = 3 Standard deviation = p n
  14. Experiment counting events during a fixed interval: Poisson distribution P

    = mne m n! γ e- e+ γ e- e+ γ e- e+ Detected: n = 3 Noise = sqrt(n) = 1.7 Standard deviation = p n
  15. Experiment counting events during a fixed interval: Poisson distribution P

    = mne m n! γ e- e+ γ e- e+ γ e- e+ Detected: n = 3 Noise = sqrt(n) = 1.7 Expected: m = 2 Standard deviation = p n
  16. L = Y i Pi = Y i mni i

    e mi ni! Likelihood for Poisson distribution
  17. L = Y i Pi = Y i mni i

    e mi ni! Likelihood for Poisson distribution
  18. L = Y i Pi = Y i mni i

    e mi ni! Likelihood for Poisson distribution L = Y i e mi Y i mni i ni! = e Npred Y i mni i ni! Binned likelihood
  19. L = Y i Pi = Y i mni i

    e mi ni! Likelihood for Poisson distribution L = Y i e mi Y i mni i ni! = e Npred Y i mni i ni! Binned likelihood L = e Npred Y i mi For bin sizes infinitesimally small, ni = 1 Unbinned likelihood
  20. L = Y i Pi = Y i mni i

    e mi ni! Likelihood for Poisson distribution L = Y i e mi Y i mni i ni! = e Npred Y i mni i ni! Binned likelihood log L = X i log mi Npred Log-likelihood L = e Npred Y i mi For bin sizes infinitesimally small, ni = 1 Unbinned likelihood
  21. log L = X i log mi Npred We need

    to maximize this likelihood function numerically total predicted number of counts Z ROI dE dp dt Z ROI dE dp dt instrument response function source model ( )( ) source model = point sources diffuse sources other stuff + +
  22. Is there a source in a given position of the

    sky? TS = 2 log ✓ L max , 0 L max , 1 ◆ maximum likelihood: model with additional source maximum likelihood: model without an additional source (the “null hypothesis”)
  23. Is there a source in a given position of the

    sky? TS = 2 log ✓ L max , 0 L max , 1 ◆ maximum likelihood: model with additional source maximum likelihood: model without an additional source (the “null hypothesis”) TS = 100 㱺 ≈10σ TS = 25 㱺 ≈5σ Rule of thumb:
  24. 1 Define your source model 2 Select data + ROI

    cuts gtselect 3 Select good time intervals (GTI) gtmktime 4 Bin data, create a count map gtbin 5 Compute useful quantities: live time cube, binned exposure cube gtltcube, gtexpcube2 6 Maximize likelihood numerically, get initial estimate of parameters gtlike 7 Optimize fit: improve parameter estimate Steps for modeling of Fermi LAT data
  25. Python wrappers make analysis of Fermi data much easier: Enrico

    and fermipy We will use Enrico: http://enrico.readthedocs.io/en/latest/
  26. Please do not download large files during the tutorial or

    the WIFI network will overload We will distribute the software and data you need via USB sticks
  27. Github Twitter Web E-mail Bitbucket Facebook Blog figshare [email protected] http://rodrigonemmen.com

    @nemmen rsnemmen http://facebook.com/rodrigonemmen nemmen http://astropython.blogspot.com http://bit.ly/2fax2cT