Bayesian Shape Measurement and Galaxy Model Fitting

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Bayesian Shape Measurement and Galaxy Model
Fitting

• Thomas Kitching
• Lance Miller, Catherine Heymans,
• Alan Heavens, Ludo Van Waerbeke
• Miller et al. (2007)
• accepted, MNRAS (background and algorithm)
  arXiv0708.2340
• Kitching et al. (2007)
• in prep. (further development and STEP analysis)

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Introduction

• Want to calculate the full (posterior)
  probability for each galaxy and use this to
  calculate the shear
• Bayesian shape measurement in general
• Bayesian vs. Frequentist
• Shear Sensitivity
• Model Fitting
• Why model fitting?
• The lensfit algorithm/implementation
• Results for individual galaxy shapes
• Results for shear measurement (STEP-1)
• ltmgt, ?c, ltqgt

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Bayesian Shape Measurement

• Applies to any shape measurement method if p(e)
  can be determined
• For a sample of galaxy with intrinsic
  distribution f(e) probability distribution of the
  data is
• For each galaxy, i (from data yi) generate a
  Bayesian posterior probability distribution

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• Want the true distribution of intrinsic
  ellipticities to be obtained from the data by
  considering the summation over the data
• Insrinsic p(e) recovered if ?(ye) L(ey),
  P(e)f(e)

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• Using we have
• The other definition can e used but extra
  non-linear terms in ltegt have to be included
• Calculating ltegt
• We know p(e), and hence ltegt for each galaxy
• For N galaxies we have

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Bayesian or Frequentist?

• Frequentist can shapes be measured using
  Likelihoods alone?
• Bayesian and Likelihoods measure different things
• Suppose x has an intrinsic normal distribution of
  variance a0.3
• For each input we measure a normal distribution
  with variance b0.4

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• Likelihood unbiased in input to output regression
• No way to account for the bias from the
  likelihood alone
• Also with no prior the hard bound elt1 can
  affect likelihood estimators
• Bayesian unbiased in output to input regression
• Best estimate of input values
• Distribution narrower but each point has an
  uncertainty
• If if there are effects due to the hard elt1
  boundary the prior should contain this information

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• Even Bayesian methods will have bias especially
  in the case that a zero-shear prior is used
• However this bias can be calculated from the
  posterior probability

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Shear Sensitivity Prior

• Since we do not know the Prior distribution we
  are forced to adopt a zero-shear prior
• For low S/N galaxies the prior could dominate
  resulting in no recoverable shear, as in all
  methods
• However the magnitude of this effect can be
  determined
• Can define a weighted estimate of the shear as
• Where we call the shear
  sensitivity
• In Bayesian case this can be approximated by

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Model Fitting

• If the model is a good fit to the data then
  maximum S/N of parameters should be obtained
• If model is a good fit then all information about
  image is contained in the model
• Use realistic models based on real galaxy image
  profiles
• Long history of model fitting for non-lensing and
  lensing applications
• Galfit
• Kuijken, Im2shape
• Problem is that we need a computationally fast
  model fitting algorithm for lensing surveys that
  uses realistic galaxy image profiles

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lensfit

• Measure PSF ? create a model ?
  convolve with PSF ? determine the
  likelihood of the fit
• Simplest galaxy model (if form is fixed) has 6
  free parameters
• Brightness, size, ellipticity (x2), position (x2)
• It is straightforward to marginalise over
  position and brightness if the model fitting is
  done in Fourier space
• Key advances and differences
• FAST model fitting technique
• Bias is taken into account in a Bayesian way

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• Choice of model is not key, we assume a de
  Vaucouleurs profile
• Free parameters are then length scale (r), e1, e2
• We use a grid in (e1,e2)
• Could use MCMC approach
• Found convergence for ?e0.1 ? lt100 points
• We adopt a uniform prior for the distribution of
  galaxy scale-length.
• This could be replaced by a prior close to the
  actual distribution of galaxy sizes, although
  such a prior would need to be magnitude-dependent
• Tested the algorithm
• Convergence in e and r resolutions
• Robust to galaxy position error up to a 10
  pixel random offset

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Estimate rms noise in each pixel from entire
image
Estimate PSF on same pixel scale as models and
FT
Generate set of models in 3D grid of e1, e2, r
and FT
Take each possible model in turn, multiply by the
transposed PSF and model transforms and carry out
the cross correlation
Measure nominal galaxy positions
Measure the amplitude, width and position of the
maximum of the resulting cross-correlation, and
hence evaluate the likelihood for this model and
galaxy
Isolate a sub-image around a galaxy and FT
Numerically marginalise over the length scale
Sum the posterior probabilities
Repeat for each galaxy
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Tests on STEP1

• Use a grid sampling of ?e0.1
• Found numerical convergence
• Use 32x32 sub images sizes
• Optimal for close pairs rejection and fitting
  every galaxy with the sub image
• Close pairs rejection if two galaxies lie in a
  sub image
• Working on a S/N based rejection criterion
• We assume the pixel noise is uncorrelated, which
  is appropriate for shot noise in CCD detectors
• The PSF was created by stacking stars from the
  simulation allowing sub-pixel registration using
  sinc-function interpolation
• Method of PSF characterisation not crucial as
  long as the PSF is a good match to the actual PSF
• Sub pixel variation in PSF not taken into account
  may lead to high spatial frequencies which are
  not included

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Prior

• Use the lens0 STEP1 input catalogue
  (zero-sheared) to generate the input prior for
  each STEP1 image and psf
• In reality could iterate the method especially
  since in reality the intrinsic p(e) will be
  approximately zero-centered
• The method should return the intrinsic p(e)
• Calculate p(e)
• Substitute p(e) for the prior and iterate until
  convergence is reached

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Individual galaxy ellipticities

• Use lens1, psf0 as an example
• Some individual galaxy probability surfaces

Maglt22
Maggt22
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• psf 0 lens 1

Maggt22
Maglt22
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• The prior is recovered
• In this zero-shear case where the prior is the
  actual input distribution

Maglt22
Maggt22

• Speed
• Approximately 1 second per galaxy on 1 2GHz CPU
• Trivially Parallelisable (e.g. 1 galaxy per CPU)
• Scales with square of number of e1,e2 points
  sampled

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Shear Results for STEP-1

• Full STEP-1 64 images, 6 PSF, 5 shear values
• Present ltmgt, ltqgt and ?c values
• Importance of knowing the shear sensitivity
• Example psf1
• Average bias of 0.88 much larger effect for faint
  galaxies

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• Results for psf1

?1 m -0.0205 c -0.0006
?2 m -0.0001 c 0.0001
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• ltmgt-0.022
• 0.0035
• ?c 0.0004
• Best performing linear method
• We have performed iterations BUT
• This was used to correct coding errors only
• NOT to tune the method or fix ad hoc parameters

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• ltmgt-0.066
• 0.003
• ?c 0.0004
• ltqgt 0.46
• Results are limited by PSF characterisation
• Results expected to improve if PSF is known more
  accurately
• Sub pixel variation?

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Conclusions

• Given a shape measurement method that can produce
  p(e) Bayesian shape measurement has the potential
  to yield an unbiased shear estimator
• Even in reality, assuming a zero-sheared prior,
  the shear sensitivity can be calculated to
  correct for any bias
• We presented a fast model fitting method lensfit
• lensfit can accurately find individual galaxy
  ellipticities
• Performance is good in the STEP1 simulations with
  small values of m, c and q
• Better PSF characterisation could improve results

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Radio STEP

• SKA could produce a very competitive weak lensing
  survey
• 50 km baseline implies angular resolution 1
  arcsec at 1.4 GHz
• The shear map constructed from continuum shape
  measurements of star-forming disk galaxies
• A subset of spectroscopic redshifts from HI
  detections
• 22 per arcmin2 at 0.3 µJy, 20,000 sqdeg survey
• No photometric redshift uncertainties
• Have software MEQtrees
• Can simulate realistic Radio images
• With realistic Radio PSFs
• Note that the PSF is complicated but
  deterministic (apart from atmospherics)
• Radio STEP
• Can measure shapes directly in the (u,v) Fourier
  plane
• Using the simulated Radio images
• Progressively complicated PSFs