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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) arXiv:0708.2340 Kitching et al. (2007) in prep. (further development and STEP analysis). - PowerPoint PPT Presentation
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Thomas Kitching
Bayesian Shape Measurement and
Galaxy Model Fitting
Bayesian Shape Measurement and
Galaxy Model Fitting
Thomas KitchingLance Miller, Catherine Heymans, Alan Heavens, Ludo Van Waerbeke
Miller et al. (2007) accepted, MNRAS (background and algorithm) arXiv:0708.2340
Kitching et al. (2007) in prep. (further development and STEP analysis)
Thomas KitchingLance Miller, Catherine Heymans, Alan Heavens, Ludo Van Waerbeke
Miller et al. (2007) accepted, MNRAS (background and algorithm) arXiv:0708.2340
Kitching et al. (2007) in prep. (further development and STEP analysis)
Thomas Kitching
IntroductionIntroduction 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) <m>, c, <q>
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) <m>, c, <q>
Thomas Kitching
Bayesian Shape Measurement
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:
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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Thomas Kitching
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 (y|e) =L(e|y), P(e)=f(e)
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 (y|e) =L(e|y), P(e)=f(e)
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Thomas Kitching
Using we have
The other definition can e used but extra non-linear terms in <e> have to be included
Calculating <e> We know p(e), and hence <e> for each galaxy
For N galaxies we have
Using we have
The other definition can e used but extra non-linear terms in <e> have to be included
Calculating <e> We know p(e), and hence <e> for each galaxy
For N galaxies we have
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Thomas Kitching
Frequentist : can shapes be measured using Likelihoods alone?
Bayesian and Likelihoods measure different things
Suppose x has an intrinsic normal distribution of variance a=0.3
For each input we measure a normal distribution with variance b=0.4
Frequentist : can shapes be measured using Likelihoods alone?
Bayesian and Likelihoods measure different things
Suppose x has an intrinsic normal distribution of variance a=0.3
For each input we measure a normal distribution with variance b=0.4
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Bayesian or Frequentist?Bayesian or Frequentist?
Thomas Kitching
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 |e|<1 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 |e|<1 boundary the prior should contain this information
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 |e|<1 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 |e|<1 boundary the prior should contain this information
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Thomas Kitching
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
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
e
Prior Posterior
etrue
Thomas Kitching
Shear Sensitivity & PriorShear 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
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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Thomas Kitching
Thomas Kitching
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
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
Model FittingModel Fitting
Thomas Kitching
lensfitlensfit 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
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
Thomas Kitching
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 e=0.1 <100 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
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 e=0.1 <100 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
Isolate a sub-image around a galaxy and FT
Take each possible model in turn, multiply by the transposed PSF and model transforms and carry out the cross correlation
Measure the amplitude, width and position of the maximum of the resulting cross-correlation, and hence evaluate the likelihood for this model and galaxy
Sum the posterior
probabilities
Numerically marginalise over the length scale
Repeat for each galaxy
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
Measure nominal galaxy positions
Thomas Kitching
Tests on STEP1Tests on STEP1 Use a grid sampling of e=0.1
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
Use a grid sampling of e=0.1 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
Thomas Kitching
PriorPrior 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
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
Thomas Kitching
Use lens1, psf0 as an example Some individual galaxy probability surfaces
Use lens1, psf0 as an example Some individual galaxy probability surfaces
Individual galaxy ellipticitiesIndividual galaxy ellipticities
Mag<22Mag>22
Thomas Kitching
psf 0 lens 1
psf 0 lens 1
Mag>22
Mag<22
Thomas Kitching
The prior is recovered In this zero-shear case where the prior is the actual input distribution
The prior is recovered In this zero-shear case where the prior is the actual input distribution
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
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
Mag<22 Mag>22
Thomas Kitching
Shear Results for STEP-1Shear Results for STEP-1 Full STEP-1 64 images,
6 PSF, 5 shear values Present <m>, <q>
and c values
Importance of knowing the shear sensitivity Example psf1 Average bias of 0.88 much
larger effect for faint galaxies
Full STEP-1 64 images, 6 PSF, 5 shear values
Present <m>, <q> and c values
Importance of knowing the shear sensitivity Example psf1 Average bias of 0.88 much
larger effect for faint galaxies
Thomas Kitching
Results for psf1 Results for psf1
1
m =-0.0205c =-0.0006
2
m =-0.0001c = 0.0001
Thomas Kitching
<m>=-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
<m>=-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
Thomas Kitching
<m>=-0.066 + 0.003 c = 0.0004 <q> = 0.46
Results are limited by PSF characterisation Results expected to
improve if PSF is known more accurately
Sub pixel variation?
<m>=-0.066 + 0.003 c = 0.0004 <q> = 0.46
Results are limited by PSF characterisation Results expected to
improve if PSF is known more accurately
Sub pixel variation?
Thomas Kitching
ConclusionsConclusions 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
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
Thomas Kitching
Thomas Kitching
Radio STEPRadio 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 PSF’s 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 PSF’s
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 PSF’s 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 PSF’s
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