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

Thomas Kitching 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) ,  c,  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) ,  c,

Thomas Kitching 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 y i ) 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 y i ) generate a Bayesian posterior probability distribution:

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)

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

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 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

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 PriorPosterior e true

Thomas Kitching 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  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

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 Fitting

Thomas Kitching 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  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), e 1, e 2  We use a grid in (e 1,e 2 )  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), e 1, e 2  We use a grid in (e 1,e 2 )  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 e 1, e 2, r and FT Measure nominal galaxy positions

Thomas Kitching Tests 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 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  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 ellipticities Mag<22Mag>22

Thomas Kitching  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 e 1,e 2 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 e 1,e 2 points sampled Mag<22Mag>22

Thomas Kitching Shear Results for STEP-1  Full STEP-1 64 images, 6 PSF, 5 shear values  Present, 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, 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  1 m = c =  2 m = c =

Thomas Kitching  =   c =  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  =   c =  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  =   c =  = 0.46  Results are limited by PSF characterisation  Results expected to improve if PSF is known more accurately  Sub pixel variation?  =   c =  = 0.46  Results are limited by PSF characterisation  Results expected to improve if PSF is known more accurately  Sub pixel variation?

Thomas Kitching 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  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

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 arcmin 2 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 arcmin 2 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