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Statistical analysis of multiStatistical analysis of multi‐‐source source CalibrationCalibrationCalibrationCalibration

SS. . KazemiKazemi

KapteynKapteyn Astronomical Institute Astronomical Institute [email protected]@astro.rug.nl

Presentation in CALIM, August Presentation in CALIM, August 20102010

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(Under supervision of S. (Under supervision of S. ZaroubiZaroubi, and , and S.YatawattaS.Yatawatta))

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IntroductionMeasurement equation

The LS, EM, and SAGE algorithmsSolver noise

Illustrative example

CalibrationNoise in solutions

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faIllustrative exampleConclusions and future work

IntroductionIntroductionThe scientific goals of new radio arrays (e.g. LOFAR & SKA) require extreme sensitivity and dynamic range (e.g. LOFAR EOR)

Calibration

E i i k i d h k d i h b f i i

Calibration challenges:• Accuracy of calibration algorithms becomes crucial for achievement of scientific goals

Estimating unknown instrument and the sky parameters and correcting them before imaging.

• Accuracy of calibration algorithms becomes crucial for achievement of scientific goals•The large number of data requires fast algorithms

Input noise output noise“I t i ”

p“Instrumental noise”

“Sky noise”“RFI”

“Input noise”+

Noise of the calibration algorithm“solver noise”

Calibration process

Here comes your footerPresentation in CALIM August 2010

The aim is to devise new calibration methods which minimize the solver noiseThe aim is to devise new calibration methods which minimize the solver noise

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2 Measurement equation

3 The LS, EM, and SAGE algorithms

4 Solver noise

6 Conclusions and future work

5 Illustrative example

6 Conclusions and future work

Here comes your footer Presentation in CALIM August 2010

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Initial assumptionsCalibration problem modeling

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Measurement equationMeasurement equationInitial assumptions Every antenna

has dual l i d f d

X

YRadio frequency sky is decomposed to K separated

sources far away from the array

polarized feeds Y

Baseline pqCalibration problem modeling Hamaker‐Bregman‐Sault, 1996.

Coherency: describes polarization state of the source i

Jones matrix: describes amplifier gains, beam shapes, ionospheric effects, and etc

V i d f

Noise matrix of

Vectorized form

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Noise matrix of the baseline pq

Measured visibility

Presentation in CALIM August 2010

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The Least Squares (LS, Normal) calibration methodExpectation Maximization (EM) algorithmSpace Alternating Generalized Expectation Maximization algorithm (SAGE)

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The LS, EM, and SAGE algorithmsThe LS, EM, and SAGE algorithmsThe Least Squares (LS, Normal) calibration method ( AIPS, AIPS++, CASA)

Levenberg Marquardt

•Easy to program But,

g q

technique

•Heavy computational cost of O((KN)2 ) •Slow rate of convergence

Expectation Maximization (EM) algorithm

Y Ob d d t

Feder and Weinstein, 1988.

Y : Observed dataX : Complete data: Parameter value

Jensen’s inequality: this expected value will be decreased for all


will be decreased for all


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Expectation‐Step: Compute the expected value of the log‐likelihood function, with respect to conditional distribution of the complete data X given observed data y under the current

Maximization‐Step: Maximize the log‐likelihood with respect to

conditional distribution of the complete data X given observed data y under the current parameter estimate

Assuming that the contribution of source i depends only on a subset of parameters and partitioning the parameter vector as

Yatawatta et al. 2009, Kazemi et al. in prep.

E‐ Step:

M‐ Step: min

Fessler and Herro, 1994.

M Step: min

Space Alternating Generalized Expectation Maximization (SAGE) algorithm

• Dedicating whole the noise to only one sourcel h l h

Fessler and Herro, 1994.


• Utilizing the EM algorithm


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Yatawatta et al. 2009, Kazemi et al. in prep.

E StE‐ Step:

M‐ Step: min

Advantages of the EM algorithm over the Normal algorithm:•Breaking the likelihood maximization into smaller computational steps•Breaking the likelihood maximization into smaller computational steps•Computational cost equal to KO(N2 )

•Increasing the likelihood at each iteration step•Increasing the likelihood at each iteration step•Improving the speed of convergence compared with LS calibrationBut still,•Possibility of converging to a local optimum•Implementation of the algorithm is complicated compared with LS method•Implementation of the algorithm is complicated compared with LS method

Additional advantages of using the SAGE algorithm:•Improving the speed of convergence


Improving the speed of convergence•Improving the accuracy of calibration results


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Initial assumptionsKullback ‐Leibler divergence (KLD)Likelihood Ratio Test (LRT)

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Solver noiseSolver noiseInitial assumptions

The goal is to minimize the “distance” between the real gainsThe goal is to minimize the distance between the real gains and calculated solutions = Minimizing the solver noise

x := gain solutions for the source s at the antenna q

• Sky noise for two different directions is uncorrelated• Thermal noise for any number of directions from the same station is the same

xq,s:= gain solutions for the source s at the antenna q

• Solver noise is the same for all stations and directions

Kullback‐Leibler Divergence (KLD)

The higher KLD between two different sources solutions at the same antenna, the solver noise is lessW fit G i i t d l t th l ti b EM l ith


•We fit a Gaussian mixture model to the solutions by EM algorithm•KLD is calculated via Monte‐Carlo algorithm


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Initial assumptionsKullback ‐Leibler Divergence (KLD)Likelihood Ratio Test (LRT)

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Likelihood Ratio Test (LRT)Likelihood Ratio Test (LRT)

Null model: Independent solutions for different sources for the same antennaAlternative model: Dependent solutions for different sources for the same antenna

Probability of accepting null modelrather than alternative model

The less LRT between two different sources solutions at the same antenna, the solver noise is less


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Comparison between the SAGE and the Normal calibration algorithms

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Illustrative exampleIllustrative exampleOne channel simulated observation of three sources by fourteen antennas

Clean image Image having gain errors and


Clean image Image having gain errors and white Gaussian noise

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Residual ErrorsSAGE, 9 iterations NORMAL, 9 iterations

=1.2186

=1.2179

STD

STD


RMSE= 3.9259 RMSE= 3.9505

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LOFAR CS1 images from CasA and CygA


Yatawatta et al. 2009.

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C l i d f t kC l i d f t k

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faConclusions and future workConclusions and future work

Conclusions:•The SAGE algorithm is superior in terms of accuracy and speed of convergence•The non‐linear problem requires suitable regularization •The KLD and the LRT could be used to study the influence of solver noise

Future work:A l th l ith t l d t f LOFAR•Apply the algorithms to real data of LOFAR

•Study different regularization methods•Implement different noise models


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The LM, EM, and SAGE algorithmsNoise in solutions


LOFARCalibrationSolver noise

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i i d /RAI P j t SKA ht l

LOFARLOFAR Thank youThank yourai.ncsa.uiuc.edu/RAI_Projects_SKA.html

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SimulationDirection dependent gainsDetecting solver noise via KLD and LRT

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Illustrative exampleIllustrative examplerai.ncsa.uiuc.edu/RAI_Projects_SKA.html

Detection solver noise via KLD and LRTSolutions of antenna seven are given sin(t) noise where t is integration time

Normal calibrationThe SAGE calibrationg ( ) g

Real Solutions Solution added sin(t) noise


The KLD of the noisy antenna becomes zero

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Detection solver noise via KLD and LRT Normal calibrationThe SAGE calibration



The LRT of the noisy antenna becomes higher The LRT of the noisy antenna becomes higher

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KLD comparison LRT comparisonThe Normal and the SAGE algorithms with 16 number of iterations


The SAGE algorithm solutions have superior accuracy

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The LRT of noisy antenna is highly increased

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The KLD of noisy antenna becomes almost zero

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Initial assumptionsKullback ‐Leibler divergence (KLD)Likelihood Ratio Test (LRT)

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Noise in solutionsNoise in solutionsrai.ncsa.uiuc.edu/RAI_Projects_SKA.html

Initial assumptions

• Sky noise for two different directions is uncorrelated• Thermal noise for any number of directions from the same station is the same• Solver noise is the same for all stations and directions

The goal is to minimize the “distance” between the real gains d l l d l i Mi i i i h l i

Kullback‐Leibler Divergence (KLD)

and calculated solutions = Minimizing the solver noise

The higher KLD between two different sources solutions at the same antenna, the solver noise is lessW fit G i i t d l t th l ti b EM l ith


•We fit a Gaussian mixture model to the solutions by EM algorithm•KLD is calculated via Monte‐Carlo algorithm

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CalibrationSolver noise

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IntroductionIntroduction

Calibration

rai.ncsa.uiuc.edu/RAI_Projects_SKA.html

•Estimating unknown instrument and the sky parameters and correcting them before imaging.•Finding the Maximum Likelihood estimate of the sky and the Instrument Unknown parameters.

Calibration challenges:•The scientific goals of LOFAR require extreme sensitivity and dynamic range (LOFAR EOR)•Intrinsically polarized feeds (dipoles) ‐ polarized measurement equation Wid fi ld f i•Wide fields of view

•Pronounced direction‐dependent effects

Input noise “instrumental noise”

output noiseNoise in calibration solutions

Solver noise


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Detection solver noise via KLD and LRT

Solutions of antenna seven are added sin(t) noise where t is integration time:


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Direction dependent gains

Jy Jy


Random solutions Sin(t) solutions

Xq,s


Statistical analysis of multiStatistical analysis of multi‐‐sourcesourceStatistical analysis of multiStatistical analysis of multi source source CalibrationCalibration

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S. S. KazemiKazemi

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KapteynKapteyn Astronomical Institute Astronomical Institute [email protected]@astro.rug.nl

Presentation in CALIM, August Presentation in CALIM, August 20102010(Under supervision of S(Under supervision of S ZaroubiZaroubi andand S YatawattaS Yatawatta))(Under supervision of S. (Under supervision of S. ZaroubiZaroubi, and , and S.YatawattaS.Yatawatta))

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