Time Delay Estimation for Gravitationally Lensed Light Curves SAMSI - - PowerPoint PPT Presentation

Time Delay Estimation for Gravitationally Lensed Light Curves

SAMSI Interdisciplinary Workshop Luna Bozeman, Nicholas Christoffersen, Anthony Coniglio, Katia Lélé Lagmago, Shenghao Wu May 19, 2017

Outline

• Introduction: Gravitational Lensing
• Nonparametric methods with bootstrapping

○ 2 Optimization ○ Pelt’s Method ○ Cross-correlation

• Parametric methods

○ Maximum Likelihood Estimation ○ Bayesian

• Conclusion: Comparison of All Methods

Gravitational Lensing

• Gravitational field of a galaxy can act as a lens and deflect light.
• Gravitational lensing causes duplicate images of the same object in the sky.
• Fluctuations in brightness are observed in the images at different times.
• Goal: Estimate time delays for gravitationally lensed light curves using

various non-parametric and parametric methods.

Image Credit: German Aerospace Center

Conceptualization

Image Credit: Rochester Institute of Technology

Non-parametric Method: 2 Optimization

Define a function: Then, find the value of Δ that minimizes: Utilizes bootstrapping to calculate the uncertainty quantification.

Our Data

Non-parametric Method: Pelt Method

• Very similar to the 2 method.
• Vertically shift the time series ontop of one another, then shift it by some Δ
• Now check the sum of the square differences from one data point to the next,

not interpolating between data points

• Find the value of delta that minimizes this sum of square differences

Non-parametric Method: Pelt’s second order method

• A more complicated model: taking into consideration the measurement error

and which light curve two consecutive points belong to

• Find Δ to minimize the loss function D (a function of Δ)

2 vs Pelt Method

Non-parametric Method: Cross Correlation

• A method that takes two time series and shifts one of them by a delay, Δ, then

calculates how similar in behavior they are at every Δ by the formula below.

• Use the Δ that provides the highest cross correlation value as our estimate for

the true time delay between the two time series.

• Utilize bootstrapping to create a tolerance on the estimate of time delay.

Time Delay Estimation

When Can Cross Correlation Fail?

Good! Bad

• Clear patterns with lots of movement.
• Minimal seasonal gaps.
• No clear pattern. Looks like “white noise”.
• Minimal fluctuations.
• Large seasonal gaps.

When Can Cross Correlation Fail?

Good! Bad

• One clearly defined peak.
• High maximum cross correlation value.
• A lot of noise with no defined peak.
• Low maximum cross correlation value.

• Likelihood function：joint probability density function; five parameters
• Goal: to find the optimal five-tuple that maximizes the likelihood function.

Parametric Method: Maximum Likelihood Estimation (MLE)

MLE as an optimization problem

• Hard to do optimization over a 5-d space: nonconvex, multimodal objective

function; hill-climbing optimization trapped at local optimum

• Fails unless reasonable initial point is given
• Instead, we maximize the profile likelihood
• Grid search: set the grid for Δ in (-100, 100) with grid size=0.01
• First maximize the likelihood for each Δi (compute )
• Then find the Δ that maximizes

Profile likelihood maximization

• Mean of the Estimate:
• Variance:

Performance

MLE may fail when the estimate is close to the marginal value (e.g. ≅-100)

When does it fail: noisy dataset with large seasonal gaps

Bad Data:

• low SNR
• large seasonal gap
• Shifted time series may lie

right below the seasonal gap

• f the other

Normalized Log Profile likelihood:

• Could blow up when one timeseries is

right below the gap of the other (when delta is about -100)

• Optimal at the margin.

Parametric Model: Bayesian Approach

• Using programming language R, we were able to install its “timedelay”

package to evaluate the different data sets given.

• Using the code and the data provided with the specific instructions, we
• btained the posterior distribution of the time delay (mean and standard

deviation values)

• Mean value and standard deviation representing estimate and uncertainty,

respectively

• The obtained values were then compared to the true values
• The mean squared errors were obtained with the formula:

The Bayesian Approach

• Other methods follow a separate procedure

for point estimation and uncertainty quantification

• Bayesian derives both the point estimate

and uncertainty from the posterior

• distribution. (see histogram)
• Downfall: Takes a lot of time

Conclusion: Comparison of All Methods

Conclusion: Comparison of All Methods

Acknowledgements

• We would like to first and foremost

thank SAMSI for providing us with the opportunity to participate in this workshop.

• We would also like to thank NCSU

for hosting us during the workshop.

• Last but not least, we would like to

thank Hyungsuk Tak for his mentorship and guidance through this project.

Reference

[1] Fassnacht, C. D., et al. "A determination of H0 with the CLASS gravitational lens B1608+ 656. I. Time delay measurements with the VLA." The Astrophysical Journal 527.2 (1999): 498. [2] H. Tak, K. Mandel, D. A. van Dyk, V. Kashyap, X. Meng, and A. Siemiginowska (2017+) “Bayesian and Profile Likelihood Strategies for Time Delay Estimation from Stochastic Time Series of Gravitationally Lensed Quasars” [3] H. Tak, K. Mandel, D. A. van Dyk, V. Kashyap, X. Meng, and A. Siemiginowska (2015) “time delay”, R package. [4] Pelt, Jaan, et al. "The light curve and the time delay of QSO 0957+ 561." arXiv preprint astro-ph/9501036 (1995).