Bayesian Approach to Time DelayEstimation

Hyungsuk Tak

Stat300

23 Sep 2014

Joint work with Xiao-Li Meng, David van Dyk, Kaisey Mandel, AnetaSiemiginowska, and Vinay Kashyap

Outline

I Introduction

I Data and Difficulties

I Bayesian ApproachI MotivationI State-space ModelingI LikelihoodI PriorI Hyper-priorI Full Posterior and Sampler based on ASIS

I Example 1: Simulated data

I Example 2: Real Data (Q0957+561)

I Discussion with Reference

Introduction

Image Credit: NASA/JPL-Caltech

Introduction

Light rays are bent by a strong gravitational field of a lensed galaxy.

I Each route has different length.

I Different arrival times of light rays

Why time delay?

I Mass structure of the lens galaxy

I Cosmological parameters, e.g., Hubble constant, H0

Data and Difficulties

Data comprise of two time series with measurement errors.

I Observation times t′ ≡ {t1, t2, . . . , tn}I Observed magnitudes x(t)′ ≡ {x(t1), x(t2), . . . , x(tn)}, and y(t)

I Measurement errors (se) δ(t)′ ≡ {δ(t1), δ(t2), . . . , δ(tn)} and η(t)

Data and Difficulties

Some difficulties occur in estimating the time delay

1. Irregular observation times (∵ weather conditions)

2. Seasonal gaps (∵ rotation of the earth)

3. Magnitude shift (∵ different gravitational potentials)

4. Measurement errors

Our job is to estimate time delay (shift in x-axis) between two time series.

Bayesian Approach: Motivation

Grid optimization methods are dominating this field!

I eg. Cross-correlation method

1. Shift one light curve by ∆ in x-axis

2. Calculate r∆, sample cross-correlation function

3. Find ∆ that maximizes r∆ on the grid of ∆

I SE (∆̂) by computationally expensive repeated sampling procedure

I Grid of ∆ (6= the whole space of ∆)

Non-grid-based Bayesian approach

I Principled way of model construction: likelihood-based

I Computational efficiency: simple and fast posterior sampling scheme

I The whole space of ∆

Bayesian Approach: State-space modeling

I Assumption 1: ∃ unobserved underlying processes representing thetrue magnitudes in continuous time (red and blue dashed curves)

I X(t)′ = (X (t1),X (t2), . . . ,X (tn)) and Y(t), values on curves at t

I Assumption 2: Y(t) = X(t−∆) + c (Harva, 2006)

Bayesian Approach: Likelihood

Independent Gaussian measurement errors

I x(tj)|X (tj) ∼ N [X (tj), δ2(tj)]

I y(tj)|Y (tj) ∼ N [Y (tj), η2(tj)]

y(tj)|X (tj −∆) + c ,∆, c ∼ N [X (tj −∆) + c , η2(tj)].

Likelihood function

I Suppose t∗ = sort(t1, t2, . . . , tn, t1 −∆, t2 −∆, . . . , tn −∆)

I L(X(t∗),∆, c) ∝∏n

j=1 p(x(tj)|X (tj)) · p(y(tj)− c |X (tj −∆),∆, c)

Bayesian Approach: Prior

I Ornstein-Uhlenbeck process for X(·) (Kelly et al., 2009)I Intrinsic variability of quasar → stochastic process in continuous time

I Easy way to sample true values at irregularly-spaced times

I dX (t) = − 1τ

(X (t)− µ

)dt + σdB(t)

I Markovian propertyX (t∗j )|X (t∗j−1), µ, σ, τ ∼ N

[mean: µ+ e−(t∗j −t∗j−1)/τ

(X (t∗j−1)− µ

),

variance: τσ2

2(1− e−2(t∗j −t∗j−1)/τ )

]I p(X(t∗)|µ, σ, τ,∆) =

p(X (t∗1 )|µ, σ, τ,∆) ·∏2n

j=2 p(X (t∗j )|X (t∗j−1), µ, σ, τ,∆)

I p(∆, c) = p(∆)p(c) ∝ I{|∆| ∈ [0, (tn−t1)]}

Bayesian Approach: Hyper-prior

I µ is a mean parameter of the underlying process

I σ is a scale parameter of the underlying process

I τ is a relaxation time of the underlying process

I Naively informative hyper-prior distribution:

p(µ, σ2, τ)= p(µ)p(σ2)p(τ) ∝ e−0.01/σ2

(σ2)1.01e−1/τ

τ 2

I Flat on µ, InvGamma(0.01, 0.01) on σ2, and InvGamma(1, 1) on τ

Full Posterior and Sampler based on ASIS

Suppose θhyp ≡ (µ, σ, τ) and Dobs ≡ (x(t), y(t))

I Full Posterior: p(X(t∗),∆, c , θ|Dobs)

∝ L(X(t∗),∆, c) · p(X(t∗),∆, c |θhyp) · p(θhyp)

Likelihood Prior Hyper-prior

I Metropolis-Hastings within GibbsI p(X(t∗),∆|Dobs , c, θhyp)I p(c|Dobs ,X(t

∗),∆, θhyp)I p(θhyp|Dobs , c,X(t

∗),∆)

I Ancillarity-Sufficiency Interweaving Strategy (Yu and Meng, 2011)I p(X(t∗),∆|Dobs , c, θhyp)!I Interweaving p(c|Dobs ,X(t

∗),∆, θhyp) with p(c|Dobs ,S(t∗),∆, θhyp)

I p(θhyp|Dobs , c,X(t∗),∆)

Example 1: Simulated Data

Summary of posterior ∆AB with the blinded truth 30.98Post. Mean Post. Median Post. SD Half-length of 68% PI

30.94 30.89 0.694 0.423

Example 2: Real Data (Q0957+561)

Data observed at the United States Naval Observatory (Hainline et al., 2012)

Researchers Method ∆̂AB SE(∆̂AB )

Oscoz et al. (1997) Discrete cross-correlation & Dispersion 424 3Serra-Ricart et al. (1999) Cross-correlation functions 425 4

Tak et al. (?) Bayesian 423.16 1.22

Example 2: Real Data (Q0957+561) (cont.)Convergence ChecksEach row: Traceplot, ACF, and histogram from the topEach column: ∆, c , µ, log(σ), log(τ) from the left

MHwG

ASIS

Discussion with Reference

I Prior on ∆

I Quadruply-lensed quasar data

I Microlensing

I Reference

1. L. Hainline, C. Morgan, J. Beach, C. Kochanek, H. Harris, T. Tilleman, R. Fadely, E. Falco, and T. Le (2012) “A new

microlensing event in the doubly imaged quasar Q 0957+561” The Astrophysical Journal, 744:104(9pp)

2. M. Harva and S. Raychaudhury (2006) “Bayesian estimation of time delays between unevenly sampled signals” IEEE

Machine Learning for Signal Processing, ISSN 1551-2541

3. B. Kelly, J. Bechtold, and A. Siemiginowska (2009) “Are the variations in quasar optical flux driven by thermal

fluctuation?” The Astrophysical Journal, 698, 895 - 910.

4. A. Oscoz, E. Mediavilla, L. Goicoechea, M. Serra-Ricart, and J. Buitrago (1997) “Time delay of QSO 0957+561 and

cosmological implications” Astronomy and Astrophysics, 479, L89-L92

5. M. Serra-Ricart, A. Oscoz, T. Sanchis, E. Mediavilla, L. Goicoechea, J. Licandro, D. Alcalde, and R. Gil-Merino (1999)

“BVRI photometry of QSO 0957+561A, B: observations, new reduction method, and time delay” Astronomy and

Astrophysics, 526, 40-51