bayesian.multiband {timedelay}

Estimating the time delay between doubly-lensed multi-band light curves in a Bayesian way

Description

bayesian.multiband produces posterior samples of all the model parameters including the time delay between doubly-lensed multi-band light curves.

Usage

bayesian.multiband(data.band1, data.band2, n.bands = 2,
  theta.ini = c(0.01, 0.01, 100, 100, 0.5),
  delta.ini, delta.uniform.range = c(-500, 500), delta.proposal.scale = 1,
  tau.proposal.scale = 1, tau.prior.shape = 1, tau.prior.scale = 1,
  sigma2.proposal.scale = 0.5, sigma2.prior.shape = 1, sigma2.prior.scale = 1e-7,
  rho.proposal.scale = 0.1, beta.prior.diag = 10 * c(0.1, 0.01, 1e-3, 1e-5)^2,
  micro = 3, timesc = 100, adaptive.frequency = 100,
  adaptive.delta.factor = 0.1, adaptive.tau.factor = 0.1,
  adaptive.sigma2.factor = 0.1, adaptive.rho.factor = 0.1,
  sample.size, warmingup.size)

Arguments

data.band1

A (n_1 by 5) matrix composed of the data for doubly-lensed images oberved in band 1; the first column contains n_1 observation times, the second column contains n_1 magnitudes for image A, the third column has n_1 measurement error standard deviations for image A, the fourth column include n_1 magnitudes for image B, and the fifth column contains n_1 measurement error standard deviations for image B.

data.band2

A (n_2 by 5) matrix composed of the data for doubly-lensed images oberved in band 2; the first column contains n_2 observation times, the second column contains n_2 magnitudes for image A, the third column has n_2 measurement error standard deviations for image A, the fourth column include n_2 magnitudes for image B, and the fifth column contains n_2 measurement error standard deviations for image B.

n.bands

The number of bands used to obtain the data. For now, only two bands are allowed. The authors plan to modify the code to allow more bands in the near future.

theta.ini

A vector for initial values of the OU processes for two bands, i.e., (\sigma_1, \sigma_2, \tau_1, \tau_2, \rho), where \sigma_1 is the short-term variability (standard deviation) for band 1, \sigma_2 is the short-term variability (standard deviation) for band 2, \tau_1 is the timescale for band 1, and \tau_2 is the timescale for band 2, and \rho is the cross-correlation parameter between two bands. Defaults are (0.01, 0.01, 100, 100, 0.5).

delta.ini

Initial values of the time delay.

delta.uniform.range

The range of the Uniform prior distribution for the time delay. Default range is set to (-500, 500).

delta.proposal.scale

The proposal scale of the Metropolis step for the time delay. Default is 1.

tau.proposal.scale

The proposal scale of the Metropolis-Hastings step for \tau_j (j=1, 2) Default is 1.

tau.prior.shape

The shape parameter of the Inverse-Gamma hyper-prior distribution for \tau_j. Default is 1.

tau.prior.scale

The scale parameter of the Inverse-Gamma hyper-prior distribution for \tau_j. Default is 1.

sigma2.proposal.scale

The proposal scale of the Metropolis-Hastings step for \sigma^2_j (j=1, 2). Default is 0.5.

sigma2.prior.shape

The shape parameter of the Inverse-Gamma hyper-prior distribution for \sigma^2_j. Default is 1.

sigma2.prior.scale

The scale parameter of the Inverse-Gamma hyper-prior distribution for \sigma^2_j. If no prior information is available, we recommend using 1e-7 (default).

rho.proposal.scale

The proposal scale of the Metropolis-Hastings step for \rho. Default is 0.1.

beta.prior.diag

The diagonal elements of the covariance matrix in the multivariate Gaussian prior for \beta (polynomial regression coefficients for microlensing adjustment). If such information is not available, these are set to 10 * c(0.1)^2 if micro = 0, 10 * c(0.1, 0.01)^2 if micro = 1, 10 * c(0.1, 0.01, 1e-3)^2 if micro = 2, and 10 * c(0.1, 0.01, 1e-3, 1e-5)^2 if micro = 3.

micro

A non-negative integer less than or equal to 3. It determines the order of a polynomial regression model that accounts for the long-term trend of microlensing effect. Default is 3.

timesc

It scales the observation time for fitting polynomial of microlensing, i.e., time / timesc, so that the coefficients are not too small. Default is 100.

adaptive.frequency

The adaptive MCMC is applied for every specified frequency. If it is specified as 500, the adaptive MCMC is applied to every 500th iterstion. Default is 100.

adaptive.delta.factor

The factor, exp(\pmadaptive.delta.factor), multiplied by the proposal scale of the time delay for adaptive MCMC. Default is 0.1.

adaptive.tau.factor

The factor, exp(\pmadaptive.tau.factor), multiplied by the proposal scale of \tau_j for adaptive MCMC. Default is 0.1.

adaptive.sigma2.factor

The factor, exp(\pmadaptive.tau.factor), multiplied by the proposal scale of \sigma^2_j for adaptive MCMC. Default is 0.1.

adaptive.rho.factor

The factor, exp(\pmadaptive.tau.factor), multiplied by the proposal scale of \rho for adaptive MCMC. Default is 0.1.

sample.size

The number of the posterior samples of each model parameter.

warmingup.size

The number of burn-in samples for MCMC.

Details

The function bayesian.multiband produces posterior samples of the model parameters, where the time delay (delta) is of primary interest. For now, this function only supports doubly-lensed data observed in two bands. The authors plan to generalize this code to account for more than two bands and more than two lens.

Please note that when astronomical time series data are loaded on R by read.table, read.csv, etc., some decimal places of the the observation times are automatically rounded because R's default is to load seven effective digits. For example, R will load the observation time 51075.412789 as 51075.41. This default will produce many ties in observation times even though there is actually no tie in observation times. To prevent this, please type "options(digits = 11)" before loading the data if the observation times are in seven effective digits.

Value

The outcomes of bayesian.multiband are composed of:

delta

A vector for m posterior samples of the time delay.

beta

An m by k + 1 matrix containing posterior samples of the polynomial regression coefficients, where m is the size of the posterior sample, and k is the polynomial order for microlensing.

rho

A vector for m posterior samples of the cross-correlation parameter.

sigma

An m by 2 matrix containing posterior samples of the short-term variability (standard deviation) of the O-U process. The first column is composed ot the m posterior samples of \sigma_1 in band 1, and the second column contains the m posterior samples of \sigma_2 in band 2.

tau

An m by 2 matrix containing posterior samples of the timescale of the O-U process. The first column is composed ot the m posterior samples of \tau_1 in band 1, and the second column contains the m posterior samples of \tau_2 in band 2.

tau.accept.rate

The acceptance rate of the MCMC for \tau_1 and \tau_2.

sigma.accept.rate

The acceptance rate of the MCMC for \sigma_1 and \sigma_2.

delta.accept.rate

The acceptance rate of the MCMC for the time delay

rho.accept.rate

The acceptance rate of the MCMC for \rho.

Author(s)

Zhirui Hu and Hyungsuk Tak

References

Hyungsuk Tak, Kaisey Mandel, David A. van Dyk, Vinay L. Kashyap, Xiao-Li Meng, and Aneta Siemiginowska (2017). "Bayesian Estimates of Astronomical Time Delays between Gravitationally Lensed Stochastic Light Curves," The Annals of Applied Statistics, 11 (3), 1309-1348. Hyungsuk Tak, Xiao-Li Meng, and David A. van Dyk (2018), "A Repelling-Attracting Metropolis Algorithm for Multimodality", Journal of Computational and Graphical Statistics, 27 (3), 479-490. Zhirui Hu and Hyungsuk Tak (2020+), "Modeling Stochastic Variability in Multi-Band Time Series Data," arXiv:2005.08049.

Examples

  # Loading datasets 
  data(simple.band1)
  data(simple.band2)

  # Doubly-lensed quasar data set observed in two bands
  # Each data set contains doubly-lensed light curves observed in one band. 
  head(simple.band1)
  head(simple.band2)

  # The length of each data set (i.e., number of observation times)
  # do not need to be the same.

  dim(simple.band1)
  dim(simple.band2)

  ###############################################
  # Time delay estimation via Bayesian approach #
  ###############################################

  # Cubic microlensing model (m = 3)
  
  output <- bayesian.multiband(data.band1 = simple.band1, 
              data.band2 = simple.band2, n.bands = 2, 
              theta.ini = c(0.01, 0.01, 100, 100, 0.5),
              delta.ini = 100, delta.uniform.range = c(-500, 500), 
              tau.proposal.scale = 1, tau.prior.shape = 1, tau.prior.scale = 1, 
              sigma2.proposal.scale = 0.5, sigma2.prior.shape = 1, sigma2.prior.scale = 1e-7, 
              rho.proposal.scale = 0.1, beta.prior.diag = 10 * c(0.1, 0.01, 1e-3, 1e-5)^2, 
              micro = 3, timesc = 100, adaptive.frequency = 100,
              adaptive.delta.factor = 0.1, adaptive.tau.factor = 0.1,
              adaptive.sigma2.factor = 0.1, adaptive.rho.factor = 0.1,
              sample.size = 100, warmingup.size = 100)
  names(output)
  

  # hist(output$delta, 20)
  # plot(output$delta, type = "l")
  # acf(output$delta)
  # output$delta.accept.rate