timedelay (version 1.0.2)

bayesian: Estimating the time delay via the Bayesian method

Description

bayesian produces posterior samples of all the model parameters one of which is the time delay. This function has options for three MCMC techniques, ancillarity-sufficiency interweaving strategy (ASIS), adaptive MCMC, and tempered transition, to improve the convergence rate of the MCMC and to handle the multimodality of the time delay.

Usage

bayesian(data, data.flux, theta.ini,  delta.ini, delta.uniform.range, delta.proposal.scale,  tau.proposal.scale, tau.prior.shape, tau.prior.scale, sigma.prior.shape, sigma.prior.scale, asis = TRUE, micro, adaptive.freqeuncy, adaptive.delta =  TRUE, adaptive.delta.factor, adaptive.tau = TRUE, adaptive.tau.factor, sample.size, warmingup.size)

Arguments

data

A (n by 1) matrix; the first column has n observation times, the second column has n flux (or magnitude) values of light A, the third column has n measurement errors of light A, the fourth column has n flux (or magnitude) values of light B, and the fifth column has n measurement errors of light B.

data.flux

"True" if data are recorded on flux scale or "FALSE" if data are on magnitude scale.

theta.ini

Initial values of theta = (mu, sigma, tau) for MCMC.

delta.ini

Initial values of the time delay for MCMC.

delta.uniform.range

The range of the Uniform prior distribution for the time delay. The feasible entire support is c(min(simple[, 1]) - max(simple[, 1]), max(simple[, 1]) - min(simple[, 1])).

delta.proposal.scale

The proposal scale of the Metropolis step for the time delay.

tau.proposal.scale

The proposal scale of the Metropolis-Hastings step for tau.

tau.prior.shape

The shape parameter of the Inverse-Gamma hyper-prior distribution for tau.

tau.prior.scale

The scale parameter of the Inverse-Gamma hyper-prior distribution for tau.

sigma.prior.shape

The shape parameter of the Inverse-Gamma hyper-prior distribution for sigma^2.

sigma.prior.scale

The scale parameter of the Inverse-Gamma hyper-prior distribution for sigma^2. If no prior information is available, we recommend using 2 * 10^(-7).

asis

(Optional) "TRUE" if we use the ancillarity-sufficiency interweaving strategy (ASIS) for c (always recommended). Default is "TRUE".

micro

It determines the order of a polynomial regression model that accounts for the difference between microlensing trends. Default is 3. When zero is assigned, the Bayesian model fits a curve-shifted model.

adaptive.freqeuncy

(If "adaptive.delta = TRUE" or "adaptive.tau = TRUE") The adaptive MCMC is applied for every specified frequency. If it is specified as 500, the adaptive MCMC is applied to every 500th iterstion.

adaptive.delta

(Optional) "TRUE" if we use the adaptive MCMC for the time delay. Default is "TRUE".

adaptive.delta.factor

(If "adaptive.delta = TRUE") The factor, exp(adaptive.delta.factor) or exp(-adaptive.delta.factor), multiplied to the proposal scale of the time delay for adaptive MCMC.

adaptive.tau

(Optional) "TRUE" if we use the adaptive MCMC for tau. Default is "TRUE".

adaptive.tau.factor

(If "adaptive.tau = TRUE") The factor, exp(adaptive.tau.factor) or exp(-adaptive.tau.factor), multiplied to the proposal scale of tau for adaptive MCMC.

sample.size

The number of the posterior samples of each model parameter.

warmingup.size

The number of burn-ins for MCMC.

Value

The outcome of bayesian comprises of:

delta

Posterior samples of the time delay

beta

Posterior samples of the polynomial regression coefficients, beta

mu

Posterior samples of the mean parameter of the O-U process, mu

sigma

Posterior samples of the short term variability of the O-U process, sigma

tau

Posterior samples of the mean reversion time of the O-U process, tau

tau.accept.rate

The acceptance rate of the MCMC for tau

delta.accept.rate

The acceptance rate of the MCMC for the time delay

Details

The function bayesian produces posterior samples of the model parameters one of which is the time delay.

References

Hyungsuk Tak, Kaisey Mandel, David A. van Dyk, Vinay L. Kashyap, Xiao-Li Meng, and Aneta Siemiginowska (in progress). Bayesian and Profile Likelihood Approaches to Time Delay Estimation for Stochastic Time Series of Gravitationally Lensed Quasars

Examples

  # Loading datasets
  data(simple)
  head(simple)
  library(mnormt)

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

  # Cubic microlensing model (m = 3)
  output = bayesian(dat = simple, data.flux = FALSE, theta.ini = c(0, 0.03, 100), 
                    delta.ini = 50, delta.uniform.range = c(0, 100), 
                    delta.proposal.scale = 1, 
                    tau.proposal.scale = 3, tau.prior.shape = 1, tau.prior.scale = 1,
                    sigma.prior.shape = 1, sigma.prior.scale = 2 / 10^7, asis = TRUE, micro = 3,
                    sample.size = 100, warmingup.size = 50)

  names(output)

  # hist(output$delta)
  # plot(output$delta, type = "l")
  # acf(output$delta)