# entirelogprofilelikelihood: Calculating the entire profilel likelihood curve over the... In timedelay: Time Delay Estimation for Stochastic Time Series of Gravitationally Lensed Quasars

## Description

`entirelogprofilelikelihood` calculates the entire profilel likelihood curve over the given grid values of the time delay.

## Usage

 ```1 2 3``` ```entirelogprofilelikelihood(data.lcA, data.lcB, grid, initial, data.flux, delta.uniform.range, micro) ```

## Arguments

 `data.lcA` A (n by 3) matrix; the first column has n observation times of light curve A, the second column has n flux (or magnitude) values of light curve A, the third column has n measurement errors of light curve A. `data.lcB` A (w by 3) matrix; the first column has w observation times of light curve B, the second column has w flux (or magnitude) values of light curve B, the third column has w measurement errors of light curve B. `grid` A vector containing values of the time delay on which the profile likelihood values are calculated. We recommend using the grid interval equal to 0.1. `initial` The initial values of the other model parameters (mu, log(sigma), log(tau), beta). We take log on sigma and tau for numerical stability. `data.flux` "True" if data are recorded on flux scale or "FALSE" if data are on magnitude scale. `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])). `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.

## Details

The function `entirelogprofilelikelihood` is used to obtain the entire profile likelihood curve over the given grid values of the time delay.

## Value

The outcome of `entirelogprofilelikelihood` is the values of the log profile likelihood function over the given grid values of the time delay.

Hyungsuk Tak

## References

Hyungsuk Tak, Kaisey Mandel, David A. van Dyk, Vinay L. Kashyap, Xiao-Li Meng, and Aneta Siemiginowska (2016+). "Bayesian Estimates of Astronomical Time Delays between Gravitationally Lensed Stochastic Light Curves," tentatively accepted in Annals of Applied Statistics (ArXiv 1602.01462).

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51``` ``` # Loading datasets data(simple) head(simple) ################################################ # Time delay estimation via profile likelihood # ################################################ # Subset (data for image A) of the typical quasar data set lcA <- simple[, 1 : 3] # Another subset (data for image B) of the typical quasar data set # The observation times for image B are not necessarily the same as those for image A lcB <- simple[, c(1, 4, 5)] # The two subsets do not need to have the same number of observations # For example, here we add one more observation time for image B lcB <- rbind(lcB, c(290, 1.86, 0.006)) dim(lcA) dim(lcB) ###### The entire profile likelihood values on the grid of values of the time delay. # Cubic microlensing model ti1 <- lcB[, 1] ti2 <- lcB[, 1]^2 ti3 <- lcB[, 1]^3 ss <- lm(lcB[, 2] - mean(lcA[, 2]) ~ ti1 + ti2 + ti3) initial <- c(mean(lcA[, 2]), log(0.01), log(200), ss\$coefficients) delta.uniform.range <- c(0, 100) grid <- seq(0, 100, by = 0.1) # grid interval "by = 0.1" is recommended, # but users can set a finer grid of values of the time delay. ### Running the following codes takes more time than CRAN policy ### Please type the following lines without "#" to run the function and to see the results # logprof <- entirelogprofilelikelihood(data.lcA = lcA, data.lcB = lcB, grid = grid, # initial = initial, data.flux = FALSE, # delta.uniform.range = delta.uniform.range, micro = 3) # plot(grid, logprof, type = "l", # xlab = expression(bold(Delta)), # ylab = expression(bold(paste("log L"[prof], "(", Delta, ")")))) # prof <- exp(logprof - max(logprof)) # normalization # plot(grid, prof, type = "l", # xlab = expression(bold(Delta)), # ylab = expression(bold(paste("L"[prof], "(", Delta, ")")))) ```

timedelay documentation built on May 29, 2017, 9:37 a.m.