# Calculating the entire profilel likelihood curve over the given grid values of the time delay `entirelogprofilelikelihood` calculates the entire profilel likelihood curve over the given grid values of the time delay. ```r entirelogprofilelikelihood(data.lcA, data.lcB, grid, initial, data.flux, delta.uniform.range, micro) ``` ## Arguments - `data.lcA`: A ($n_1$ by 3) matrix for image A (light curve A); the first column has $n_1$ observation times, the second column contains $n_1$ flux (or magnitude) values, the third column includes $n_1$ measurement errors. - `data.lcB`: A ($n_2$ by 3) matrix for image B (light curve B); the first column has $n_2$ observation times, the second column contains $n_2$ flux (or magnitude) values, the third column includes $n_2$ measurement errors. - `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. - `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. ## Returns The outcome of `entirelogprofilelikelihood` is the values of the log profile likelihood function over the given grid values of the time delay. ## 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. ## Author(s) Hyungsuk Tak ## Examples ```r # 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. 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, ")")))) ```

entirelogprofilelikelihood function