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Introduction

This vignettes contains the analysis and code that compares the results obtained from MATHPOP and those from the standard method, specifically for the UDGs — R27 and R84, as these two UDGs have the largest discrepancy between the two methods. We also reproduces Figure 5 in the original Li et al. (2024) paper.

Data For R27 and R84

R27 is in the field V10-WFC3 and R84 is in the field V7-ACS. We first read in the GC data for R27 from both DOLPHOT and SExtractor in below:

library(tidyverse)
library(sf)
library(sp)
library(raster)
library(parallel)
library(Rcpp)
library(RcppArmadillo)
library(posterior)
library(coda)
library(loo)
library(xtable)
library(priorsense)
library(ggpubr)
library(wesanderson)
library(reshape2)
library(tikzDevice)

sourceCpp('code/cpp_help_func.cpp')
source('code/fit_mod_MCMC.R')

# DOLPHOT V10WFC3
V10WFC <- read_csv('data/prob_GC_data/v10wfc3_pGC.csv')
# SExtractor V10WFC3
V10WFC_J <- read_csv('data/prob_GC_data/V10WFC3_pGC_Jans.csv')

# transform pixel coordinates to pyhsical
V10WFC <- as.data.frame(V10WFC)
V10WFC[,c('x','y')] <- 62*V10WFC[,c('x','y')]/4400
V10WFC_J <- as.data.frame(V10WFC_J)
V10WFC_J[,c('x','y')] <- 62*V10WFC_J[,c('x','y')]/4400

# locations of UDGs in V10WFC3
c <- 62*rbind(c(808, 744), c(1930, 2653), c(2695, 2132))/4400

Next, generate cutout data for R27:

# cutout GC data (within 7.5 kpc radius) for R27 from DOLPHOT 
R27_dat_harris <- filter(V10WFC, (x - c[1, 1])^2 + (y - c[1,2])^2 < 7.5^2)
R27_dat_harris$p <- rowMeans(R27_dat_harris[,8:507])
R27_dat_harris <- R27_dat_harris %>%
  filter(p > 0.05) %>%
  dplyr::select(x, y, M, C, p) %>%
  mutate(DATA = 'L24 ($p(\\mathrm{GC}) > 0.05$)')

# cutout probabilistic GC data (within 7.5 kpc radius) for R27 from SExtractor
R27_dat_Jans <- filter(V10WFC_J, (x - c[1, 1])^2 + (y - c[1,2])^2 < 7.5^2)
R27_dat_Jans$p <- rowMeans(R27_dat_Jans[,8:507])
R27_dat_Jans <- R27_dat_Jans %>%
  filter(p > 0.05) %>%
  dplyr::select(x, y, M, C, p) %>%
  mutate(DATA = 'Prob J24 ($p(\\mathrm{GC}) > 0.05$)')

# cutout binary GC data (within 7.5 kpc radius) for R27 from SExtractor (J24)
R27_dat_Jans_bin <- filter(V10WFC_J, (x - c[1, 1])^2 + (y - c[1,2])^2 < 7.5^2)
R27_dat_Jans_bin$p <- 1
R27_dat_Jans_bin <- R27_dat_Jans_bin %>%
  filter(p > 0 & M < 26.3 & C > 0.8 & C < 2.4) %>%
  dplyr::select(x, y, M, C, p) %>%
  mutate(DATA = 'Binary J24')

# combine all data sources
R27_dat <- bind_rows(R27_dat_harris, R27_dat_Jans, R27_dat_Jans_bin)

# construct annotation text in the figure
ann_text_R27 <- data.frame(x = c(9, 9, 16.5, 9, 9, 16, 8.5, 16.5), y = c(17, 16, 5.5, 17, 16, 5.5, 16.5, 5.5), 
                           lab = c("$N_{\\mathrm{GC}}: 26_{-9}^{+11}$ (Ours)",
                                   '$\\mu_{\\mathrm{TO}}: 25.35_{-0.28}^{+0.38}$ mag (Ours)',
                                   '$N_{\\mathrm{cand}} = 26.3$',
                                   "$N_{\\mathrm{GC}}: 37_{-13}^{+10}$ (Ours)",
                                   '$\\mu_{\\mathrm{TO}}: 25.75_{-0.25}^{+0.32}$ mag (Ours)',
                                   '$N_{\\mathrm{cand}} = 27.6$',
                                   '$N_{\\mathrm{GC}}: 52\\pm 8$ (J24)',
                                   '$N_{\\mathrm{cand}} = 37$'),
                           DATA = c(rep(c('Prob J24 ($p(\\mathrm{GC}) > 0.05$)', 'L24 ($p(\\mathrm{GC}) > 0.05$)'), each = 3), rep('Binary J24', 2)))

R27_dat$DATA <- factor(R27_dat$DATA, levels = c("L24 ($p(\\mathrm{GC}) > 0.05$)", "Binary J24", "Prob J24 ($p(\\mathrm{GC}) > 0.05$)"))
ann_text_R27$DATA <-  factor(ann_text_R27$DATA, levels = c("L24 ($p(\\mathrm{GC}) > 0.05$)", "Binary J24", "Prob J24 ($p(\\mathrm{GC}) > 0.05$)"))
ann_text_R27$ID <- 'R27'

R27_dat$ID <- 'R27'

Now do the same thing for R84:

# grab V7ACS GC data
V7ACS <- read_csv('data/prob_GC_data/v7acs_pGC.csv')
V7ACS_J <- read_csv('data/prob_GC_data/V7ACS_pGC_Jans.csv')

V7ACS <- as.data.frame(V7ACS)
V7ACS[,c('x','y')] <- 76*V7ACS[,c('x','y')]/4300
V7ACS_J <- as.data.frame(V7ACS_J)
V7ACS_J[,c('x','y')] <- 76*V7ACS_J[,c('x','y')]/4300

# UDG locations in V7ACS
c <- 76*rbind(c(661, 2987), c(1756, 752))/4300

# DOLPHOT GC data
R84_dat_harris <- filter(V7ACS, (x - c[1, 1])^2 + (y - c[1,2])^2 < 7.5^2)
R84_dat_harris$p <- rowMeans(R84_dat_harris[,8:507])
R84_dat_harris <- R84_dat_harris %>%
  filter(p > 0.05) %>%
  dplyr::select(x, y, M, C, p) %>%
  mutate(DATA = 'L24 ($p(\\mathrm{GC}) > 0.05$)')

# SExtractor probabilistic GC data
R84_dat_Jans <- filter(V7ACS_J, (x - c[1, 1])^2 + (y - c[1,2])^2 < 7.5^2)
R84_dat_Jans$p <- rowMeans(R84_dat_Jans[,8:507])
R84_dat_Jans <- R84_dat_Jans %>%
  filter(p > 0.05) %>%
  dplyr::select(x, y, M, C, p) %>%
  mutate(DATA = 'Prob J24 ($p(\\mathrm{GC}) > 0.05$)')

# SExtractor binary GC data
R84_dat_Jans_bin <- filter(V7ACS_J, (x - c[1, 1])^2 + (y - c[1,2])^2 < 7.5^2)
R84_dat_Jans_bin$p <- 1
R84_dat_Jans_bin <- R84_dat_Jans_bin %>%
  filter(p > 0 & M < 26.3 & C > 0.8 & C < 2.4) %>%
  dplyr::select(x, y, M, C, p) %>%
  mutate(DATA = 'Binary J24')

#combine all data
R84_dat <- bind_rows(R84_dat_harris, R84_dat_Jans, R84_dat_Jans_bin) %>%
  filter(C > 0.8)

# construct annotation text
ann_text_R84 <- data.frame(x = c(14.7, 14.7, 7.5, 14.4, 14.4, 7.5, 15.5, 7.5), y = c(59.5, 58.7, 47.5, 59.5, 58.7, 47.5, 59.1, 47.5), 
                       lab = c("$N_{\\mathrm{GC}}: 19\\pm 9$ (Ours)",
                               '$\\mu_{\\mathrm{TO}}: 25.96_{-0.46}^{+0.43}$ mag (Ours)',
                               '$N_{\\mathrm{cand}} = 18.0$',
                               "$N_{\\mathrm{GC}}: 32\\pm 13$ (Ours)",
                               '$\\mu_{\\mathrm{TO}}: 26.12_{-0.39}^{+0.43}$ mag (Ours)',
                               '$N_{\\mathrm{cand}} = 17.4$',
                               '$N_{\\mathrm{GC}}: 43\\pm 6$ (J24)',
                               '$N_{\\mathrm{cand}} = 26$'),
                       DATA = c(rep(c('Prob J24 ($p(\\mathrm{GC}) > 0.05$)', 'L24 ($p(\\mathrm{GC}) > 0.05$)'), each = 3), rep('Binary J24',2)))

R84_dat$DATA <- factor(R84_dat$DATA, levels = c("L24 ($p(\\mathrm{GC}) > 0.05$)", "Binary J24", "Prob J24 ($p(\\mathrm{GC}) > 0.05$)"))
ann_text_R84$DATA <-  factor(ann_text_R84$DATA, levels = c("L24 ($p(\\mathrm{GC}) > 0.05$)", "Binary J24", "Prob J24 ($p(\\mathrm{GC}) > 0.05$)"))
ann_text_R84$ID <- 'R84'

ann_text <- bind_rows(ann_text_R27, ann_text_R84)

R84_dat$ID <- 'R84'

# combine all things together
LSBG_dat <- bind_rows(R27_dat, R84_dat)

Now we can plot the cutouts of GC candidates from different catalogs:

ggplot(LSBG_dat, aes(x, y)) + geom_point(aes(color = M, size = p*1.5)) + facet_grid(ID~DATA, scales = 'free', switch = 'y') +
  scale_alpha_identity() + scale_size_identity() +
  geom_text(aes(x = x, y = y, label = lab), data = ann_text, size = 2) +
  scale_color_viridis_c(direction = -1, name = 'F814W') + 
  theme_bw() + 
  theme(axis.title=element_blank(), 
        axis.text=element_blank(),
        axis.ticks=element_blank(),
        strip.background = element_blank(),
        panel.grid.minor = element_blank(),
        panel.grid.major = element_blank(),
        legend.justification = "left",
        legend.box.margin = margin(l = 0.1, unit = "cm"),
        aspect.ratio = 1) +
  guides(fill = guide_legend(title.position = "top", title.hjust = 0.5))
GC catalogs comparison for R27 and R84. (See Figure 5 in the original Li et al. (2024) paper for more details)

GC catalogs comparison for R27 and R84. (See Figure 5 in the original Li et al. (2024) paper for more details)

Version Author Date
388826c david.li 2024-07-05

Posterior Probability that a GC belongs to R27

We here show the analysis done to obtain the posterior probability that a GC candidate indeed belongs to R27. Note that the figure produced here was not included in the original paper.

# read in the MCMC results for V10WFC3
res_V10WFC <- readRDS('data/v10wfc3/res_prob_v10wfc3.RDS')

# thin the chain for easier computation
res_V10WFC_thinned <- as.data.frame(res_V10WFC[seq(1, nrow(res_V10WFC), by = 405),])

# locations of UDGs in V10WFC3
c <- 62*rbind(c(808, 744), c(1930, 2653), c(2695, 2132))/4400
# fixed parameters
e <- c(0.88, 1.35, 0.75)
theta <- c(0, -pi/12, 0)

# spatial grid for computing posterior GC membership probabilities
loc <- as.matrix(expand.grid(x = seq(11.38545 - 7.5, 11.38545 + 7.5, by = 0.24), y = seq(10.48364 - 7.5, 10.48364 + 7.5, by = 0.24)))

# function to compute the R27 membership probability for regions around R27
compute_mem_prob <- function(loc, res_prob, e, theta){
  prob <- numeric(nrow(loc))
  loc <- matrix(loc, ncol = 2)
  for (i in 1:nrow(res_prob)) {
    tot <- Sersic_ints(loc, c[1, ], res_prob[i, 'N_R27'], res_prob[i, 'R_R27'],
                       e[1], res_prob[i, 'n_R27'], theta[1]) + 
           Sersic_ints(loc, c[2, ], res_prob[i, 'N_W84'], res_prob[i, 'R_W84'],
                       e[2], res_prob[i, 'n_W84'], theta[2]) +
           Sersic_ints(loc, c[3, ], res_prob[i, 'N_W83'], res_prob[i, 'R_W83'],
                       e[3], res_prob[i, 'n_W83'], theta[3]) + res_prob[i, 'b0']
    
    prob <- prob + Sersic_ints(loc, c[1, ], res_prob[i, 'N_R27'], res_prob[i, 'R_R27'],
                        e[1], res_prob[i, 'n_R27'], theta[1])/tot 
  }
  prob <- prob/nrow(res_prob)
  loc <- data.frame(x = loc[,1], y = loc[,2]) %>%
         mutate(p = prob)
  return(loc)
}

# compute the probability
R27_dat_loc <- compute_mem_prob(loc, res_V10WFC_thinned, e, theta)

# plot it
ggplot(R27_dat_loc, aes(x,y)) + geom_contour_filled(aes(z = p)) +
  scale_fill_manual(values = wes_palette("Zissou1", n = 18, type = "continuous")[seq(1,18, by = 2)], 
                    name = '$\\hat{\\pi}_{\\mathrm{R27}}(s_i)$') +
  geom_point(data = R27_dat_harris, aes(x, y), size = 0.25) + coord_fixed() + 
  theme_minimal() + 
  theme(axis.title=element_blank(), 
        axis.text=element_blank(), 
        axis.ticks=element_blank(),
        strip.background = element_blank(),
        legend.justification = "left",
        legend.box.margin = margin(l = 0.1, unit = "cm")) +
  guides(fill = guide_legend(title.position = "top", title.hjust = 0.5, reverse = TRUE))
Probabilities that a GC candidates belong to R27.

Probabilities that a GC candidates belong to R27.

Version Author Date
388826c david.li 2024-07-05

From the above figure, we see that the posterior probabilities for GC candidates in the outer region of R27 are only around \(50\%\). This means that the membership of these GC candidates are highly uncertain, which is one factor that contributes to the big difference between the estimates from MATHPOP and the standard method.


sessionInfo()
R version 4.3.2 (2023-10-31)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Sonoma 14.1.1

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

time zone: America/Toronto
tzcode source: internal

attached base packages:
[1] parallel  stats     graphics  grDevices utils     datasets  methods  
[8] base     

other attached packages:
 [1] progress_1.2.2           tikzDevice_0.12.5        reshape2_1.4.4          
 [4] wesanderson_0.3.7        ggpubr_0.6.0             priorsense_0.0.0.9000   
 [7] xtable_1.8-4             loo_2.6.0                coda_0.19-4             
[10] posterior_1.5.0          RcppArmadillo_0.12.6.6.0 Rcpp_1.0.11             
[13] raster_3.6-26            sp_2.1-1                 sf_1.0-14               
[16] lubridate_1.9.3          forcats_1.0.0            stringr_1.5.1           
[19] dplyr_1.1.4              purrr_1.0.2              readr_2.1.4             
[22] tidyr_1.3.0              tibble_3.2.1             ggplot2_3.4.4           
[25] tidyverse_2.0.0          workflowr_1.7.1         

loaded via a namespace (and not attached):
 [1] DBI_1.1.3            rlang_1.1.4          magrittr_2.0.3      
 [4] git2r_0.33.0         matrixStats_1.1.0    e1071_1.7-13        
 [7] compiler_4.3.2       getPass_0.2-4        png_0.1-8           
[10] callr_3.7.3          vctrs_0.6.5          pkgconfig_2.0.3     
[13] crayon_1.5.2         fastmap_1.2.0        magick_2.8.3        
[16] backports_1.4.1      labeling_0.4.3       utf8_1.2.4          
[19] promises_1.2.1       rmarkdown_2.25       tzdb_0.4.0          
[22] ps_1.7.5             tinytex_0.48         bit_4.0.5           
[25] xfun_0.41            cachem_1.0.8         jsonlite_1.8.7      
[28] highr_0.10           later_1.3.1          terra_1.7-55        
[31] broom_1.0.5          prettyunits_1.2.0    R6_2.5.1            
[34] bslib_0.5.1          stringi_1.8.4        car_3.1-2           
[37] jquerylib_0.1.4      assertthat_0.2.1     knitr_1.45          
[40] klippy_0.0.0.9500    filehash_2.4-5       httpuv_1.6.12       
[43] timechange_0.2.0     tidyselect_1.2.0     rstudioapi_0.15.0   
[46] abind_1.4-5          yaml_2.3.7           codetools_0.2-19    
[49] processx_3.8.2       qpdf_1.3.3           lattice_0.22-5      
[52] plyr_1.8.9           withr_2.5.2          askpass_1.2.0       
[55] evaluate_0.23        isoband_0.2.7        units_0.8-4         
[58] proxy_0.4-27         pillar_1.9.0         carData_3.0-5       
[61] tensorA_0.36.2       whisker_0.4.1        KernSmooth_2.23-22  
[64] checkmate_2.3.1      distributional_0.3.2 generics_0.1.3      
[67] vroom_1.6.4          rprojroot_2.0.4      hms_1.1.3           
[70] munsell_0.5.0        scales_1.3.0         class_7.3-22        
[73] glue_1.6.2           tools_4.3.2          ggsignif_0.6.4      
[76] pdftools_3.4.0       fs_1.6.3             grid_4.3.2          
[79] colorspace_2.1-0     cli_3.6.1            fansi_1.0.6         
[82] viridisLite_0.4.2    zipfR_0.6-70         gtable_0.3.4        
[85] rstatix_0.7.2        sass_0.4.7           digest_0.6.36       
[88] classInt_0.4-10      farver_2.1.1         htmltools_0.5.8.1   
[91] lifecycle_1.0.4      httr_1.4.7           bit64_4.0.5