Posterior Results Analysis
Last updated: 2024-07-15
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Introduction
This vignette provides the code for the analysis of the posterior results of MATHPOP model fitted to the 40 Perseus LSBGs considered in the original Li et al. (2024) paper. It also reproduces Figure 4 and provides the data contained in Table 3 of the paper. To fit the MATHPOP model, see here.
Note that the MCMC results in the MATHPOP paper were obtained using
an older version of the code contained in the source/
folder of this repo. The current version of the code, i.e., the main
fit_MATHPOP function was rewritten so that it is more
user-friendly. Due to the update, it is unavoidable that the results
from fit_MATHPOP do not match exactly with the results from
the older code since various tuning and starting parameter values are
specified individually for each data set that was fitted, while these
are completely automatized in the current code. Because of the
computational constraints, we did not rerun everything using the new
code, but our testing shows that the new code provides entirely
consistent results as the old code. If you wish to use the old code,
please see the source files in the source/ folder.
Post Inference Results
Due to the computational time to fit the MATHPOP model and the size
of the MCMC results, we will not be fitting the model in any of the
vignettes in this Github repo. We instead have post-processed and saved
three individual runs of the MCMC runs of the MATHPOP model for each
individual image we considered. These are contained in the
data/v#### folders, where v### is the imaging
field ID. For example, data/v11acs/res_prob_v11acs.RDS
contains the MCMC results for the MATHPOP model fitted to the DOLPHOT GC
data.
We first load all required packages and read in the MCMC results for MATHPOP models fitted to probabilistic GC catalog from DOLPHOT:
library(tidyverse)
library(ggpubr)
library(ggridges)
library(tikzDevice)
library(posterior)
library(xtable)
library(wesanderson)
library(ggnewscale)
library(modeest)
library(reshape2)
library(HDInterval)
library(MATHPOP)
# read in file paths
fnames <- list.files('data/')
fnames <- fnames[grepl('v', fnames)]
file_paths <- paste0('data/',fnames,'/')
# construct data frames for each model parameter of the GC system of a UDG
summary_NGC <- data.frame()
summary_RGC <- data.frame()
summary_muGC <- data.frame()
# generate a data frame containing all summary statistics of the model parameters
set.seed(123456)
# loop through each image
for(i in 1:length(fnames)){
res_i_name <- paste0(file_paths[i], 'res_prob_', fnames[i], '.RDS')
file_read <- readRDS(res_i_name)
# GC count summary
N_GC <- as.data.frame(apply(file_read[, grepl( "N_" , names(file_read)) & !grepl( "gal" , names(file_read))], 2,
function(x) rpois(nrow(file_read), x)))
# Half-number radius summary
R_GC <- as.data.frame(file_read[, grepl( "R_" , names(file_read)) & !grepl( "gal" , names(file_read))])
# GCLF TO summary
mu_GC <- as.data.frame(file_read[, grepl( "mu_" , names(file_read)) & !grepl( "gal" , names(file_read)) & !grepl( "_0" , names(file_read))])
UDG_ID <- gsub('N_', '', colnames(N_GC))
colnames(N_GC) <- UDG_ID
colnames(R_GC) <- UDG_ID
colnames(mu_GC) <- UDG_ID
# obtain summary statistics for GC counts
N_GC <- melt(N_GC)
summary_N <- N_GC %>%
rename(ID = variable) %>%
group_by(ID) %>%
summarise_all(list(N_mode = mlv, N_median = median, N_mean = mean,
N_HDI_0.68_lower = function(x) hdi(x, 0.68)[1],
N_HDI_0.68_upper = function(x) hdi(x, 0.68)[2],
`P(N_GC = 0)` = function(x) mean(x == 0)))
# obtain summary statistics for half-number radius
R_GC <- melt(R_GC)
summary_R <- R_GC %>%
rename(ID = variable) %>%
group_by(ID) %>%
summarise_all(list(R_mode = mlv, R_median = median, R_mean = mean,
R_HDI_0.68_lower = function(x) hdi(x, 0.68)[1],
R_HDI_0.68_upper = function(x) hdi(x, 0.68)[2]))
# obtain summary statistics for GCLF TO
mu_GC <- melt(mu_GC)
summary_mu <- mu_GC %>%
rename(ID = variable) %>%
group_by(ID) %>%
summarise_all(list(mu_mode = mlv, mu_median = median, mu_mean = mean,
mu_HDI_0.68_lower = function(x) hdi(x, 0.68)[1],
mu_HDI_0.68_upper = function(x) hdi(x, 0.68)[2],
mu_HDI_0.95_lower = function(x) hdi(x, 0.95)[1],
mu_HDI_0.95_upper = function(x) hdi(x, 0.95)[2]))
summary_NGC <- bind_rows(summary_NGC, summary_N)
summary_RGC <- bind_rows(summary_RGC, summary_R)
summary_muGC <- bind_rows(summary_muGC, summary_mu)
}
summary_NGC$ID <- as.character(summary_NGC$ID)
summary_RGC$ID <- as.character(summary_RGC$ID)
summary_muGC$ID <- as.character(summary_muGC$ID)
# W7 is observed in both V12-ACS and V14-ACS
summary_NGC$ID[13] <- summary_RGC$ID[13] <- summary_muGC$ID[13] <- 'W7 (V12ACS)'
summary_NGC$ID[22] <- summary_RGC$ID[22] <- summary_muGC$ID[22] <- 'W7 (V14ACS)'
# post-processing
summary_muGC <- summary_muGC %>%
mutate(minus_mu = mu_mean - mu_HDI_0.68_lower, plus_mu = mu_HDI_0.68_upper - mu_mean)
summary_RGC <- summary_RGC %>%
mutate(minus_R = R_mode - R_HDI_0.68_lower, plus_R = R_HDI_0.68_upper - R_mode)
summary_NGC <- summary_NGC %>%
mutate_at(2:7, as.numeric) %>%
arrange(`P(N_GC = 0)`)
# GC count summary from Janssens et al. 2024
Jans_NGC <- data.frame(ID = summary_NGC$ID) %>%
mutate(Janssens = c(36, 52, 43, 20, 24, 23, 13, 20, 4, 9, 6, 13, 2, -2, 0, 8, 1, 9, 7, 9, 0, -2, 3,
0, 7, -2, 0, 1, -4, -1, 1, -4, 4, 2, 6, 4, -1, -5, -2, -1, -1),
J_se = c(8, 8, 6, 6, 13, 7, 6, 14, 8, 9, 8, 5, 7, 8, 10, 10, 7, 6, 6, 7, 10, 8, 12, 7,
8, 10, 9, 10, 9, 7, 8, 9, 8, 9, 7, 6, 5, 6, 7, 7, 7),
Re = c(2.16, 2.23, 1.99, 1.42, 5.44, 4.57, 2.72, 1.83, 1.15, 1.5, 2.22, 2.31, 1.59, 2.11,
1.17, 1.8, 1.31, 2.58, 1.92, 1.99, 1.17, 1.72, 1.67, 1.82, 1.5, 0.83, 0.97, 3.7,
3.13, 1.5, 0.91, 1.5, 0.96, 0.63, 1, 1.31, 1.54, 1.84, 1.31, 0.64, 0.81))
summary_NGC <- merge(summary_NGC, Jans_NGC)
summary_dat <- merge(merge(summary_NGC, summary_muGC, by = 'ID'), summary_RGC, by = 'ID') %>%
arrange(desc(N_mode)) %>%
mutate(data = 'DOLPHOT')We will do the same for results obtained using SExtractor (J24) probabilistic GC catalog, and combine the summary table with the one from DOLPHOT:
# everything here is the same as above, only changing the object names to identify it is based on J24.
summary_NGC_J <- data.frame()
summary_RGC_J <- data.frame()
summary_muGC_J <- data.frame()
set.seed(123456)
for(i in 1:length(fnames)){
res_i_name <- paste0(file_paths[i], 'res_prob_', fnames[i], '_Jans.RDS')
file_read <- readRDS(res_i_name)
N_GC <- as.data.frame(apply(file_read[, grepl( "N_" , names(file_read)) & !grepl( "gal" , names(file_read))], 2,
function(x) rpois(nrow(file_read), x)))
R_GC <- as.data.frame(file_read[, grepl( "R_" , names(file_read)) & !grepl( "gal" , names(file_read))])
mu_GC <- as.data.frame(file_read[, grepl( "mu_" , names(file_read)) & !grepl( "gal" , names(file_read)) & !grepl( "_0" , names(file_read))])
UDG_ID <- gsub('N_', '', colnames(N_GC))
colnames(N_GC) <- UDG_ID
colnames(R_GC) <- UDG_ID
colnames(mu_GC) <- UDG_ID
N_GC <- melt(N_GC)
summary_N <- N_GC %>%
rename(ID = variable) %>%
group_by(ID) %>%
summarise_all(list(N_mode = mlv, N_median = median, N_mean = mean,
N_HDI_0.68_lower = function(x) hdi(x, 0.68)[1],
N_HDI_0.68_upper = function(x) hdi(x, 0.68)[2],
`P(N_GC = 0)` = function(x) mean(x == 0)))
R_GC <- melt(R_GC)
summary_R <- R_GC %>%
rename(ID = variable) %>%
group_by(ID) %>%
summarise_all(list(R_mode = mlv, R_median = median, R_mean = mean,
R_HDI_0.68_lower = function(x) hdi(x, 0.68)[1],
R_HDI_0.68_upper = function(x) hdi(x, 0.68)[2]))
mu_GC <- melt(mu_GC)
summary_mu <- mu_GC %>%
rename(ID = variable) %>%
group_by(ID) %>%
summarise_all(list(mu_mode = mlv, mu_median = median, mu_mean = mean,
mu_HDI_0.68_lower = function(x) hdi(x, 0.68)[1],
mu_HDI_0.68_upper = function(x) hdi(x, 0.68)[2],
mu_HDI_0.95_lower = function(x) hdi(x, 0.95)[1],
mu_HDI_0.95_upper = function(x) hdi(x, 0.95)[2]))
summary_NGC_J <- bind_rows(summary_NGC_J, summary_N)
summary_RGC_J <- bind_rows(summary_RGC_J, summary_R)
summary_muGC_J <- bind_rows(summary_muGC_J, summary_mu)
}
summary_NGC_J$ID <- as.character(summary_NGC_J$ID)
summary_RGC_J$ID <- as.character(summary_RGC_J$ID)
summary_muGC_J$ID <- as.character(summary_muGC_J$ID)
summary_NGC_J$ID[13] <- summary_RGC_J$ID[13] <- summary_muGC_J$ID[13] <- 'W7 (V12ACS)'
summary_NGC_J$ID[22] <- summary_RGC_J$ID[22] <- summary_muGC_J$ID[22] <- 'W7 (V14ACS)'
summary_NGC_J <- summary_NGC_J %>%
mutate_at(2:7, as.numeric) %>%
arrange(`P(N_GC = 0)`)
summary_NGC_J <- merge(summary_NGC_J, Jans_NGC)
summary_dat_J <- merge(merge(summary_NGC_J, summary_muGC_J, by = 'ID'), summary_RGC_J, by = 'ID') %>%
arrange(desc(N_mode)) %>%
mutate(data = 'SExtractor (J24)')
## save the combined summary results in a data object (only do it once)
# summary_all_dat <- merge(summary_dat, summary_dat_J, by = 'ID')
# saveRDS(summary_all_dat, 'data/summary_results.RDS')Results Comparison
Now we can plot the GC count comparison results using the two summary data sets:
# grabbing the summary statistics of GC counts from the two summary datasets
N_sum <- bind_rows(summary_dat, summary_dat_J) %>%
dplyr::select(ID, N_mean, N_mode, N_HDI_0.68_lower, N_HDI_0.68_upper, Janssens, J_se, data) %>%
melt(., id = c('ID', 'data', 'Janssens', 'J_se','N_HDI_0.68_lower', 'N_HDI_0.68_upper')) %>%
mutate(variable = ifelse(variable == 'N_mean', 'Mean', 'Mode'))
# plot it
color <- c('Mean' = "#E1AF00", 'Mode' = "#F21A00")
ggplot(N_sum, aes(value, Janssens, color = variable)) +
geom_rect(aes(xmin = N_HDI_0.68_lower, xmax = N_HDI_0.68_upper, ymin = Janssens - J_se, ymax = Janssens + J_se), fill = '#3B9AB2', color = 'black', alpha = 0.1, size = 0.1) +
geom_point(size = 1) + geom_abline(slope = 1, intercept = 0, linetype = 'dashed') +
xlab('$N_{\\mathrm{GC}}^{\\mathrm{est}}$ (Ours)') + ylab('$N_{\\mathrm{GC}}$ (J24)') + coord_fixed() + labs(color = 'Estimator') +
theme(panel.grid.major = element_blank(), panel.background = element_blank(),
panel.grid.minor = element_blank(), axis.line = element_line(colour = "gray"),
strip.background = element_blank()) +
scale_color_manual(values = color) +
facet_wrap(.~data)
GC count estimates comparison. (See Figure 4 in the original Li et al. (2024) paper for more details)
Note that Table 3 in the paper are not generated through code,
although the numerical results are all contained in
summary_dat and summary_dat_J.
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] stats graphics grDevices utils datasets methods base
other attached packages:
[1] MATHPOP_0.0.0.9000 HDInterval_0.2.4 reshape2_1.4.4 modeest_2.4.0
[5] ggnewscale_0.4.10 wesanderson_0.3.7 xtable_1.8-4 posterior_1.6.0
[9] tikzDevice_0.12.6 ggridges_0.5.6 ggpubr_0.6.0 lubridate_1.9.3
[13] forcats_1.0.0 stringr_1.5.1 dplyr_1.1.4 purrr_1.0.2
[17] readr_2.1.5 tidyr_1.3.1 tibble_3.2.1 ggplot2_3.5.1
[21] tidyverse_2.0.0 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] splines_4.3.2 later_1.3.2 klippy_0.0.0.9500
[4] polyclip_1.10-6 rpart_4.1.21 lifecycle_1.0.4
[7] sf_1.0-16 rstatix_0.7.2 rprojroot_2.0.4
[10] processx_3.8.4 lattice_0.22-5 MASS_7.3-60
[13] backports_1.5.0 magrittr_2.0.3 plotly_4.10.4
[16] sass_0.4.9 rmarkdown_2.27 jquerylib_0.1.4
[19] yaml_2.3.9 httpuv_1.6.12 askpass_1.2.0
[22] sp_2.1-4 spatstat.sparse_3.1-0 DBI_1.2.3
[25] abind_1.4-5 spatstat.model_3.2-11 mixtools_2.0.0
[28] tensorA_0.36.2.1 git2r_0.33.0 rmutil_1.1.10
[31] spatstat.utils_3.0-5 terra_1.7-78 units_0.8-5
[34] goftest_1.2-3 spatial_7.3-17 spatstat.random_3.2-3
[37] codetools_0.2-19 pdftools_3.4.0 tidyselect_1.2.1
[40] filehash_2.4-6 raster_3.6-26 farver_2.1.2
[43] stable_1.1.6 matrixStats_1.3.0 stats4_4.3.2
[46] spatstat.explore_3.2-7 jsonlite_1.8.8 e1071_1.7-14
[49] survival_3.5-7 segmented_2.1-0 tools_4.3.2
[52] progress_1.2.3 ggstar_1.0.4 Rcpp_1.0.12
[55] glue_1.7.0 gridExtra_2.3 xfun_0.45
[58] mgcv_1.9-0 distributional_0.4.0 priorsense_1.0.1
[61] loo_2.7.0 withr_3.0.0 timeSeries_4032.109
[64] fastmap_1.2.0 fansi_1.0.6 callr_3.7.6
[67] digest_0.6.36 timechange_0.3.0 R6_2.5.1
[70] qpdf_1.3.3 colorspace_2.1-0 tensor_1.5
[73] spatstat.data_3.1-2 utf8_1.2.4 generics_0.1.3
[76] data.table_1.15.4 class_7.3-22 prettyunits_1.2.0
[79] httr_1.4.7 htmlwidgets_1.6.4 whisker_0.4.1
[82] pkgconfig_2.0.3 gtable_0.3.5 timeDate_4032.109
[85] htmltools_0.5.8.1 carData_3.0-5 clue_0.3-65
[88] scales_1.3.0 knitr_1.48 rstudioapi_0.16.0
[91] tzdb_0.4.0 spatstat.linnet_3.1-5 statip_0.2.3
[94] checkmate_2.3.1 nlme_3.1-163 proxy_0.4-27
[97] cachem_1.1.0 KernSmooth_2.23-22 parallel_4.3.2
[100] fBasics_4032.96 pillar_1.9.0 grid_4.3.2
[103] vctrs_0.6.5 VGAM_1.1-11 promises_1.3.0
[106] car_3.1-2 cluster_2.1.4 evaluate_0.24.0
[109] tinytex_0.51 magick_2.8.3 cli_3.6.3
[112] compiler_4.3.2 rlang_1.1.4 crayon_1.5.3
[115] ggsignif_0.6.4 labeling_0.4.3 classInt_0.4-10
[118] ps_1.7.7 getPass_0.2-4 plyr_1.8.9
[121] fs_1.6.4 stringi_1.8.4 viridisLite_0.4.2
[124] deldir_2.0-4 assertthat_0.2.1 munsell_0.5.1
[127] lazyeval_0.2.2 spatstat.geom_3.2-9 Matrix_1.6-3
[130] hms_1.1.3 stabledist_0.7-1 highr_0.11
[133] spatstat_3.0-8 kernlab_0.9-32 broom_1.0.6
[136] bslib_0.7.0