Last updated: 2022-01-07

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Here we compare the quality of the fits obtained from the different updates (EM and SCD, with and without extrapolation), and with different numbers of topics, \(K\).

Load the packages used in the analysis below, as well as some additional functions for creating the plots.

library(Matrix)
library(fastTopics)
library(ggplot2)
library(cowplot)
library(ggrepel)
set.seed(1)

Load the 68k PBMC data, and the results of running fit_poisson_nmf on the 68k PBMC data, with different algorithms, and for various settings of \(K\).

load("../data/pbmc_68k.RData")
load("../output/pbmc68k/fits-pbmc68k.RData")
fits <- lapply(fits,poisson2multinom)

This plot shows the improvement in the log-likelihood as the rank, \(K\), is increased. The log-likelihoods are shown relative to the log-likelihood at \(K = 2\).

plot_loglik_vs_rank(fits) +
  theme_cowplot(font_size = 12)

Version Author Date
279ad71 Peter Carbonetto 2021-04-07

The next set of plots shows the improvement in the fit over time, for \(K\) from 2 to 12, using EM or SCD, with and without extrapolation. The quality of the fit is measured by the log-likelihood relative to the best log-likelihood that was identified among all methods compared.

prune_prefit_iters <- function (fit) {
  n <- nrow(fit$progress)
  fit$progress <- fit$progress[1000:n,]
  fit$progress <- transform(fit$progress,timing = timing/60^2)
  return(fit)
}
create_progress_plot <- function (fits, k, y = "loglik")
  plot_progress(fits,y = y,add.point.every = 100,shapes = 21,
                colors = c("dodgerblue","red","dodgerblue","red"),
                fills = c("dodgerblue","red","white","white")) +
  scale_y_continuous(trans = "log10",breaks = 10^seq(-8,8)) +
  guides(color = "none",fill = "none",size = "none",
         shape = "none",linetype = "none") +
  labs(x = "runtime (h)",title = paste("K =",k)) +
  theme_cowplot(font_size = 10) +
  theme(plot.title = element_text(size = 10,face = "plain"))
fits <- lapply(fits,prune_prefit_iters)
p <- vector("list",12)
for (i in 2:12)
  p[[i]] <- create_progress_plot(fits[dat$k == i],i)
plot_grid(p[[2]],p[[3]],p[[4]],p[[5]],
          p[[6]],p[[7]],p[[8]],p[[9]],
          p[[10]],p[[11]],p[[12]],
          nrow = 3,ncol = 4)

Version Author Date
279ad71 Peter Carbonetto 2021-04-07

These plots shows the evolution of the KKT residuals over time.

for (i in 2:12)
  p[[i]] <- create_progress_plot(fits[dat$k == i],i,y = "res")
plot_grid(p[[2]],p[[3]],p[[4]],p[[5]],
          p[[6]],p[[7]],p[[8]],p[[9]],
          p[[10]],p[[11]],p[[12]],
          nrow = 3,ncol = 4)

Version Author Date
279ad71 Peter Carbonetto 2021-04-07

An example in which the EM and (extrapolated) CD estimates greatly differ:

topic_colors <- c("#a6cee3","#1f78b4","#b2df8a","#33a02c",
                  "#fb9a99","#e31a1c","#fdbf6f")
fit1 <- fits[["fit-pbmc68k-em-k=7"]]
fit2 <- fits[["fit-pbmc68k-scd-ex-k=7"]]
n    <- nrow(fit1$L)
pdat <- data.frame(x = as.vector(fit1$L),
                   y = as.vector(fit2$L),
                   k = factor(rep(1:7,each = n)))
pdat <- pdat[sample(7*n),]
p1 <- ggplot(pdat,aes(x = x,y = y,fill = k)) +
  geom_point(color = "white",shape = 21,size = 2) +
  scale_fill_manual(values = topic_colors) +
  labs(x = "EM estimate",y = "extrapolated CD estimate") +
  theme_cowplot(font_size = 12)
print(p1 + geom_abline(color = "black",linetype = "dotted"))

Version Author Date
50a3fca Peter Carbonetto 2021-04-10

An example in which the EM and (extrapolated) CD estimates largely agree:

topic_colors <- c("#8dd3c7","#ffffb3","#bebada","#fb8072","#80b1d3","#fdb462",
                  "#b3de69","#fccde5","#d9d9d9","#bc80bd","#ccebc5","#ffed6f")
fit1 <- fits[["fit-pbmc68k-em-k=12"]]
fit2 <- fits[["fit-pbmc68k-scd-ex-k=12"]]
n    <- nrow(fit1$L)
pdat <- data.frame(x = as.vector(fit1$L),
                   y = as.vector(fit2$L),
                   k = factor(rep(1:12,each = n)))
pdat <- pdat[sample(12*n),]
p2 <- ggplot(pdat,aes(x = x,y = y,fill = k)) +
  geom_point(color = "white",shape = 21,size = 2) +
  scale_fill_manual(values = topic_colors) +
  labs(x = "EM estimate",y = "extrapolated CD estimate") +
  theme_cowplot(font_size = 12)
print(p2 + geom_abline(color = "black",linetype = "dotted"))

Version Author Date
50a3fca Peter Carbonetto 2021-04-10

From the PCs of the mixture proportions, we define four clusters, roughly corresponding to (1) B cells, (2) dendritic and CD14+ monocytes, (3) CD34+ cells, and (4) T cells and natural killer cells:

fit  <- fits[["fit-pbmc68k-scd-ex-k=7"]]
pca  <- prcomp(fit$L)$x
n    <- nrow(pca)
x    <- rep("T+NK",n)
pc4  <- pca[,4]
pc6  <- pca[,6]
x[pc4 > 0.25] <- "B"
x[pc4 < -0.25] <- "CD14+dendritic"
x[pc6 > 0.8] <- "CD34+"
clusters <- factor(x)
table(clusters)
# clusters
#              B CD14+dendritic          CD34+           T+NK 
#           3733           4028            163          60655

The Structure plot summarizes the topic proportions in the cells. Here we subsample the larger clusters in order to better visualize the rarer cell types.

set.seed(1)
topics <- c(1:5,7,6)
topic_colors <- c("#a6cee3","#1f78b4","#b2df8a","#33a02c",
                  "#fb9a99","#e31a1c","#fdbf6f")
rows <- sort(c(sample(which(clusters == "B"),746),
               sample(which(clusters == "CD14+dendritic"),806),
               which(clusters == "CD34+"),
               sample(which(clusters == "T+NK"),1213)))
fit1 <- select_loadings(fit,loadings = rows)
tsne <- tsne_from_topics(fit1,dims = 1,verbose = FALSE)
p3 <- structure_plot(fit1,loadings_order = order(tsne[,1]),
                     grouping = clusters[rows],topics = topics,
                     colors = topic_colors[topics],gap = 30)
print(p3)

Version Author Date
72ca1a7 Peter Carbonetto 2021-04-12

Let’s compare this to the Structure plot we would have obtained had we used the model fit obtained by runnning the EM updates without any extrapolation:

fit2 <- fits[["fit-pbmc68k-em-k=7"]]
fit2 <- select_loadings(fit2,loadings = rows)
p3b <- structure_plot(fit2,loadings_order = order(tsne[,1]),
                      grouping = clusters[rows],topics = topics,
                      colors = topic_colors[topics],gap = 30)
print(p3b)

Many of the “driving genes” in topic 6—that is, genes showing the largest increase in expression relative to other topics—are ribosomal protein genes:

pdat <- data.frame(gene = genes$symbol,
                   x    = pmax(1e-6,apply(fit$F[,-6],1,max)),
                   y    = pmax(1e-6,fit$F[,6]),
                   stringsAsFactors = FALSE)
pdat <- transform(pdat,fc = y/x)
pdat[with(pdat,!(fc > 1.6 & x > 0.0005)),"gene"] <- ""
p4 <- ggplot(pdat,aes(x = x,y = y,label = gene)) +
  geom_point(color = "white",fill = "royalblue",shape = 21) +
  geom_text_repel(color = "darkgray",size = 2.25,segment.color = "darkgray",
                  segment.size = 0.25,min.segment.length = 0,
                  max.overlaps = Inf) +
  geom_abline(intercept = 0,slope = 1,color = "black",linetype = "dotted") +
  scale_x_continuous(trans = "log10",limits=10^c(-6,-1),breaks=10^seq(-6,-1)) +
  scale_y_continuous(trans = "log10",limits=10^c(-6,-1),breaks=10^seq(-6,-1)) +
  labs(x = "highest frequency in another topic",
       y = "frequency in topic 6") +
  theme_cowplot(font_size = 10)
print(p4)

Version Author Date
a2f3c07 Peter Carbonetto 2021-04-20
b099034 Peter Carbonetto 2021-04-15
5311f12 Peter Carbonetto 2021-04-15 

sessionInfo()
# R version 3.6.2 (2019-12-12)
# Platform: x86_64-apple-darwin15.6.0 (64-bit)
# Running under: macOS Catalina 10.15.7
# 
# Matrix products: default
# BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
# 
# locale:
# [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
# 
# attached base packages:
# [1] stats     graphics  grDevices utils     datasets  methods   base     
# 
# other attached packages:
# [1] ggrepel_0.9.1     cowplot_1.0.0     ggplot2_3.3.5     fastTopics_0.6-96
# [5] Matrix_1.2-18    
# 
# loaded via a namespace (and not attached):
#  [1] httr_1.4.2         tidyr_1.1.3        jsonlite_1.7.2     viridisLite_0.3.0 
#  [5] RcppParallel_4.4.2 assertthat_0.2.1   mixsqp_0.3-46      yaml_2.2.0        
#  [9] progress_1.2.2     pillar_1.6.2       backports_1.1.5    lattice_0.20-38   
# [13] quantreg_5.54      glue_1.4.2         quadprog_1.5-8     digest_0.6.23     
# [17] promises_1.1.0     colorspace_1.4-1   htmltools_0.4.0    httpuv_1.5.2      
# [21] pkgconfig_2.0.3    invgamma_1.1       SparseM_1.78       purrr_0.3.4       
# [25] scales_1.1.0       whisker_0.4        later_1.0.0        Rtsne_0.15        
# [29] MatrixModels_0.4-1 git2r_0.26.1       tibble_3.1.3       farver_2.0.1      
# [33] generics_0.0.2     ellipsis_0.3.2     withr_2.4.2        ashr_2.2-51       
# [37] pbapply_1.5-1      lazyeval_0.2.2     magrittr_2.0.1     crayon_1.4.1      
# [41] mcmc_0.9-6         evaluate_0.14      fs_1.3.1           fansi_0.4.0       
# [45] MASS_7.3-51.4      truncnorm_1.0-8    tools_3.6.2        data.table_1.12.8 
# [49] prettyunits_1.1.1  hms_1.1.0          lifecycle_1.0.0    stringr_1.4.0     
# [53] MCMCpack_1.4-5     plotly_4.9.2       munsell_0.5.0      irlba_2.3.3       
# [57] compiler_3.6.2     systemfonts_1.0.2  rlang_0.4.11       grid_3.6.2        
# [61] htmlwidgets_1.5.1  labeling_0.3       rmarkdown_2.3      gtable_0.3.0      
# [65] DBI_1.1.0          R6_2.4.1           knitr_1.26         dplyr_1.0.7       
# [69] uwot_0.1.10        utf8_1.1.4         workflowr_1.6.2    rprojroot_1.3-2   
# [73] ragg_0.3.1         stringi_1.4.3      parallel_3.6.2     SQUAREM_2017.10-1 
# [77] Rcpp_1.0.7         vctrs_0.3.8        tidyselect_1.1.1   xfun_0.11         
# [81] coda_0.19-3