PCA vs. t-SNE and UMAP: an illustration
Peter Carbonetto
Last updated: 2021-01-03
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Knit directory: single-cell-topics/analysis/
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Here we contrast use of a simple linear dimensionality reduction technique, PCA, with nonlinear dimensionality reduction methods t-SNE and UMAP.
Load the packages used in the analysis below.
library(Matrix)
library(fastTopics)
library(Rtsne)
library(uwot)
library(ggplot2)
library(cowplot)Load the count data, the \(K = 6\) topic model fit, and the 8 clusters identified in the clustering analysis
load("../data/pbmc_purified.RData")
fit <- readRDS(file.path("../output/pbmc-purified/rds",
"fit-pbmc-purified-scd-ex-k=6.rds"))$fit
fit <- poisson2multinom(fit)
samples <- readRDS("../output/pbmc-purified/clustering-pbmc-purified.rds")To begin, draw a random subset of 2,000 cells from the B, CD14+ and CD34+ clusters identified above. (The main reason for taking a random subset is that we don’t want to wait a long time for t-SNE and UMAP to complete.)
set.seed(5)
rows <- which(with(samples,
cluster == "B" |
cluster == "CD14+" |
cluster == "CD34+"))
rows <- sort(sample(rows,2000))
fit2 <- select_loadings(fit,loadings = rows)
x <- samples$cluster[rows,drop = TRUE]Next, run PCA on the topic proportions for this random subset of 2,000 samples.
p1 <- pca_plot(fit2,fill = x) + labs(fill = "cluster")Run t-SNE on the topic proportions.
tsne <- Rtsne(fit2$L,dims = 2,pca = FALSE,normalize = FALSE,perplexity = 100,
theta = 0.1,max_iter = 1000,eta = 200,verbose = FALSE)
tsne$x <- tsne$Y
colnames(tsne$x) <- c("tsne1","tsne2")
p2 <- pca_plot(fit2,out.pca = tsne,fill = x) + labs(fill = "cluster")Then run UMAP on the topic proportions.
out.umap <- umap(fit2$L,n_neighbors = 30,metric = "euclidean",n_epochs = 1000,
min_dist = 0.1,scale = "none",learning_rate = 1,
verbose = FALSE)
out.umap <- list(x = out.umap)
colnames(out.umap$x) <- c("umap1","umap2")
p3 <- pca_plot(fit2,out.pca = out.umap,fill = x) + labs(fill = "cluster")Here are the PCA, t-SNE and UMAP 2-d embeddings, side-by-side:
plot_grid(p1,p2,p3,nrow = 1)
By the projection of the samples onto the first two PCs, the B-cells cluster is distinct from the others, whereas the CD14+ and CD34+ cells do not separate as well.
By contrast, this detail is not captured in the t-SNE and UMAP embeddings. This illustrates the tendency of t-SNE and UMAP to accentuate clusters in the data at the risk of distorting or obscuring finer scale substructure.
Note that the first 2 PCs should be sufficient for capturing the full structure in the topic proportions as they explain >96% of the variance:
summary(prcomp(fit2$L))
# Importance of components:
# PC1 PC2 PC3 PC4 PC5 PC6
# Standard deviation 0.4831 0.3269 0.10351 0.03397 0.02396 3.49e-16
# Proportion of Variance 0.6618 0.3030 0.03038 0.00327 0.00163 0.00e+00
# Cumulative Proportion 0.6618 0.9647 0.99510 0.99837 1.00000 1.00e+00
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
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# LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib
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# attached base packages:
# [1] stats graphics grDevices utils datasets methods base
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# other attached packages:
# [1] cowplot_1.0.0 ggplot2_3.3.0 uwot_0.1.8 Rtsne_0.15
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