knitr::opts_chunk$set(echo = TRUE, tidy = TRUE, tidy.opts = list(comment = FALSE)) library(epca) library(Matrix) library(tidyverse)
This vignette walks you through how you could easily use epca
to explore your data.
Using sca
and sma
is simple. The only required input are the data matrix x
and k
--the number of sparse PCs (or rank of matrix decomposition).
To illustrate, we simulated a $300 \times 50$ example rank-5 data matrix x
with some additive Gaussian noise (the steps to generate this matrix is skipped; look for the source code of this vignette for details).
## simulate a rank-5 data matrix with some additive Gaussian noise n <- 300 p <- 50 k <- 5 ## rank z <- shrinkage(svd(matrix(runif(n * k), n, k))$u, gamma = sqrt(n)) b <- diag(5) * 3 y <- shrinkage(svd(matrix(runif(p * k), p, k))$u, gamma = sqrt(p)) e <- matrix(rnorm(n * p, sd = .01), n, p) x <- scale(z %*% b %*% t(y) + e)
## Sparse PCA sca(x, k = 5) ## Sparse matrix approximation sma(x, k = 5)
pitprops
dataIn this example, we apply sca
to the pitprops
data set. This data is a Gram matrix (i.e., covariance matrix). For this, we just need to set is.Cov = TRUE
. We look for 6 sparse PCs and set the sparsity parameter gamma = 6
. Here, the sparsity parameter controls the L1 norm of the returned PC loadings. The default of gamma
(if absent) is sqrt(p * k)
, where p
is the number of original variables. This default of gamma
is usually well sufficiently large.
data("pitprops", package = "epca") ## find 6 sparse PCs s.sca <- sca(pitprops, k = 6, gamma = 6)
There are two ways to inspect the sparse PC loadings. The first way is to directly extract the loadings
component in the output object: s.sca$loadings
. The second option is to call the print
generic function with the verbose = TRUE
option. It prints the original variables with non-zero loadings for each PC sequentially.
print(s.sca, verbose = TRUE)
load("scrnaseq.rda")
This example shows a large-scale application of sparse PCA to a single-cell RNA-seq data. For this example, we use the human/mouse pancreas single-cell RNA-seq data from Baron et al. (2017).
Fe used the single-cell RNA-seq data with the scRNAseq
package. We removed the genes that do not have any variation across samples (i.e., zero standard deviation) and the cell types that contain fewer than 100 cells. This resulted in a sparse data matrix pancreas
of 17499 genes (rows) and 8451 cells (columns) across nine cell types.
# library(scRNAseq) dat <- BaronPancreasData('human') # dim(dat) ## 20125 8569 gene.select <- !!apply(counts(dat), 1, sd) ## remove non-variance gene label.select <- colData(dat) %>% data.frame() %>% dplyr::count(label) %>% filter(n > 100) # label n # 1 acinar 958 # 2 activated_stellate 284 # 3 alpha 2326 # 4 beta 2525 # 5 delta 601 # 6 ductal 1077 # 7 endothelial 252 # 8 gamma 255 # 9 quiescent_stellate 173 dat1 <- dat[gene.select, colData(dat)$label %in% label.select$label]
For SCA, we use the expression count matrix (count
) as the input, where count[i,j]
is the expression level of gene j in cell i, with 10.8% being non-zero.
count <- counts(dat1) # dim(count) ## 17499 8451 # length(count@i) / length(count) ## %(nnz) ## 10.80605% non-zeros
The dataset contains labels for each cell.
label <- setNames(factor(dat1$label), colnames(dat1))
Next, We applied sca
to the transpose of count
to find k = 9
sparse gene PCs. Aiming for a small number of genes (i.e., non-zero loadings) in individual PCs, we set the sparsity parameter to gamma = log(pk)
, which is approximately 12.
scar <- sca(t(count), k = 9, gamma = 12, center = F, scale = F, epsilon = 1e-3)
We can exam the number of original genes included by each gene PC.
n.gene <- apply(!!scar$loadings, 2, sum) n.gene
Each gene PC uses a handful of original genes.
We can plot the component scores of the nine PCs, with dplyr
and ggplot2
packages. Each panel displays one of nine cell types with the names of cell types and the number of cells reported on the top strips. For each cell type, a box depicts the component scores for nine sparse gene PCs.
scar$scores %>% reshape2::melt(varnames = c("cell", "PC"), value.name = "scores") %>% mutate(PC = factor(PC), label = label[cell]) %>% ggplot(aes(PC, scores / 1000, fill = PC)) + geom_boxplot(color = "grey30", outlier.shape = NA, show.legend = FALSE) + labs(x = "gene PC", y = bquote("scores ("~10^3~")")) + scale_x_discrete(labels = 1:9) + facet_wrap(~ label, nrow = 3) + scale_fill_brewer(palette = "Set3") + theme_classic()
We observed that most of the gene PCs consist of one or a handful of genes, yet the component scores showed that these PCs distinguish different cell types effectively . For example, the PC 2 consists of only one gene (named SST), and the expression of the gene marks the "delta" cells among others. This result highlights power of scRNA-seq in capture cell-type specific information and suggests the applicability of our methods to biological data.
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