PCAtools: everything Principal Component Analysis
Kevin Blighe, Aaron Lun 2020-05-31
Introduction
Principal Component Analysis (PCA) is a very powerful technique that has wide applicability in data science, bioinformatics, and further afield. It was initially developed to analyse large volumes of data in order to tease out the differences/relationships between the logical entities being analysed. It extracts the fundamental structure of the data without the need to build any model to represent it. This ‘summary’ of the data is arrived at through a process of reduction that can transform the large number of variables into a lesser number that are uncorrelated (i.e. the ‘principal components’), while at the same time being capable of easy interpretation on the original data (Blighe and Lun 2019) (Blighe 2013).
PCAtools provides functions for data exploration via PCA, and allows the user to generate publication-ready figures. PCA is performed via BiocSingular (Lun 2019) - users can also identify optimal number of principal components via different metrics, such as elbow method and Horn’s parallel analysis (Horn 1965) (Buja and Eyuboglu 1992), which has relevance for data reduction in single-cell RNA-seq (scRNA-seq) and high dimensional mass cytometry data.
Installation
1. Download the package from Bioconductor
if (!requireNamespace('BiocManager', quietly = TRUE))
install.packages('BiocManager')
BiocManager::install('PCAtools')
Note: to install development version:
devtools::install_github('kevinblighe/PCAtools')
2. Load the package into R session
library(PCAtools)
Quick start
For this vignette, we will load breast cancer gene expression data with recurrence free survival (RFS) from Gene Expression Profiling in Breast Cancer: Understanding the Molecular Basis of Histologic Grade To Improve Prognosis.
First, let’s read in and prepare the data:
library(Biobase)
library(GEOquery)
# load series and platform data from GEO
gset <- getGEO('GSE2990', GSEMatrix = TRUE, getGPL = FALSE)
x <- exprs(gset[[1]])
# remove Affymetrix control probes
x <- x[-grep('^AFFX', rownames(x)),]
# extract information of interest from the phenotype data (pdata)
idx <- which(colnames(pData(gset[[1]])) %in%
c('age:ch1', 'distant rfs:ch1', 'er:ch1',
'ggi:ch1', 'grade:ch1', 'size:ch1',
'time rfs:ch1'))
metadata <- data.frame(pData(gset[[1]])[,idx],
row.names = rownames(pData(gset[[1]])))
# tidy column names
colnames(metadata) <- c('Age', 'Distant.RFS', 'ER', 'GGI', 'Grade',
'Size', 'Time.RFS')
# prepare certain phenotypes
metadata$Age <- as.numeric(gsub('^KJ', NA, metadata$Age))
metadata$Distant.RFS <- factor(metadata$Distant.RFS, levels=c(0,1))
metadata$ER <- factor(gsub('\\?', NA, metadata$ER), levels=c(0,1))
metadata$ER <- factor(ifelse(metadata$ER == 1, 'ER+', 'ER-'), levels = c('ER-', 'ER+'))
metadata$GGI <- as.numeric(metadata$GGI)
metadata$Grade <- factor(gsub('\\?', NA, metadata$Grade), levels=c(1,2,3))
metadata$Grade <- gsub(1, 'Grade 1', gsub(2, 'Grade 2', gsub(3, 'Grade 3', metadata$Grade)))
metadata$Grade <- factor(metadata$Grade, levels = c('Grade 1', 'Grade 2', 'Grade 3'))
metadata$Size <- as.numeric(metadata$Size)
metadata$Time.RFS <- as.numeric(gsub('^KJX|^KJ', NA, metadata$Time.RFS))
# remove samples from the pdata that have any NA value
discard <- apply(metadata, 1, function(x) any(is.na(x)))
metadata <- metadata[!discard,]
# filter the expression data to match the samples in our pdata
x <- x[,which(colnames(x) %in% rownames(metadata))]
# check that sample names match exactly between pdata and expression data
all(colnames(x) == rownames(metadata))
## [1] TRUE
Conduct principal component analysis (PCA)
p <- pca(x, metadata = metadata, removeVar = 0.1)
## -- removing the lower 10% of variables based on variance
A scree plot
screeplot(p, axisLabSize = 20, titleLabSize = 22)
A bi-plot
Different interpretations of the biplot exist. In the OMICs era, for most general users, a biplot is a simple representation of samples in a 2-dimensional space:
biplot(p)
However, the original definition of a biplot by Gabriel KR (Gabriel 1971) is a plot that plots both variables and observatinos (samples) in the same space. The variables are indicated by arrows drawn from the origin, which indicate their ‘weight’ in different directions. We touch on this later via the plotLoadings function.
biplot(p, showLoadings = TRUE, lab = NULL)
One of the probes pointing downward is 205225_at, which targets the ESR1 gene. This is already a useful validation, as the oestrogen receptor, which is in part encoded by ESR1, is strongly represented by PC2 (y-axis), with negative-to-positive receptor status going from top-to-bottom. More on this later in this vignette.
A pairs plot
pairsplot(p)
A loadings plot
If the biplot was previously generated with showLoadings = TRUE, check how this loadings plot corresponds to the biplot loadings - they should match up for the top hits.
plotloadings(p, labSize = 3)
## -- variables retained:
## 215281_x_at, 214464_at, 211122_s_at, 210163_at, 204533_at, 205225_at, 209351_at, 205044_at, 202037_s_at, 204540_at, 215176_x_at, 214768_x_at, 212671_s_at, 219415_at, 37892_at, 208650_s_at, 206754_s_at, 205358_at, 205380_at, 205825_at
An eigencor plot
eigencorplot(p,
metavars = c('Age','Distant.RFS','ER','GGI','Grade','Size','Time.RFS'))
Access the internal data
The rotated data that represents the observatinos / samples is stored in rotated, while the variable loadings are stored in loadings
p$rotated[1:5,1:5]
## PC1 PC2 PC3 PC4 PC5
## GSM65752 -30.24272 43.826310 3.781677 -39.536149 18.612835
## GSM65753 -37.73436 -15.464421 -4.913100 -5.877623 9.060108
## GSM65755 -29.95155 7.788280 -22.980076 -15.222649 23.123766
## GSM65757 -33.73509 1.261410 -22.834375 2.494554 13.629207
## GSM65758 -40.95958 -8.588458 4.995440 14.340150 0.417101
p$loadings[1:5,1:5]
## PC1 PC2 PC3 PC4 PC5
## 206378_at -0.0024336244 -0.05312797 -0.004809456 0.04045087 0.0096616577
## 205916_at -0.0051057533 0.00122765 -0.010593760 0.04023264 0.0285972617
## 206799_at 0.0005723191 -0.05048096 -0.009992964 0.02568142 0.0024626261
## 205242_at 0.0129147329 0.02867789 0.007220832 0.04424070 -0.0006138609
## 206509_at 0.0019058729 -0.05447596 -0.004979062 0.01510060 -0.0026213610
Advanced features
All plots in PCAtools are highly configurable and should cover virtually all general usage requirements. The following sections take a look at some of these advanced features, and form a somewhat practical example of how one can use PCAtools to make a clinical interpretation of data.
First, let’s sort out the gene annotation by mapping the probe IDs to gene symbols. The array used for this study was the Affymetrix U133a, so let’s use the hgu133a.db Bioconductor package:
suppressMessages(require(hgu133a.db))
newnames <- mapIds(hgu133a.db,
keys = rownames(p$loadings),
column = c('SYMBOL'),
keytype = 'PROBEID')
## 'select()' returned 1:many mapping between keys and columns
# tidy up for NULL mappings and duplicated gene symbols
newnames <- ifelse(is.na(newnames) | duplicated(newnames),
names(newnames), newnames)
rownames(p$loadings) <- newnames
Determine optimum number of PCs to retain
A scree plot on its own just shows the accumulative proportion of explained variation, but how can we determine the optimum number of PCs to retain? PCAtools provides two metrics for this purpose: elbow method and Horn’s parallel analysis (Horn 1965) (Buja and Eyuboglu 1992).
Let’s perform Horn’s parallel analysis first:
horn <- parallelPCA(x)
horn$n
## [1] 11
Now the elbow method:
elbow <- findElbowPoint(p$variance)
elbow
## PC8
## 8
In most cases, the identified values will disagree. This is because finding the correct number of PCs is a difficult task and is akin to finding the ‘correct’ number of clusters in a dataset - there is no correct answer.
Taking the value from Horn’s parallel analyis, we can produce a new scree plot:
library(ggplot2)
screeplot(p,
components = getComponents(p, 1:20),
vline = c(horn$n, elbow)) +
geom_label(aes(x = horn$n + 1, y = 50, label = "Horn's", vjust = -1, size = 8)) +
geom_label(aes(x = elbow + 1, y = 50, label = "Elbow", vjust = -1, size = 8))
If all else fails, one can simply take the number of PCs that contributes to a pre-selected total of explained variation, e.g., in this case, 27 PCs account for >80% explained variation.
Modify bi-plots
The bi-plot comparing PC1 versus PC2 is the most characteristic plot of PCA. However, PCA is much more than the bi-plot and much more than PC1 and PC2. This said, PC1 and PC2, by the very nature of PCA, are indeed usually the most important parts of PCA.
In a bi-plot, we can shade the points by different groups and add many more features.
Colour by a factor from the metadata, use a custom label, add lines through center, and add legend
biplot(p,
lab = paste0(p$metadata$Age, ' años'),
colby = 'ER',
hline = 0, vline = 0,
legendPosition = 'right')
Supply custom colours, add more lines, and increase legend size
biplot(p,
colby = 'ER', colkey = c('ER+'='forestgreen', 'ER-'='purple'),
hline = 0, vline = c(-25, 0, 25),
legendPosition = 'top', legendLabSize = 16, legendIconSize = 8.0)
Change shape based on tumour grade, remove connectors, and add titles
biplot(p,
colby = 'ER', colkey = c('ER+'='forestgreen', 'ER-'='purple'),
hline = 0, vline = c(-25, 0, 25),
legendPosition = 'top', legendLabSize = 13, legendIconSize = 8.0,
shape = 'Grade', shapekey = c('Grade 1'=15, 'Grade 2'=17, 'Grade 3'=8),
drawConnectors = FALSE,
title = 'PCA bi-plot',
subtitle = 'PC1 versus PC2',
caption = '27 PCs ≈ 80%')
Remove labels, modify line types, remove gridlines, and increase point size
biplot(p,
lab = NULL,
colby = 'ER', colkey = c('ER+'='royalblue', 'ER-'='red3'),
hline = 0, vline = c(-25, 0, 25),
vlineType = c('dotdash', 'solid', 'dashed'),
gridlines.major = FALSE, gridlines.minor = FALSE,
pointSize = 5,
legendPosition = 'left', legendLabSize = 14, legendIconSize = 8.0,
shape = 'Grade', shapekey = c('Grade 1'=15, 'Grade 2'=17, 'Grade 3'=8),
drawConnectors = FALSE,
title = 'PCA bi-plot',
subtitle = 'PC1 versus PC2',
caption = '27 PCs ≈ 80%')
Let’s plot the same as above but with loadings:
biplot(p,
# loadings parameters
showLoadings = TRUE,
lengthLoadingsArrowsFactor = 1.5,
sizeLoadingsNames = 4,
colLoadingsNames = 'red4',
# other parameters
lab = NULL,
colby = 'ER', colkey = c('ER+'='royalblue', 'ER-'='red3'),
hline = 0, vline = c(-25, 0, 25),
vlineType = c('dotdash', 'solid', 'dashed'),
gridlines.major = FALSE, gridlines.minor = FALSE,
pointSize = 5,
legendPosition = 'left', legendLabSize = 14, legendIconSize = 8.0,
shape = 'Grade', shapekey = c('Grade 1'=15, 'Grade 2'=17, 'Grade 3'=8),
drawConnectors = FALSE,
title = 'PCA bi-plot',
subtitle = 'PC1 versus PC2',
caption = '27 PCs ≈ 80%')
Colour by a continuous variable and plot other PCs
There are two ways to colour by a continuous variable. In the first way, we simply ‘add on’ a continuous colour scale via scale_colour_gradient:
# add ESR1 gene expression to the metadata
p$metadata$ESR1 <- x["205225_at",]
biplot(p,
x = 'PC2', y = 'PC3',
lab = NULL,
colby = 'ESR1',
shape = 'ER',
hline = 0, vline = 0,
legendPosition = 'right') +
scale_colour_gradient(low = 'gold', high = 'red2')
We can also just permit that the internal ggplot2 engine picks the colour scheme - here, we also plot PC10 versus PC50:
biplot(p, x = 'PC10', y = 'PC50',
lab = NULL,
colby = 'Age',
hline = 0, vline = 0,
hlineWidth = 1.0, vlineWidth = 1.0,
gridlines.major = FALSE, gridlines.minor = TRUE,
pointSize = 5,
legendPosition = 'left', legendLabSize = 16, legendIconSize = 8.0,
shape = 'Grade', shapekey = c('Grade 1'=15, 'Grade 2'=17, 'Grade 3'=8),
drawConnectors = FALSE,
title = 'PCA bi-plot',
subtitle = 'PC10 versus PC50',
caption = '27 PCs ≈ 80%')
Quickly explore potentially informative PCs via a pairs plot
The pairs plot in PCA unfortunately suffers from a lack of use; however, for those who love exploring data and squeezing every last ounce of information out of data, a pairs plot provides for a relatively quick way to explore useful leads for other downstream analyses.
As the number of pairwise plots increases, however, space becomes limited. We can shut off titles and axis labeling to save space. Reducing point size and colouring by a variable of interest can additionally help us to rapidly skim over the data.
pairsplot(p,
components = getComponents(p, c(1:10)),
triangle = TRUE, trianglelabSize = 12,
hline = 0, vline = 0,
pointSize = 0.4,
gridlines.major = FALSE, gridlines.minor = FALSE,
colby = 'Grade',
title = 'Pairs plot', plotaxes = FALSE,
margingaps = unit(c(-0.01, -0.01, -0.01, -0.01), 'cm'))
We can arrange these in a way that makes better use of the screen space by setting ‘triangle = FALSE’. In this case, we can further control the layout with the ‘ncol’ and ‘nrow’ parameters, although, the function will automatically determine these based on your input data.
pairsplot(p,
components = getComponents(p, c(4,33,11,1)),
triangle = FALSE,
hline = 0, vline = 0,
pointSize = 0.8,
gridlines.major = FALSE, gridlines.minor = FALSE,
colby = 'ER',
title = 'Pairs plot', titleLabSize = 22,
axisLabSize = 14, plotaxes = TRUE,
margingaps = unit(c(0.1, 0.1, 0.1, 0.1), 'cm'))
Determine the variables that drive variation among each PC
If, on the bi-plot or pairs plot, we encounter evidence that 1 or more PCs are segregating a factor of interest, we can explore further the genes that are driving these differences along each PC.
For each PC of interest, ‘plotloadings’ determines the variables falling within the top/bottom 5% of the loadings range, and then creates a final consensus list of these. These variables are then plotted.
The loadings plot, like all others, is highly configurable. To modify the cut-off for inclusion / exclusion of variables, we use ‘rangeRetain’, where 0.01 equates to the top/bottom 1% of the loadings range per PC. We can also add a title, subtitle, and caption, and alter the shape and colour scheme.
plotloadings(p,
rangeRetain = 0.01,
labSize = 4.0,
title = 'Loadings plot',
subtitle = 'PC1, PC2, PC3, PC4, PC5',
caption = 'Top 1% variables',
shape = 24,
col = c('limegreen', 'black', 'red3'),
drawConnectors = TRUE)
## -- variables retained:
## POGZ, CDC42BPA, CXCL11, ESR1, SFRP1, EEF1A2, IGKC, GABRP, CD24, PDZK1
At least one interesting finding is 205225_at / ESR1, which is by far the gene most responsible for variation along PC2. The previous bi-plots showed that this PC also segregated ER+ from ER- patients. The other results could be explored. Also, from the biplots with loadings that we have already generated, this results is also verified in these.
With the loadings plot, in addition, we can instead plot absolute values and modify the point sizes to be proportional to the loadings. We can also switch off the line connectors and plot the loadings for any PCs
plotloadings(p,
components = getComponents(p, c(4,33,11,1)),
rangeRetain = 0.1,
labSize = 4.0,
absolute = FALSE,
title = 'Loadings plot',
subtitle = 'Misc PCs',
caption = 'Top 10% variables',
shape = 23, shapeSizeRange = c(1, 16),
col = c('white', 'pink'),
drawConnectors = FALSE)
## -- variables retained:
## CXCL11, IGKC, CXCL9, 210163_at, 214768_x_at, 211645_x_at, 211644_x_at, IGHA1, 216491_x_at, 214777_at, 216576_x_at, 212671_s_at, IL23A, PLAAT4, 212588_at, 212998_x_at, KRT14, GABRP, SOX10, PTX3, TTYH1, CPB1, KRT15, MYBPC1, DST, CXADR, GALNT3, CDH3, TCIM, DHRS2, MMP1, CRABP1, CST1, MAGEA3, ACOX2, PRKAR2B, PLCB1, HDGFL3, CYP2B6, ORM1, 205040_at, HSPB8, SCGB2A2, JCHAIN, POGZ, 213872_at, DYNC2LI1, CDC42BPA
Correlate the principal components back to the clinical data
Further exploration of the PCs can come through correlations with clinical data. This is also a mostly untapped resource in the era of ‘big data’ and can help to guide an analysis down a particular path (or not!).
We may wish, for example, to correlate all PCs that account for 80% variation in our dataset and then explore further the PCs that have statistically significant correlations.
‘eigencorplot’ is built upon another function by the PCAtools developers, namely CorLevelPlot. Further examples can be found there.
eigencorplot(p,
components = getComponents(p, 1:27),
metavars = c('Age','Distant.RFS','ER','GGI','Grade','Size','Time.RFS'),
col = c('darkblue', 'blue2', 'black', 'red2', 'darkred'),
cexCorval = 0.7,
colCorval = 'white',
fontCorval = 2,
posLab = 'bottomleft',
rotLabX = 45,
posColKey = 'top',
cexLabColKey = 1.5,
scale = TRUE,
main = 'PC1-27 clinical correlations',
colFrame = 'white',
plotRsquared = FALSE)
We can also supply different cut-offs for statistical significance, apply p-value adjustment, plot R-squared values, and specify correlation method:
eigencorplot(p,
components = getComponents(p, 1:horn$n),
metavars = c('Age','Distant.RFS','ER','GGI','Grade','Size','Time.RFS'),
col = c('white', 'cornsilk1', 'gold', 'forestgreen', 'darkgreen'),
cexCorval = 1.2,
fontCorval = 2,
posLab = 'all',
rotLabX = 45,
scale = TRUE,
main = bquote(Principal ~ component ~ Pearson ~ r^2 ~ clinical ~ correlates),
plotRsquared = TRUE,
corFUN = 'pearson',
corUSE = 'pairwise.complete.obs',
corMultipleTestCorrection = 'BH',
signifSymbols = c('****', '***', '**', '*', ''),
signifCutpoints = c(0, 0.0001, 0.001, 0.01, 0.05, 1))
Clearly, PC2 is coming across as the most interesting PC in this experiment, with highly statistically significant correlation (p<0.0001) to ER status, tumour grade, and GGI (genomic Grade Index), an indicator of response. It comes as no surprise that the gene driving most variationn along PC2 is ESR1, identified from our loadings plot.
This information is, of course, not new, but shows how PCA is much more than just a bi-plot used to identify outliers!
Plot the entire project on a single panel
pscree <- screeplot(p, components = getComponents(p, 1:30),
hline = 80, vline = 27, axisLabSize = 14, titleLabSize = 20,
returnPlot = FALSE) +
geom_label(aes(20, 80, label = '80% explained variation', vjust = -1, size = 8))
ppairs <- pairsplot(p, components = getComponents(p, c(1:3)),
triangle = TRUE, trianglelabSize = 12,
hline = 0, vline = 0,
pointSize = 0.8, gridlines.major = FALSE, gridlines.minor = FALSE,
colby = 'Grade',
title = '', plotaxes = FALSE,
margingaps = unit(c(0.01, 0.01, 0.01, 0.01), 'cm'),
returnPlot = FALSE)
pbiplot <- biplot(p,
# loadings parameters
showLoadings = TRUE,
lengthLoadingsArrowsFactor = 1.5,
sizeLoadingsNames = 4,
colLoadingsNames = 'red4',
# other parameters
lab = NULL,
colby = 'ER', colkey = c('ER+'='royalblue', 'ER-'='red3'),
hline = 0, vline = c(-25, 0, 25),
vlineType = c('dotdash', 'solid', 'dashed'),
gridlines.major = FALSE, gridlines.minor = FALSE,
pointSize = 5,
legendPosition = 'none', legendLabSize = 16, legendIconSize = 8.0,
shape = 'Grade', shapekey = c('Grade 1'=15, 'Grade 2'=17, 'Grade 3'=8),
drawConnectors = FALSE,
title = 'PCA bi-plot',
subtitle = 'PC1 versus PC2',
caption = '27 PCs ≈ 80%',
returnPlot = FALSE)
ploadings <- plotloadings(p, rangeRetain = 0.01, labSize = 4,
title = 'Loadings plot', axisLabSize = 12,
subtitle = 'PC1, PC2, PC3, PC4, PC5',
caption = 'Top 1% variables',
shape = 24, shapeSizeRange = c(4, 8),
col = c('limegreen', 'black', 'red3'),
legendPosition = 'none',
drawConnectors = FALSE,
returnPlot = FALSE)
peigencor <- eigencorplot(p,
components = getComponents(p, 1:10),
metavars = c('Age','Distant.RFS','ER','GGI','Grade','Size','Time.RFS'),
#col = c('royalblue', '', 'gold', 'forestgreen', 'darkgreen'),
cexCorval = 1.0,
fontCorval = 2,
posLab = 'all',
rotLabX = 45,
scale = TRUE,
main = "PC clinical correlates",
cexMain = 1.5,
plotRsquared = FALSE,
corFUN = 'pearson',
corUSE = 'pairwise.complete.obs',
signifSymbols = c('****', '***', '**', '*', ''),
signifCutpoints = c(0, 0.0001, 0.001, 0.01, 0.05, 1),
returnPlot = FALSE)
library(cowplot)
library(ggplotify)
top_row <- plot_grid(pscree, ppairs, pbiplot,
ncol = 3,
labels = c('A', 'B Pairs plot', 'C'),
label_fontfamily = 'serif',
label_fontface = 'bold',
label_size = 22,
align = 'h',
rel_widths = c(1.10, 0.80, 1.10))
bottom_row <- plot_grid(ploadings,
as.grob(peigencor),
ncol = 2,
labels = c('D', 'E'),
label_fontfamily = 'serif',
label_fontface = 'bold',
label_size = 22,
align = 'h',
rel_widths = c(0.8, 1.2))
plot_grid(top_row, bottom_row, ncol = 1,
rel_heights = c(1.1, 0.9))
Acknowledgments
The development of PCAtools has benefited from contributions and suggestions from:
- Krushna Chandra Murmu
- Jinsheng
- Myles Lewis
- Anna-Leigh Brown
- Vincent Carey
- Vince Vu
Session info
sessionInfo()
## R version 3.6.3 (2020-02-29)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 16.04.6 LTS
##
## Matrix products: default
## BLAS: /usr/lib/atlas-base/atlas/libblas.so.3.0
## LAPACK: /usr/lib/atlas-base/atlas/liblapack.so.3.0
##
## locale:
## [1] LC_CTYPE=pt_BR.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=en_GB.UTF-8 LC_COLLATE=pt_BR.UTF-8
## [5] LC_MONETARY=en_GB.UTF-8 LC_MESSAGES=pt_BR.UTF-8
## [7] LC_PAPER=en_GB.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=en_GB.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats4 parallel stats graphics grDevices utils datasets
## [8] methods base
##
## other attached packages:
## [1] ggplotify_0.0.5 hgu133a.db_3.2.3 org.Hs.eg.db_3.10.0
## [4] AnnotationDbi_1.48.0 IRanges_2.20.2 S4Vectors_0.24.4
## [7] GEOquery_2.54.1 Biobase_2.46.0 BiocGenerics_0.32.0
## [10] PCAtools_2.1.4 cowplot_1.0.0 lattice_0.20-41
## [13] reshape2_1.4.4 ggrepel_0.8.2 ggplot2_3.3.0
##
## loaded via a namespace (and not attached):
## [1] BiocSingular_1.2.2 tidyr_1.0.2 bit64_0.9-7
## [4] DelayedMatrixStats_1.8.0 assertthat_0.2.1 BiocManager_1.30.10
## [7] rvcheck_0.1.8 highr_0.8 dqrng_0.2.1
## [10] blob_1.2.1 yaml_2.2.1 pillar_1.4.3
## [13] RSQLite_2.2.0 glue_1.4.0 limma_3.42.2
## [16] digest_0.6.25 colorspace_1.4-1 htmltools_0.4.0
## [19] Matrix_1.2-18 plyr_1.8.6 pkgconfig_2.0.3
## [22] purrr_0.3.3 scales_1.1.0 BiocParallel_1.20.1
## [25] tibble_3.0.0 farver_2.0.3 ellipsis_0.3.0
## [28] withr_2.1.2 cli_2.0.2 magrittr_1.5
## [31] crayon_1.3.4 memoise_1.1.0 evaluate_0.14
## [34] fansi_0.4.1 xml2_1.3.1 tools_3.6.3
## [37] hms_0.5.3 lifecycle_0.2.0 matrixStats_0.56.0
## [40] stringr_1.4.0 munsell_0.5.0 DelayedArray_0.12.3
## [43] irlba_2.3.3 compiler_3.6.3 rsvd_1.0.3
## [46] gridGraphics_0.5-0 rlang_0.4.5 grid_3.6.3
## [49] labeling_0.3 rmarkdown_2.1 gtable_0.3.0
## [52] DBI_1.1.0 curl_4.3 R6_2.4.1
## [55] knitr_1.28 dplyr_0.8.5 bit_1.1-15.2
## [58] readr_1.3.1 stringi_1.4.6 Rcpp_1.0.4.6
## [61] vctrs_0.2.4 tidyselect_1.0.0 xfun_0.13
References
Blighe and Lun (2019)
Blighe (2013)
Horn (1965)
Buja and Eyuboglu (1992)
Lun (2019)
Gabriel (1971)
Blighe, K. 2013. “Haplotype classification using copy number variation and principal components analysis.” The Open Bioinformatics Journal 7:19-24.
Blighe, K, and A Lun. 2019. “PCAtools: everything Principal Components Analysis.” https://github.com/kevinblighe/PCAtools.
Buja, A, and N Eyuboglu. 1992. “Remarks on Parallel Analysis.” Multivariate Behav. Res. 27, 509-40.
Gabriel, KR. 1971. “The Biplot Graphic Display of Matrices with Application to Principal Component Analysis 1.” Biometrika 58 (3): 453–67. http://biomet.oxfordjournals.org/content/58/3/453.short.
Horn, JL. 1965. “A rationale and test for the number of factors in factor analysis.” Psychometrika 30(2), 179-185.
Lun, A. 2019. “BiocSingular: Singular Value Decomposition for Bioconductor Packages.” R package version 1.0.0, https://github.com/LTLA/BiocSingular.