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principal_component_analysis

PCA

• Both are data reduction techniques — they allow you to capture the variance in variables in a smaller set.
• Both are usually run in stat software using the same procedure, and the output looks pretty much the same.
• The steps you take to run them are the same-extraction, interpretation, rotation, choosing the number of factors or components.
• Despite all these similarities, there is a fundamental difference between them: PCA is a linear combination of variables; Factor Analysis is a measurement model of a latent variable.

Some useful lectures

pca_gsp.csv
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## In this example, the data is in a matrix called
## data.matrix
## columns are individual samples (i.e. cells)
## rows are measurements taken for all the samples (i.e. genes)
## Just for the sake of the example, here's some made up data...
data.matrix <- matrix(nrow=100, ncol=10)
colnames(data.matrix) <- c(
paste("wt", 1:5, sep=""),
paste("ko", 1:5, sep=""))
rownames(data.matrix) <- paste("gene", 1:100, sep="")
for (i in 1:100) {
wt.values <- rpois(5, lambda=sample(x=10:1000, size=1))
ko.values <- rpois(5, lambda=sample(x=10:1000, size=1))

data.matrix[i,] <- c(wt.values, ko.values)
}
dim(data.matrix)

pca <- prcomp(t(data.matrix), scale=TRUE)

## plot pc1 and pc2
plot(pca$x[,1], pca$x[,2])

## make a scree plot
pca.var <- pca$sdev^2 pca.var.per <- round(pca.var/sum(pca.var)*100, 1) barplot(pca.var.per, main="Scree Plot", xlab="Principal Component", ylab="Percent Variation") ## now make a fancy looking plot that shows the PCs and the variation: library(ggplot2) pca.data <- data.frame(Sample=rownames(pca$x),
X=pca$x[,1], Y=pca$x[,2])
pca.data

ggplot(data=pca.data, aes(x=X, y=Y, label=Sample)) +
geom_text() +
xlab(paste("PC1 - ", pca.var.per[1], "%", sep="")) +
ylab(paste("PC2 - ", pca.var.per[2], "%", sep="")) +
theme_bw() +
ggtitle("My PCA Graph")

## get the name of the top 10 measurements (genes) that contribute
## most to pc1.
loading_scores <- pca$rotation[,1] gene_scores <- abs(loading_scores) ## get the magnitudes gene_score_ranked <- sort(gene_scores, decreasing=TRUE) top_10_genes <- names(gene_score_ranked[1:10]) top_10_genes ## show the names of the top 10 genes pca$rotation[top_10_genes,1] ## show the scores (and +/- sign)

#######
##
## NOTE: Everything that follow is just bonus stuff.
## It simply demonstrates how to get the same
## results using "svd()" (Singular Value Decomposition) or using "eigen()"
## (Eigen Decomposition).
##
#######

############################################
##
## Now let's do the same thing with svd()
##
## svd() returns three things
## v = the "rotation" that prcomp() returns, this is a matrix of eigenvectors
## u = this is similar to the "x" that prcomp() returns. In other words,
##     sum(the rotation * the original data), but compressed to the unit vector
##     You can spread it out by multiplying by "d"
## d = this is similar to the "sdev" value that prcomp() returns (and thus
##     related to the eigen values), but not
##     scaled by sample size in an unbiased way (ie. 1/(n-1)).
##     For prcomp(), sdev = sqrt(var) = sqrt(ss(fit)/(n-1))
##     For svd(), d = sqrt(ss(fit))
##
############################################

svd.stuff <- svd(scale(t(data.matrix), center=TRUE))

## calculate the PCs
svd.data <- data.frame(Sample=colnames(data.matrix),
X=(svd.stuff$u[,1] * svd.stuff$d[1]),
Y=(svd.stuff$u[,2] * svd.stuff$d[2]))
svd.data

## alternatively, we could compute the PCs with the eigen vectors and the
## original data
svd.pcs <- t(t(svd.stuff$v) %*% t(scale(t(data.matrix), center=TRUE))) svd.pcs[,1:2] ## the first to principal components svd.df <- ncol(data.matrix) - 1 svd.var <- svd.stuff$d^2 / svd.df
svd.var.per <- round(svd.var/sum(svd.var)*100, 1)

ggplot(data=svd.data, aes(x=X, y=Y, label=Sample)) +
geom_text() +
xlab(paste("PC1 - ", svd.var.per[1], "%", sep="")) +
ylab(paste("PC2 - ", svd.var.per[2], "%", sep="")) +
theme_bw() +
ggtitle("svd(scale(t(data.matrix), center=TRUE)")

############################################
##
## Now let's do the same thing with eigen()
##
## eigen() returns two things...
## values = eigen values
##
############################################
cov.mat <- cov(scale(t(data.matrix), center=TRUE))
dim(cov.mat)

## since the covariance matrix is symmetric, we can tell eigen() to just
## work on the lower triangle with "symmetric=TRUE"
eigen.stuff <- eigen(cov.mat, symmetric=TRUE)
dim(eigen.stuff$vectors) head(eigen.stuff$vectors[,1:2])