# Difference between revisions of "PCA"

Principal component analysis

# R source code

> stats:::prcomp.default
function (x, retx = TRUE, center = TRUE, scale. = FALSE, tol = NULL,
...)
{
x <- as.matrix(x)
x <- scale(x, center = center, scale = scale.)
cen <- attr(x, "scaled:center")
sc <- attr(x, "scaled:scale")
if (any(sc == 0))
stop("cannot rescale a constant/zero column to unit variance")
s <- svd(x, nu = 0)
s$d <- s$d/sqrt(max(1, nrow(x) - 1))
if (!is.null(tol)) {
rank <- sum(s$d > (s$d[1L] * tol))
if (rank < ncol(x)) {
s$v <- s$v[, 1L:rank, drop = FALSE]
s$d <- s$d[1L:rank]
}
}
dimnames(s$v) <- list(colnames(x), paste0("PC", seq_len(ncol(s$v))))
r <- list(sdev = s$d, rotation = s$v, center = if (is.null(cen)) FALSE else cen,
scale = if (is.null(sc)) FALSE else sc)
if (retx)
r$x <- x %*% s$v
class(r) <- "prcomp"
r
}
<bytecode: 0x000000003296c7d8>
<environment: namespace:stats>


# R example

## R built-in plot

pc <- prcomp(x)


# "scale. = TRUE"

By default, it centers the variable to have mean equals to zero. With parameter scale. = T, we normalize the variables to have standard deviation equals to 1.

# Removing Zero Variance Columns

# Assume df is a data frame or a list (matrix is not enough!)
removeZeroVar3 <- function(df){
df[, !sapply(df, function(x) min(x) == max(x))]
}
# Assume df is a matrix
removeZeroVar3 <- function(df){
df[, !apply(df, 2, function(x) min(x) == max(x))]
}

# benchmark
dim(t(tpmlog))  # [1]    58 28109
system.time({a <- t(tpmlog); a <- a[, apply(a, 2, sd) !=0]}) # 0.54
system.time({a <- t(tpmlog); a <- removeZeroVar3(a)})        # 0.18


# prcomp vs princomp

prcomp vs princomp from sthda. prcomp() is preferred compared to princomp().

# Relation to Multidimensional scaling/MDS

With no missing data, classical MDS (Euclidean distance metric) is the same as PCA.

Comparisons are here.

Differences are asked/answered on stackexchange.com. The post also answered the question when these two are the same.

isoMDS (Non-metric)

cmdscale (Metric)

# Matrix factorization methods

http://joelcadwell.blogspot.com/2015/08/matrix-factorization-comes-in-many.html Review of principal component analysis (PCA), K-means clustering, nonnegative matrix factorization (NMF) and archetypal analysis (AA).