# 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)
group <- as.numeric(tab\$Tissue)
plot(pc\$x[, 1], pc\$x[, 2], col = group, main = "PCA", xlab = "PC1", ylab = "PC2")
```

The meaning of colors can be found by palette().

1. black
2. red
3. green3
4. blue
5. cyan
6. magenta
7. yellow
8. gray

# PCA and SVD

Using the SVD to perform PCA makes much better sense numerically than forming the covariance matrix to begin with, since the formation of ${\displaystyle XX^{T}}$ can cause loss of precision.

# Related to Factor Analysis

In short,

1. In Principal Components Analysis, the components are calculated as linear combinations of the original variables. In Factor Analysis, the original variables are defined as linear combinations of the factors.
2. In Principal Components Analysis, the goal is to explain as much of the total variance in the variables as possible. The goal in Factor Analysis is to explain the covariances or correlations between the variables.
3. Use Principal Components Analysis to reduce the data into a smaller number of components. Use Factor Analysis to understand what constructs underlie the data.

# What does it do if we choose center=FALSE in prcomp()?

In USArrests data, use center=FALSE gives a better scatter plot of the first 2 PCA components.

```x1 = prcomp(USArrests)
x2 = prcomp(USArrests, center=F)
plot(x1\$x[,1], x1\$x[,2])  # looks random
windows(); plot(x2\$x[,1], x2\$x[,2]) # looks good in some sense
```

# "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.

# 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).