PCA

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Principal component analysis

What is PCA

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>

rank of a matrix

Matrix::rankMatrix()

R example

Principal component analysis (PCA) in R including bi-plot.

R built-in plot

http://genomicsclass.github.io/book/pages/pca_svd.html 

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

Theory with an example

Principal Component Analysis in R: prcomp vs princomp

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 [math]\displaystyle{ X X^T }[/math] can cause loss of precision.

R example

covMat <- matrix(c(4, 0, 0, 1), nr=2)
p <- 2
n <- 1000
set.seed(1)
x <- mvtnorm::rmvnorm(n, rep(0, p), covMat)
svdx <- svd(x)
result= prcomp(x, scale = TRUE) 
summary(result)
# Importance of components:
#                          PC1    PC2
# Standard deviation     1.0004 0.9996
# Proportion of Variance 0.5004 0.4996
# Cumulative Proportion  0.5004 1.0000

# It seems scale = FALSE result can reflect the original data
result2 <- prcomp(x, scale = FALSE) # DEFAULT
summary(result2)
# Importance of components:
#                           PC1    PC2
# Standard deviation     2.1332 1.0065  ==> Close to the original data
                                     #  ==> How to compute
                              # sqrt(eigen(var(x))$values ) = 2.133203 1.006460
# Proportion of Variance 0.8179 0.1821  ==> How to verify
                                        # 2.1332^2/(2.1332^2+1.0065^2) = 0.8179
# Cumulative Proportion  0.8179 1.0000
result2$sdev
# 2.133203 1.006460    #  sqrt( SS(distance)/(n-1) )
result2$sdev^2 / sum(result2$sdev^2)
# 0.8179279 0.1820721
result2$rotation
#                PC1           PC2
# [1,]  0.9999998857 -0.0004780211
# [2,] -0.0004780211 -0.9999998857
result2$sdev^2 * (n-1)
# [1] 4546.004 1011.948   #  SS(distance)

# eigenvalue for PC1 = singular value^2
svd(scale(x, center=T, scale=F))$d
# [1] 67.42406 31.81113
svd(scale(x, center=T, scale=F))$d ^ 2  # SS(distance)
# [1] 4546.004 1011.948
svd(scale(x, center=T, scale=F))$d / sqrt(nrow(x) -1)
# [1] 2.133203 1.006460     ==> This is the same as prcomp()$sdev
svd(scale(x, center=F, scale=F))$d / sqrt(nrow(x) -1)
# [1] 2.135166 1.006617     ==> So it does not matter to center or scale
svd(var(x))$d
# [1] 4.550554 1.012961
svd(scale(x, center=T, scale=F))$v # same as result2$rotation
#               [,1]          [,2]
# [1,]  0.9999998857 -0.0004780211
# [2,] -0.0004780211 -0.9999998857
sqrt(eigen(var(x))$values )
# [1] 2.133203 1.006460
eigen(t(scale(x,T,F)) %*% scale(x,T,F))$values  # SS(distance)
# [1] 4546.004 1011.948
sqrt(eigen(t(scale(x,T,F)) %*% scale(x,T,F))$values ) # Same as svd(scale(x.T,F))$d
# [1] 67.42406 31.81113.   

# So SS(distance) = eigen(t(scale(x,T,F)) %*% scale(x,T,F))$values
#                 = svd(scale(x.T,F))$d ^ 2

AIC/BIC in estimating the number of components

Consistency of AIC and BIC in estimating the number of significant components in high-dimensional principal component analysis

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.

Calculated by Hand

http://strata.uga.edu/software/pdf/pcaTutorial.pdf

Do not scale your matrix

https://privefl.github.io/blog/(Linear-Algebra)-Do-not-scale-your-matrix/

Visualization

Interactive Principal Component Analysis

Interactive Principal Component Analysis in R

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" and Mean subtraction

USArrests[1:3,]
#         Murder Assault UrbanPop Rape
# Alabama   13.2     236       58 21.2
# Alaska    10.0     263       48 44.5
# Arizona    8.1     294       80 31.0

pca1 <- prcomp(USArrests)  # inappropriate, default is scale. = FALSE
pca2 <- prcomp(USArrests, scale. = TRUE)
pca1$x[1:3, 1:3]
#               PC1        PC2       PC3
# Alabama  64.80216 -11.448007 -2.494933
# Alaska   92.82745 -17.982943 20.126575
# Arizona 124.06822   8.830403 -1.687448
pca2$x[1:3, 1:3]
#                PC1        PC2         PC3
# Alabama -0.9756604  1.1220012 -0.43980366
# Alaska  -1.9305379  1.0624269  2.01950027
# Arizona -1.7454429 -0.7384595  0.05423025

set.seed(1)
X <- matrix(rnorm(10*100), nr=10)
pca1 <- prcomp(X)
pca2 <- prcomp(X, scale. = TRUE)
range(abs(pca1$x - pca2$x))  # [1] 3.764350e-16 1.139182e+01
par(mfrow=c(1,2))
plot(pca1$x[,1], pca1$x[, 2], main='scale=F')
plot(pca2$x[,1], pca2$x[, 2], main='scale=T')
par(mfrow=c(1,1))
# rotation matrices are different
# sdev are different
# rotation matrices are different

# Same observations for a long matrix too
set.seed(1)
X <- matrix(rnorm(10*100), nr=100)
pca1 <- prcomp(X)
pca2 <- prcomp(X, scale. = TRUE)
range(abs(pca1$x - pca2$x))  # [1] 0.001915974 5.112158589

svd1 <- svd(USArrests)
svd2 <- svd(scale(USArrests, F, T))
svd1$d
# [1] 1419.06140  194.82585   45.66134   18.06956
svd2$d
# [1] 13.560545  2.736843  1.684743  1.335272
# u (or v) are also different

Variance explained by the first XX components

svd2 <- svd(Z)
variance.explained <- prop.table(svd2$d^2)

Removing Zero Variance Columns

Efficiently Removing Zero Variance Columns (An Introduction to Benchmarking) 

# 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).

Number of components

Obtaining the number of components from cross validation of principal components regression

Outlier samples

Detecting outlier samples in PCA

Reconstructing images

Reconstructing images using PCA

Misc

Data reduction prior to inference: Are there consequences of comparing groups using a t‐test based on principal component scores? Bedrick 2019

psych package

Principal component analysis (PCA) in R

glmpca: Generalized PCA for dimension reduction of sparse counts

Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model

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