PCA

Principal component analysis

What is PCA

• https://en.wikipedia.org/wiki/Principal_component_analysis
• StatQuest: Principal Component Analysis (PCA), Step-by-Step (video)
  • Singular vector/eigenvector & Loading scores/coefficients for PC1(12:23),
  • Eigenvalue (12:41),
  • Variation and eigenvalues (15:04)
• StatQuest: PCA in R
• Using R for Multivariate Analysis
• How exactly does PCA work? What PCA (with SVD) does is, it finds the best fit line for these data points which minimizes the distance between the data points and their projections on the best fit line.
• What Is Principal Component Analysis (PCA) and How It Is Used? PCA creates a visualization of data that minimizes residual variance in the least squares sense and maximizes the variance of the projection coordinates.
• Principal Component Analysis: a brief intro for biologists

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

palmerpenguins data

PCA with penguins and recipes

Theory with an example

Principal Component Analysis in R: prcomp vs princomp

Biplot

• Biplots are everywhere: where do they come from?
• ggbiplot. Principal Component Analysis in R Tutorial from datacamp.

factoextra

• https://cran.r-project.org/web/packages/factoextra/index.html
• Factoextra R Package: Easy Multivariate Data Analyses and Elegant Visualization

Tips

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

Removal of constant columns in R

Do not scale your matrix

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

Apply the transformation on test data

• How to use PCA on test set (code)
• How to use R prcomp results for prediction?
• How to perform PCA in the validation/test set?

Calculated by Hand

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

Variance explained by the first XX components

• StatQuest: Principal Component Analysis (PCA), Step-by-Step
• Singular Value Decomposition (SVD) Tutorial Using Examples in R

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

OR

pr <- prcomp(USArrests, scale. = TRUE) # PS default is scale. = FALSE
summary(pr)
# Importance of components:
#                           PC1    PC2     PC3     PC4
# Standard deviation     1.5749 0.9949 0.59713 0.41645
# Proportion of Variance 0.6201 0.2474 0.08914 0.04336  <-------
# Cumulative Proportion  0.6201 0.8675 0.95664 1.00000
pr$sdev^2/ sum(pr$sdev^2) 
# [1] 0.62006039 0.24744129 0.08914080 0.04335752

Visualization

• ggfortify or the basic method
  • PCA Visualization in ggplot2 - ggplot2::autoplot(). How To Make PCA Plot with R
  • Difference ggplot2 and autoplot() after prcomp? In the autoplot method, the principal components are scaled. We can turn off the scaling by specifying "scale =0" in autoplot().
  • Add ellipse - Difference ggplot2 and autoplot() after prcomp? (same post as above, use the basic method instead of autoplot()). We can also use factoextra package to draw a PCA plot with ellipses. Draw Ellipse Plot for Groups in PCA in R (2 Examples).
  
  ?ggfortify::autoplot.prcomp

  library(ggfortify)
  iris[1:2,]
  #   Sepal.Length Sepal.Width Petal.Length Petal.Width Species
  # 1          5.1         3.5          1.4         0.2  setosa
  # 2          4.9         3.0          1.4         0.2  setosa
  df <- iris[1:4]  # exclude "Species" column
  
  pca_res <- prcomp(df, scale. = TRUE)
  p <- autoplot(pca_res, data = iris, colour = 'Species')
  # OR turn off scaling on axes in order to get the same plot as the direct method
  # p <- autoplot(pca_res, data = iris, colour = 'Species', scale = 0) 
  p  # PC1 (72.96%), PC2 (22.85%)
  
  library(plotly)
  ggplotly(p)
  

  Basic method

  group <- as.numeric(iris$Species)
  
  # ggplot2's palette
  gg_color_hue <- function(n) {
    hues = seq(15, 375, length = n + 1)
    hcl(h = hues, l = 65, c = 100)[1:n]
  }
  n <- 3   # iris$Species has 3 levels
  cols = gg_color_hue(n)
  
  prp <- pca_res$sdev^2/ sum(pca_res$sdev^2) 
  xlab <- paste0("PC1 (", round(prp[1]*100,2), "%)")
  ylab <- paste0("PC2 (", round(prp[2]*100,2), "%)")
  plot(pca_res$x[, 1], pca_res$x[, 2], 
       xlab = xlab, ylab = ylab,
       col = cols[group], pch=16)
  legend(-1, -1.5, legend = levels(iris$Species), col=cols, pch=16, bty = "n")
  

  ? factoextra::fviz_pca_ind

  library(factoextra)
  fviz_pca_ind(pca_res, 
               habillage=iris$Species,
               label = "none",
               addEllipses=TRUE)
  

• PCA and Visualization
• Scree plots from the FactoMineR package (based on ggplot2)
• 2 lines of code, plotPCA() from DESeq2

Interactive Principal Component Analysis

Interactive Principal Component Analysis in R

center and scale

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

• PCA is sensitive to the scaling of the variables. See PCA -> Further considerations.
• https://www.rdocumentation.org/packages/stats/versions/3.6.2/topics/prcomp
• By default scale. = FALSE in prcomp()
• 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.
• Practical Guide to Principal Component Analysis (PCA) in R & Python
• What is the best way to scale parameters before running a Principal Component Analysis (PCA)?
  • As a rule of thumb, if all your variables are measured on the same scale and have the same unit, it might be a good idea *not* to scale the variables (i.e., PCA based on the covariance matrix). If you want to maximize variation, it is fair to let variables with more variation contribute more. On the other hand, If you have different types of variables with different units, it is probably wise to scale the data first (i.e., PCA based on the correlation matrix).
• Let's compare the difference of the returned objects.

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

Number of components

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

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

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.

• http://math.stackexchange.com/questions/3869/what-is-the-intuitive-relationship-between-svd-and-pca
• Relationship between SVD and PCA. How to use SVD to perform PCA?

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

Related to Factor Analysis

• http://www.aaronschlegel.com/factor-analysis-introduction-principal-component-method-r/.
• http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/multivariate/principal-components-and-factor-analysis/differences-between-pca-and-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.

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).

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

PCA for Categorical Variables

• FAMD - Factor Analysis of Mixed Data in R: Essentials
• PCA for Categorical Variables in R. Factorial Analysis of Mixed Data (FAMD) Is a PCA for Categorical Variables Alternate.

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