I've always wondered what goes on behind the scenes of a Principal Component Analysis (PCA). I found this extremely useful tutorial that explains the key concepts of PCA and shows the step by step calculations. Here, I use R to perform each step of a PCA as per the tutorial.

# use a simple two dimensional dataset to illustrate PCA x <- c(2.5, 0.5, 2.2, 1.9, 3.1, 2.3, 2, 1, 1.5, 1.1) y <- c(2.4, 0.7, 2.9, 2.2, 3.0, 2.7, 1.6, 1.1, 1.6, 0.9) plot(x, y, pch = 19)

*Our dataset visualised on the x-y coordinates*.

Next we need to work out the mean of each dimension and subtract it from each value from the respective dimensions. This is known as standardisation, where the dimensions now have a mean of zero.

mean(x) # [1] 1.81 mean(y) # [1] 1.91 x1 <- x - mean(x) x1 # [1] 0.69 -1.31 0.39 0.09 1.29 0.49 0.19 -0.81 -0.31 -0.71 summary(x1) Min. 1st Qu. Median Mean 3rd Qu. Max. -1.310 -0.610 0.140 0.000 0.465 1.290 y1 <- y - mean(y) y1 # [1] 0.49 -1.21 0.99 0.29 1.09 0.79 -0.31 -0.81 -0.31 -1.01 summary(y1) Min. 1st Qu. Median Mean 3rd Qu. Max. -1.210 -0.685 -0.010 0.000 0.715 1.090 plot(x1, y1, pch = 19)

*Our standardised dataset visualised on the x-y coordinates*.

The next step is to calculate the covariance matrix. Covariance measures how dimensions vary with respect to each other and the covariance matrix contains all covariance measures between all dimensions.

cov(x1, y1) #[1] 0.6154444 cov(x1, x1) #[1] 0.6165556 cov(y1, y1) #[1] 0.7165556 m <- matrix(c(cov(x1, x1), cov(x1, y1), cov(y1, x1),cov(y1, y1)), nrow=2, ncol=2, byrow=TRUE, dimnames=list(c("x","y"),c("x","y"))) m # x y # x 0.6165556 0.6154444 # y 0.6154444 0.7165556

Next we need to find the eigenvector and eigenvalues of the covariance matrix. An eigenvector is a direction and an eigenvalue is a number that indicates how much variance is in the data in that direction.

e <- eigen(m) e # eigen() decomposition # $values # [1] 1.2840277 0.0490834 # # $vectors # [,1] [,2] # [1,] 0.6778734 -0.7351787 # [2,] 0.7351787 0.6778734

The largest eigenvalue is the first principal component; we multiply the standardised values to the first eigenvector, which is stored in e$vectors[,1].

pc1 <- x1 * e$vectors[1,1] + y1 * e$vectors[2,1] pc1 # [1] 0.82797019 -1.77758033 0.99219749 0.27421042 1.67580142 0.91294910 -0.09910944 -1.14457216 -0.43804614 # [10] -1.22382056 pc2 <- x1 * e$vectors[1,2] + y1 * e$vectors[2,2] pc2 # [1] -0.17511531 0.14285723 0.38437499 0.13041721 -0.20949846 0.17528244 -0.34982470 0.04641726 0.01776463 # [10] -0.16267529 data.frame(PC1 = pc1, PC2 = pc2) # PC1 PC2 # 1 0.82797019 -0.17511531 # 2 -1.77758033 0.14285723 # 3 0.99219749 0.38437499 # 4 0.27421042 0.13041721 # 5 1.67580142 -0.20949846 # 6 0.91294910 0.17528244 # 7 -0.09910944 -0.34982470 # 8 -1.14457216 0.04641726 # 9 -0.43804614 0.01776463 # 10 -1.22382056 -0.16267529 plot(pc1, pc2, pch = 19)

*Our standardised dataset visualised on the first and second eigenvectors*.

Now to perform PCA using the prcomp() function.

data <- data.frame(x,y) data.pca <- prcomp(data) data.pca # Standard deviations (1, .., p=2): # [1] 1.1331495 0.2215477 # # Rotation (n x k) = (2 x 2): # PC1 PC2 # x -0.6778734 0.7351787 # y -0.7351787 -0.6778734 names(data.pca) # [1] "sdev" "rotation" "center" "scale" "x" data.pca$x # PC1 PC2 # [1,] -0.82797019 0.17511531 # [2,] 1.77758033 -0.14285723 # [3,] -0.99219749 -0.38437499 # [4,] -0.27421042 -0.13041721 # [5,] -1.67580142 0.20949846 # [6,] -0.91294910 -0.17528244 # [7,] 0.09910944 0.34982470 # [8,] 1.14457216 -0.04641726 # [9,] 0.43804614 -0.01776463 #[10,] 1.22382056 0.16267529 plot(data.pca$x[,1], data.pca$x[,2], pch = 19)

*Our standardised dataset visualised on the first and second eigenvectors*.

I'm unsure why the eigen() function returns eigenvectors that have inverted signs with respect to the eigenvectors returned by prcomp().

eigen(m) # eigen() decomposition # $values # [1] 1.2840277 0.0490834 # # $vectors # [,1] [,2] # [1,] 0.6778734 -0.7351787 # [2,] 0.7351787 0.6778734 data.pca # Standard deviations (1, .., p=2): # [1] 1.1331495 0.2215477 # # Rotation (n x k) = (2 x 2): # PC1 PC2 # x -0.6778734 0.7351787 # y -0.7351787 -0.6778734

What is even weirder is that the eigenvectors calculated in the tutorial are a combination of values calculated from prcomp() and eigen().

This work is licensed under a Creative Commons

Attribution 4.0 International License.

The sign is meaningless here. It’s just chosen at random at the beginning . YOu are absolutely right, it does not change the relationship.

Hi Stephan,

Thanks for the comment and the affirmation ðŸ™‚

Cheers,

Dave

Hi guys

Thank you for four tutorial and comment,

use correlation matrix instead of co-variance matrix.

by default, R use correlation matrix in PCA computation.

I checked it, the results are exactly as prncomp result.

Thank you

Hi again

I’m sorry, I make a mistake in previous comment.

Let me check again and I will send you the correction.

Thank you.

Thanks for the wonderful post.

I was actually reading the tutotial by Lindsay,but I wanted to implement it in R.

I was fortunate to find your post as you have used the same data used by the tutorial. Helped me a lot.

Hi,

Thank you for this simple meaningful tutorial. I have several simple questions.

Does PCA mean transform existing data frame into new data frame?

After new data frame constructed using PCA, you will need to choose n first columns as features right? Does it mean n first columns always be the first n important features?

Please advise

Does PCA mean transform existing data frame into new data frame?If you use prcomp(), the saved object is a prcomp object.

After new data frame constructed using PCA, you will need to choose n first columns as features right?Typically people choose the PCs that explain the most variance.

The first PC explains 96% of the variance.

Does it mean n first columns always be the first n important features?Yes.

I was wonder why I got a different result using SPSS.

DATASET ACTIVATE DataSet0.

FACTOR

/VARIABLES VAR00003 VAR00004

/MISSING LISTWISE

/ANALYSIS VAR00003 VAR00004

/CRITERIA FACTORS(2) ITERATE(25)

/EXTRACTION PC

/ROTATION NOROTATE

/SAVE REG(ALL)

/METHOD=CORRELATION.

.74268 .77915

-1.57839 -.62075

.84897 -1.74814

.23296 -.59230

1.49318 .92288

.79348 -.80340

-.06330 1.58016

-1.01317 -.19404

-.38776 -.07426

-1.06866 .75070

Sorry, I have never used SPSS.