Last updated: 2022-10-20
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Rmd | f9bfcec | Dave Tang | 2022-10-20 | PCA |
I have always wondered what goes on behind the scenes of a Principal Components Analysis (PCA). I found this extremely useful tutorial (that I have hosted on my server for the sake of prosperity), which explains the key concepts behind the PCA and also shows the step by step calculations. Here, I use R to perform each step of a PCA as per the tutorial.
pca_data <- data.frame(
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(
pca_data,
pch = '+',
main = "Original PCA data"
)
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.
pca_data_scaled <- apply(pca_data, 2, function(x) x - mean(x))
plot(
pca_data_scaled,
pch = '+',
main = "Scaled PCA data"
)
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.
m <- cov(pca_data_scaled)
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. Note that the eigenvalues calculated here are the same as the tutorial (but in a different order). However, the first eigenvector calculated are negative in the tutorial.
e <- eigen(m)
e
eigen() decomposition
$values
[1] 1.2840277 0.0490834
$vectors
[,1] [,2]
[1,] 0.6778734 -0.7351787
[2,] 0.7351787 0.6778734
The largest eigenvalue is the first principal component and the second largest is the second principal component; we multiply the standardised values to the eigenvectors to obtain the principal components.
my_pca <- as.matrix(pca_data_scaled) %*% e$vectors
plot(
my_pca,
pch = '+',
xlab = "PC1",
ylab = "PC2",
main = "PCA (manual) plot"
)
Now to perform PCA using the prcomp() function.
pca <- prcomp(pca_data)
summary(pca)
Importance of components:
PC1 PC2
Standard deviation 1.1331 0.22155
Proportion of Variance 0.9632 0.03682
Cumulative Proportion 0.9632 1.00000
plot(
pca$x,
pch = 16,
col = 1,
xlab = "PC1",
ylab = "PC2",
main = "PCA (prcomp) plot"
)
points(
my_pca,
pch = 16,
col = 2
)
This post
on Stack Overflow suggests that setting the symmetric
argument is the reason for the difference in the eigenvector signs and
the bottom line is that this does not matter anyway.
eigen(m)
eigen() decomposition
$values
[1] 1.2840277 0.0490834
$vectors
[,1] [,2]
[1,] 0.6778734 -0.7351787
[2,] 0.7351787 0.6778734
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
But I still can’t replicate the sign from prcomp
.
eigen(m, symmetric = TRUE)
eigen() decomposition
$values
[1] 1.2840277 0.0490834
$vectors
[,1] [,2]
[1,] 0.6778734 -0.7351787
[2,] 0.7351787 0.6778734
eigen(m, symmetric = FALSE)
eigen() decomposition
$values
[1] 1.2840277 0.0490834
$vectors
[,1] [,2]
[1,] -0.6778734 -0.7351787
[2,] -0.7351787 0.6778734
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
However, the eigenvectors calculated in the tutorial can be
replicated with symmetric = FALSE
(but with the larger
eigenvalue displayed last).
tut_eigen <- matrix(c(-0.735178656, -0.677873399, 0.677873399, -0.735178656), nrow = 2, byrow = TRUE)
eigen(m, symmetric = FALSE)
eigen() decomposition
$values
[1] 1.2840277 0.0490834
$vectors
[,1] [,2]
[1,] -0.6778734 -0.7351787
[2,] -0.7351787 0.6778734
tut_eigen
[,1] [,2]
[1,] -0.7351787 -0.6778734
[2,] 0.6778734 -0.7351787
The help page of eigen
states that if symmetric is TRUE,
the matrix is assumed to be symmetric (or Hermitian if complex) and only
its lower triangle (diagonal included) is used. Since the covariance
matrix is symmetric, we should really use
symmetric = TRUE
.
isSymmetric(m)
[1] TRUE
sessionInfo()
R version 4.2.0 (2022-04-22)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 20.04.4 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/liblapack.so.3
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] forcats_0.5.1 stringr_1.4.0 dplyr_1.0.9 purrr_0.3.4
[5] readr_2.1.2 tidyr_1.2.0 tibble_3.1.8 ggplot2_3.3.6
[9] tidyverse_1.3.1 workflowr_1.7.0
loaded via a namespace (and not attached):
[1] Rcpp_1.0.8.3 lubridate_1.8.0 getPass_0.2-2 ps_1.7.0
[5] assertthat_0.2.1 rprojroot_2.0.3 digest_0.6.29 utf8_1.2.2
[9] R6_2.5.1 cellranger_1.1.0 backports_1.4.1 reprex_2.0.1
[13] evaluate_0.15 highr_0.9 httr_1.4.3 pillar_1.8.1
[17] rlang_1.0.4 readxl_1.4.0 rstudioapi_0.13 whisker_0.4
[21] callr_3.7.0 jquerylib_0.1.4 rmarkdown_2.14 munsell_0.5.0
[25] broom_0.8.0 compiler_4.2.0 httpuv_1.6.5 modelr_0.1.8
[29] xfun_0.31 pkgconfig_2.0.3 htmltools_0.5.2 tidyselect_1.1.2
[33] fansi_1.0.3 crayon_1.5.1 tzdb_0.3.0 dbplyr_2.1.1
[37] withr_2.5.0 later_1.3.0 grid_4.2.0 jsonlite_1.8.0
[41] gtable_0.3.0 lifecycle_1.0.1 DBI_1.1.2 git2r_0.30.1
[45] magrittr_2.0.3 scales_1.2.0 cli_3.3.0 stringi_1.7.6
[49] fs_1.5.2 promises_1.2.0.1 xml2_1.3.3 bslib_0.3.1
[53] ellipsis_0.3.2 generics_0.1.3 vctrs_0.4.1 tools_4.2.0
[57] glue_1.6.2 hms_1.1.2 processx_3.5.3 fastmap_1.1.0
[61] yaml_2.3.5 colorspace_2.0-3 rvest_1.0.2 knitr_1.39
[65] haven_2.5.0 sass_0.4.1