Last updated: 2022-10-20

Checks: 7 0

Knit directory: muse/

This reproducible R Markdown analysis was created with workflowr (version 1.7.0). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.

R Markdown file: up-to-date

Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.

Environment: empty

Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.

Seed: set.seed(20200712)

The command set.seed(20200712) was run prior to running the code in the R Markdown file. Setting a seed ensures that any results that rely on randomness, e.g. subsampling or permutations, are reproducible.

Session information: recorded

Great job! Recording the operating system, R version, and package versions is critical for reproducibility.

Cache: none

Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.

File paths: relative

Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.

Repository version: f9bfcec

Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.

The results in this page were generated with repository version f9bfcec. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.

Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use wflow_publish or wflow_git_commit). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated: 


Ignored files:
    Ignored:    .Rhistory
    Ignored:    .Rproj.user/
    Ignored:    r_packages_4.1.2/
    Ignored:    r_packages_4.2.0/

Untracked files:
    Untracked:  analysis/cell_ranger.Rmd
    Untracked:  data/ncrna_NONCODE[v3.0].fasta.tar.gz
    Untracked:  data/ncrna_noncode_v3.fa

Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.

These are the previous versions of the repository in which changes were made to the R Markdown (analysis/pca.Rmd) and HTML (docs/pca.html) files. If you’ve configured a remote Git repository (see ?wflow_git_remote), click on the hyperlinks in the table below to view the files as they were in that past version.

File Version Author Date Message
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

Further reading


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