LAB: Principal Component Analysis

# We will use the following packages. 
# If needed, install them : pak::pkg_install(). 
stopifnot(
  require("corrr"),
  require("magrittr"),
  require("lobstr"),
  require("ggforce"),
  require("gt"),
  require("glue"),
  require("skimr"),
  require("patchwork"), 
  require("tidyverse"),
  require("ggfortify")
  # require("autoplotly")
)

old_theme <- theme_set(theme_minimal())

options(ggplot2.discrete.colour="viridis")
options(ggplot2.discrete.fill="viridis")
options(ggplot2.continuous.fill="viridis")
options(ggplot2.continuous.colour="viridis")

M1 MIDS/MFA/LOGOS

Université Paris Cité

Année 2024

Course Homepage

Moodle

Swiss fertility data

Dataset swiss from datasets::swiss connect fertility and social, economic data within 47 French-speaking districts in Switzerland.

• Fertility : fertility index
• Agriculture : jobs in agricultural sector
• Examination : literacy index (military examination)
• Education : proportion of people with successful secondary education
• Catholic : proportion of Catholics
• Infant.Mortality : mortality quotient at age 0

Fertility index (Fertility) is considered as the response variable

The social and economic variables are covariates (explanatory variables).

See European Fertility Project for more on this dataset.

PCA (Principal Component Analysis) is concerned with covariates.

data("swiss")

swiss %>% 
  glimpse(50)

Rows: 47
Columns: 6
$ Fertility        <dbl> 80.2, 83.1, 92.5, 85.8,…
$ Agriculture      <dbl> 17.0, 45.1, 39.7, 36.5,…
$ Examination      <int> 15, 6, 5, 12, 17, 9, 16…
$ Education        <int> 12, 9, 5, 7, 15, 7, 7, …
$ Catholic         <dbl> 9.96, 84.84, 93.40, 33.…
$ Infant.Mortality <dbl> 22.2, 22.2, 20.2, 20.3,…

Have a look at the documentation of the dataset

Describe the dataset

Question

Compute summary for each variable

Question

Display graphic summary for each variable.

Investigate pairwise correlations

Question

• Compute, display and comment the sample correlation matrix
• Display jointplots for each pair of variables

Singular Value Decomposition (SVD)

Question

• Project the swiss dataset on the covariates (all columns but Fertility)
• Center the projected data using matrix manipulation
• Center the projected data using dplyr verbs
• Compare the results with the output of scale() with various optional arguments
• Call the centered matrix Y

Question

Check that the ouput of svd(Y) actually defines a Singular Value Decomposition.

Question

Relate the SVD of \(Y\) and the eigen decomposition of \(Y^\top \times Y\)

Perform PCA on covariates

Question

Pairwise analysis did not provide us with a clear and simple picture of the French-speaking districts.

PCA (Principal Component Analysis) aims at exploring the variations of multivariate datasets around their mean (center of inertia). In the sequel, we will perform PCA on the matrix of centered covariates, with and without standardizing the centered columns.

Base R offers prcomp(). Call prcomp() on the centered covariates

Note that R also offers princomp

Question

Check that prcomp() is indeed a wrapper for svd().

Question

Check that rows and columns of component rotation of the result of prcomp() have unit norm.

Question

Check Orthogonality of \(V\) (component rotation of the prcomp object)

Question

Make a scatterplot from the first two columns of the \(x\) component of the prcomp object.

Question

Define a graphical pipeline for the screeplot.

Hint: use function tidy() from broom, to get the data in the right form from an instance of prcomp.

Question

Define a function that replicates autoplot.prcomp()

Project the dataset on the first two principal components (perform dimension reduction) and build a scatterplot. Colour the points according to the value of original covariates.

Hint: use generic function augment from broom.

Question

Apply broom::tidy() with optional argument matrix="v" or matrix="loadings" to the prcomp object.

Comment.

Question

Build the third SVD plot, the so called correlation circle.

Question

Compute PCA after standardizing the columns, draw the correlation circle.

Compare standardized and non-standardized PCA

Question

Pay attention to the correlation circles.

1. How well are variables represented?
2. Which variables contribute to the first axis?

Question

Explain the contrast between the two correlation circles.

In the sequel we focus on standardized PCA.

Provide an interpretation of the first two principal axes

Question

Which variables contribute to the two first principal axes?

Question

Analyze the signs of correlations between variables and axes?

Add the Fertility variable

Question

Plot again the correlation circle using the same principal axes as before, but add the Fertility variable.

How does Fertility relate with covariates? with principal axes?

Biplot

Question

The last svd plot (biplot) consists of overlaying the scatter plot of component x of the prcomp object and the correlation circle.

So the biplot is a graphical object built on two dataframes derived on components x and rotation of the prcomp objects.

Design a graphical pipeline.

Question

autoplot.prcomp() has optional arguments. If set to True, logical argument loadings overlays the scatterplot defined by the principal components with the correlation circle.

Generics

autoplot() is an example of S3 generic function. Let us examine this function using sloop

Use sloop::s3_dispatch() to compare autoplot(prcomp(swiss)) and autoplot(lm(Fertility ~ ., swiss))

Use sloop::s3_getmethod() to see the body of autoplot.prcomp

References

S3 classes

https://scholar.google.com/citations?user=xbCKOYMAAAAJ&hl=fr&oi=ao