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Principal Component Analysis

Source: vignettes/articles/principal-component-analysis.Rmd
principal-component-analysis.Rmd

When looking for stylistic similarities among texts, it is common to use principal component analysis to project many different features in a flattened two-dimensional space. Typically the first principal component is projected on the X-axis, and the second component is projected on the Y-axis.

The stylo package, in addition to measuring word frequencies among all documents in a corpus, will create visualizations of this type, projecting many documents into a two-dimensional chart showing these kinds of coordinates. As an example, here’s the code and output from stylo showing similarities among the eighty-five Federalist Papers, originally published pseudonymously in 1788:

library(stylo)

federalist_mfw <- 
  stylo(gui = FALSE,
        corpus.dir = system.file("extdata/federalist", package = "stylo2gg"),
        analysis.type = "PCR",
        pca.visual.flavour = "symbols",
        analyzed.features = "w",
        ngram.size = 1,
        display.on.screen = TRUE,
        sampling = "no.sampling",
        culling.max = 75,
        culling.min = 75,
        mfw.min = 900,
        mfw.max = 900)
This visualization places each part by its frequencies of 120 of the most frequent words—chosen from among words appearing in at least three-fourths of all papers The chart shows that the texts whose authorship had once been in question, shown here with red Xs, have frequency distributions most similar to those by James Madison, shown here with green crosses.

This visualization places each part by its frequencies of 120 of the most frequent words—chosen from among words appearing in at least three-fourths of all papers The chart shows that the texts whose authorship had once been in question, shown here with red Xs, have frequency distributions most similar to those by James Madison, shown here with green crosses.

As the figure suggests, most of these documents were eventually known to be written by Alexander Hamilton, John Jay, and James Madison, shown categorized here by their last names. Although most had known authorship, some were disputed or had joint authorship, shown here by the “NA” category. From their placement along the X- and Y-axes, the disputed papers seem closest in style to those by James Madison. The analysis here uses some of the same measures Frederick Mosteller and David Wallace famously used in their 1963 study, and it arrives at similar conclusions, but the ease and usefulness of tools like stylo means that preparing this quick visualization demands far less time and sweat.

In saving this output to a named object federalist_mfw, stylo makes it possible to access the frequency tables to study them in other ways. By taking advantage of this object, the stylo2gg package makes it very easy to try out different visualizations. Without any changed parameters, the stylo2gg() function will import defaults from the call used to run stylo():

library(stylo2gg)
federalist_mfw |> 
  stylo2gg()
Using selected `ggplot2` defaults for shapes and colors, the visualization created by `stylo2gg` nevertheless shows the same patterns of style, presenting a figure drawn from the same principal components. Here, the disputed papers are marked by purple diamonds, and they seem closest in style to the parts known to be by Madison, marked by blue Xs.

Using selected ggplot2 defaults for shapes and colors, the visualization created by stylo2gg nevertheless shows the same patterns of style, presenting a figure drawn from the same principal components. Here, the disputed papers are marked by purple diamonds, and they seem closest in style to the parts known to be by Madison, marked by blue Xs.

In the simplest conversion of a stylo object, stylo2gg() tries as closely as is reasonable to recreate the analytic and aesthetic decisions that went into the creation of that object, creating a chart comparing first and second principal components with axes marked by each’s distributed percentage of variation; the caption shows the feature count, feature type, culling percentage, and matrix type; and the legend is shown at the top of the graph. As shown, stylo2gg even honors the choice from the original stylo() call to show principal components derived from a correlation matrix, although other options are available.

Labeling points

From here, it’s easy to change options to clarify an analysis without having to call stylo() again. Files prepared for stylo typically follow a specific naming convention: in the case of this corpus, Federalist No. 10 is prepared in a text file called Madison_10.txt, indicating metadata offset by underscores, with the author name coming first and the title or textual marker coming next. The stylo package already uses the first part of this metadata to apply color to different authors or class groupings of texts. Likewise, stylo2gg follows suit, but it can also choose among these aspects to apply a label. For this chart, it might make sense to replace symbols with the number of each paper it represents:

federalist_mfw |> 
  stylo2gg(shapes = FALSE, 
           labeling = 2)
The option `shapes=FALSE` turns off the symbols that would otherwise also appear; simultaneously, the option `labeling=2` selects the second metadata element from corpus filenames---in this case the number of the specific paper---as a label for the visualization. When a chosen label consists of nothing but numbers, as it does here, the legend key changes to a number sign. If a label includes any other characters, it becomes the letter 'a', ggplot2's default key for showing color of text.

The option shapes=FALSE turns off the symbols that would otherwise also appear; simultaneously, the option labeling=2 selects the second metadata element from corpus filenames—in this case the number of the specific paper—as a label for the visualization. When a chosen label consists of nothing but numbers, as it does here, the legend key changes to a number sign. If a label includes any other characters, it becomes the letter ‘a’, ggplot2’s default key for showing color of text.

Displaying these labels makes it possible further to study Mosteller and Wallace’s findings on the papers jointly authored by Madison and Hamilton: in this principal component analysis of 120 most frequent words, papers 18, 19, and 20 seem closer in style to Madison than to Hamilton, and Mosteller and Wallace’s work using different techniques seems to show the same finding for two of these three papers, with mixed results for number 20.

If it were preferred instead to label the author names, we could set labeling=1. If we wanted to show everything, replicating stylo’s option pca.visual.flavour="labels", we can set labeling=0:

federalist_mfw |> 
  stylo2gg(shapes = FALSE, 
           labeling = 0)
The option `labeling=0` shows entire file names for items in the corpus, excepting the extension. This option also turns off the legend by default, since that information is already indicated.

The option labeling=0 shows entire file names for items in the corpus, excepting the extension. This option also turns off the legend by default, since that information is already indicated.

Highlighting groups

In addition to recreating some of the visualizations offered by stylo, stylo2gg takes advantage of ggplot2’s extensibility to offer additional options. If, for instance, we want to emphasize the overlap of style among the disputed papers and those by Madison, it’s easy to show a highlight of the 3rd and 4th categories of texts, corresponding to their ordering on the legend:

federalist_mfw |> 
  stylo2gg(shapes = FALSE, 
           labeling = 2,
           highlight = c(3, 4))
The `highlight` option accepts numbers corresponding to categories shown on the legend. Highlights on principal component charts can include 1 or more categories, but highlights for hierarchical clusters can only accept one category. To draw these loops around points on a scatterplot, stylogg relies on the <a href='https://cran.r-project.org/web/packages/ggalt/index.html'>ggalt</a> package.” width=“700” /> <p class= The highlight option accepts numbers corresponding to categories shown on the legend. Highlights on principal component charts can include 1 or more categories, but highlights for hierarchical clusters can only accept one category. To draw these loops around points on a scatterplot, stylogg relies on the ggalt package.

Overlay loadings

With these texts charted, we might want to communicate something about the underlying word frequencies that inform their placement. The top.loadings option allows us to show a number of words—ordered from the most frequent to the least frequent—overlaid with scaled vectors as alternative axes on the principal component chart:

federalist_mfw |> 
  stylo2gg(shapes = FALSE, 
           labeling = 2,
           highlight = 4,
           top.loadings = 10)
Set `top.loadings` to a number `n` to overlay loadings for the most frequent words, from 1 to `n`. This chart shows loadings and scaled vectors for the 10 most frequent words.

Set top.loadings to a number n to overlay loadings for the most frequent words, from 1 to n. This chart shows loadings and scaled vectors for the 10 most frequent words.

Alternatively, show loadings by nearest principal components, by the middle point of a given category, by a specific word, or all of the above:

federalist_mfw |> 
  stylo2gg(shapes = FALSE, 
           labeling = 2,
           select.loadings = list(c(-4, -6),
                                  "Madison",
                                  call("word", 
                                       c("the", "a", "an"))))
In a list form, the `select.loadings` option accepts coordinates, category names, and words. Here, `c(-4,-6)` indicates that the code should find the loading nearest to -4 on the first principal component and -6 on the second principal component; *Madison* indicates that the function should find coordinates at the middle of papers by Madison and then find the loading nearest those coordinates; and three articles *the*, *a*, and *an* indicate, using `call('word')`, that these specific loadings should be shown.

In a list form, the select.loadings option accepts coordinates, category names, and words. Here, c(-4,-6) indicates that the code should find the loading nearest to -4 on the first principal component and -6 on the second principal component; Madison indicates that the function should find coordinates at the middle of papers by Madison and then find the loading nearest those coordinates; and three articles the, a, and an indicate, using call('word'), that these specific loadings should be shown.

These words and lines are gray by default. As of fall 2022, other colors can be specified, and the letters can be converted to uppercase for legibility:1

federalist_mfw |> 
  stylo2gg(top.loadings = 6,
           loadings.line.color = "blue",
           loadings.word.color = "navy",
           loadings.upper = TRUE)
Set `loadings.line.color` and `loadings.word.color` to change the coloring of loadings. Optionally toggle `loadings.upper` to show uppercase letters, or set `loadings.spacer` to define the character shown in lieu of spaces when measuring bigrams and other n-grams.

Set loadings.line.color and loadings.word.color to change the coloring of loadings. Optionally toggle loadings.upper to show uppercase letters, or set loadings.spacer to define the character shown in lieu of spaces when measuring bigrams and other n-grams.

Narrowing things down

One beauty of using a saved frequency list is that it becomes possible to select a subset from the data to inform an analysis. By counting all words that appear in 75% of the texts for this analysis, stylo prepares a frequency table of 120 words. From there, stylo2gg can select a subset of these using the features option, for selecting a specific vector of words, or the num.features option, for automatically selecting a given number of the most frequent words.

One might, for instance, hypothesize that words shorter than four characters are sufficient to differentiate style in these English texts.2 Using the features option, this hypothesis can be tested by choosing a smaller subset from the full list of 120 most frequent words:

short_words <- 
  federalist_mfw$features.actually.used[federalist_mfw$features.actually.used |> nchar() < 4]

federalist_mfw |> 
  stylo2gg(shapes = TRUE, 
           features = short_words,
           top.loadings = 10)
Selecting a subset of features will also cause the caption to update from *120 MFW* to *42 W,* reflecting the changed number of features and the type: they are no longer most frequent words (MFW) but are now just words (W).

Selecting a subset of features will also cause the caption to update from 120 MFW to 42 W, reflecting the changed number of features and the type: they are no longer most frequent words (MFW) but are now just words (W).

Results here suggest that the hypothesis would have been mostly correct, as it is possible still to see patterns in clusters. But limiting an analysis solely to these shorter words makes it harder to differentiate the styles of Hamilton and Madison. Interestingly, Jay’s style remains distinct in this consideration. Interesting, too, the overlay of the top ten loadings shows that papers with positive values in the second principal component in this chart—above a center line—are strongest in first-person plural features like “us” and “our” and “we.” And perhaps most interesting, just quickly looking at the top ten loadings suggests that Hamilton’s papers may have been less likely to use past-tense constructions like “was” and “had,” preferring infinitive forms marked by “to” and “be.”

If instead of manually selecting features one preferred to choose a subset by number, the num.features option makes it possible to do so.

library(stringr)

federalist_mfw |> 
  stylo2gg(shapes = FALSE, 
           labeling = federalist_mfw$table.with.all.freqs |> 
             rownames() |> 
             str_extract("^[A-Z]") |> 
             paste0(".",
                    federalist_mfw$table.with.all.freqs |> 
                      rownames() |> 
                      str_extract("[0-9]+")),
           legend = FALSE,
           highlight = 2,
           num.features = 50)
Setting `num.features` to 50 will limit a chart to the 50 most frequent words. The caption updates to reflect this choice.

Setting num.features to 50 will limit a chart to the 50 most frequent words. The caption updates to reflect this choice.

The code for this last visualization also shows that, in addition to a number corresponding to the metadata from a filename, the labeling option can also accept a vector of the same length as the number of texts. Here, the result shows the first letter of each author category, a dot, and the text’s corresponding number; additionally, the legend has been turned off by setting its option to FALSE.

Emphasizing with contrast

By default, stylo2gg uses symbols that ought to be distinguishable when printing in gray scale. Use the black= option with the number of a given category to further optimize for gray-scale printing or to employ contrast to emphasize a particular group.

federalist_mfw |> 
  stylo2gg(black = 4)

Withholding texts from a PCA projection

In cases of disputed authorship, it can be desirable to understand relationships among known texts and authors before considering those of unknown provenance. New in version 1.0, stylo2gg’s withholding parameter allows for certain classes to be left out from defining the base projection of a principal component analysis. These texts are then projected into a space they did not help define:

federalist_mfw |> 
  stylo2gg(withholding = "NA",
           highlight = 3,
           black = 4)
Defining `withholding` makes it possible to ignore certain classes of texts from the underlying projection.

Defining withholding makes it possible to ignore certain classes of texts from the underlying projection.

Choosing principal components

Following stylo’s lead, stylo2gg shows the first two principal components by default, but it may often be necessary to show more. Introduced earlier in 2023, the pc.x and pc.y parameters make it possible to map other components to the X-axis and Y-axis, simultaneously updating axis labels to indicate components and variance.3

federalist_mfw |> 
  stylo2gg(pc.x = 3, pc.y = 4)

Other options for principal component analysis

In addition to the options shown above, principal component analysis can be directed with a covariance matrix (viz="PCV") or correlation matrix (viz="PCV"), and a given chart can be flipped horizontally (with invert.x=TRUE) or vertically (invert.y=TRUE). Additionally, the caption below the chart can be removed using caption=FALSE. Alternatively, setting viz="pca" will choose a minimal set of changes from which one might choose to build up selected additions: turning on captions (caption=TRUE), moving the legend or calling on other Ggplot2 commands, adding a title (using title="Title Goes Here"), or other matters.

library(ggplot2)

federalist_mfw |> 
  rename_category("NA", "unknown") |> 
  stylo2gg(viz = "pca") + 
  theme(legend.position = "bottom") + 
  scale_shape_manual(values = 15:18) +
  scale_size_manual(values = c(8.5, 9, 7, 10)) +
  scale_alpha_manual(values = rep(.5, 4)) + 
  scale_color_manual(values = c("#E69F00", "#56B4E9", "#009E73", "#CC79A7")) +
  labs(title = "Larger, solid points can make relationships easier to understand.",
       subtitle = "Setting alpha values is a good idea when solid points overlap.") +
  theme(plot.title.position = "plot")
Setting `viz='pca'` rather than the stylo-flavored `viz='PCR'` or `viz='PCV'` prepares a minimal visualization of a principal component analysis derived from a correlation matrix. This might be a good setting to use if further customizing the figure by adding refinements provided by ggplot2 functions---at which point it will become necessary to load that package explicitly. The example here also shows the utility of the stylo2gg function for adjusting labels, `rename_category()`.

Setting viz='pca' rather than the stylo-flavored viz='PCR' or viz='PCV' prepares a minimal visualization of a principal component analysis derived from a correlation matrix. This might be a good setting to use if further customizing the figure by adding refinements provided by ggplot2 functions—at which point it will become necessary to load that package explicitly. The example here also shows the utility of the stylo2gg function for adjusting labels, rename_category().