4 Analyzing word and document frequency: tf-idf

A central question in text mining and natural language processing is how to quantify what a document is about. Can we do this by looking at the words that make up the document? One measure of how important a word may be is its term frequency (tf), how frequently a word occurs in a document; we have examined how to measure word frequency using tidy data principles in Chapter 2. There are words in a document, however, that occur many times but may not be important; in English, these are probably words like “the”, “is”, “of”, and so forth. We might take the approach of adding words like these to a list of stop words and removing them before analysis, but it is possible that some of these words might be more important in some documents than others. A list of stop words is not a very sophisticated approach to adjusting term frequency for commonly used words.

Another approach is to look at a term’s inverse document frequency (idf), which decreases the weight for commonly used words and increases the weight for words that are not used very much in a collection of documents. This can be combined with term frequency to calculate a term’s tf-idf (the two quantities multiplied together), the frequency of a term adjusted for how rarely it is used. It is intended to measure how important a word is to a document in a collection (or corpus) of documents. It is a rule-of-thumb or heuristic quantity; while it has proved useful in text mining, search engines, etc., its theoretical foundations are considered less than firm by information theory experts. The inverse document frequency for any given term is defined as

\[idf(\text{term}) = \ln{\left(\frac{n_{\text{documents}}}{n_{\text{documents containing term}}}\right)}\]

We can use tidy data principles, as described in Chapter 2, to approach tf-idf analysis and use consistent, effective tools to quantify how important various terms are in a document that is part of a collection.

4.1 Term frequency in Jane Austen’s novels

Let’s start by looking at the published novels of Jane Austen and examine first term frequency, then tf-idf. We can start just by using dplyr verbs such as group_by and join. What are the most commonly used words in Jane Austen’s novels? (Let’s also calculate the total words in each novel here, for later use.)

library(dplyr)
library(janeaustenr)
library(tidytext)

book_words <- austen_books() %>%
  unnest_tokens(word, text) %>%
  count(book, word, sort = TRUE) %>%
  ungroup()

total_words <- book_words %>% 
  group_by(book) %>% 
  summarize(total = sum(n))

book_words <- left_join(book_words, total_words)

book_words
## # A tibble: 40,379 × 4
##                 book  word     n  total
##               <fctr> <chr> <int>  <int>
## 1     Mansfield Park   the  6206 160460
## 2     Mansfield Park    to  5475 160460
## 3     Mansfield Park   and  5438 160460
## 4               Emma    to  5239 160996
## 5               Emma   the  5201 160996
## 6               Emma   and  4896 160996
## 7     Mansfield Park    of  4778 160460
## 8  Pride & Prejudice   the  4331 122204
## 9               Emma    of  4291 160996
## 10 Pride & Prejudice    to  4162 122204
## # ... with 40,369 more rows

There is one row in this data frame for each word-book combination; n is the number of times that word is used in that book and total is the total words in that book. The usual suspects are here, “the”, “and”, “to”, and so forth. Let’s look at the distribution of n/total for each novel, the number of times a word appears in a novel divided by the total number of terms (words) in that novel. This is exactly what term frequency is.

library(ggplot2)

ggplot(book_words, aes(n/total, fill = book)) +
  geom_histogram(alpha = 0.8, show.legend = FALSE) +
  xlim(NA, 0.0009) +
  labs(title = "Term Frequency Distribution in Jane Austen's Novels") +
  facet_wrap(~book, ncol = 2, scales = "free_y")

There are very long tails to the right for these novels (those extremely common words!) that we have not shown in these plots. These plots exhibit similar distributions for all the novels, with many words that occur rarely and fewer words that occur frequently.

4.2 Zipf’s law

TODO

freq_by_rank <- book_words %>% 
  group_by(book) %>% 
  mutate(rank = row_number(), 
         `term frequency` = n/total)

freq_by_rank
## Source: local data frame [40,379 x 6]
## Groups: book [6]
## 
##                 book  word     n  total  rank `term frequency`
##               <fctr> <chr> <int>  <int> <int>            <dbl>
## 1     Mansfield Park   the  6206 160460     1       0.03867631
## 2     Mansfield Park    to  5475 160460     2       0.03412065
## 3     Mansfield Park   and  5438 160460     3       0.03389007
## 4               Emma    to  5239 160996     1       0.03254118
## 5               Emma   the  5201 160996     2       0.03230515
## 6               Emma   and  4896 160996     3       0.03041069
## 7     Mansfield Park    of  4778 160460     4       0.02977689
## 8  Pride & Prejudice   the  4331 122204     1       0.03544074
## 9               Emma    of  4291 160996     4       0.02665284
## 10 Pride & Prejudice    to  4162 122204     2       0.03405780
## # ... with 40,369 more rows
freq_by_rank %>% 
  ggplot(aes(rank, `term frequency`, color = book)) + 
  geom_line(size = 1.2, alpha = 0.8) + 
  scale_x_log10() +
  scale_y_log10()

Notice that this plot is in log-log coordinates.

lower_rank <- freq_by_rank %>% 
  filter(rank < 500)

lm(log10(`term frequency`) ~ log10(rank), data = lower_rank)
## 
## Call:
## lm(formula = log10(`term frequency`) ~ log10(rank), data = lower_rank)
## 
## Coefficients:
## (Intercept)  log10(rank)  
##     -0.7727      -1.0491
freq_by_rank %>% 
  ggplot(aes(rank, `term frequency`, color = book)) + 
  geom_abline(intercept = -0.77, slope = -1.05, color = "gray50", linetype = 2) +
  geom_line(size = 1.2, alpha = 0.8) + 
  scale_x_log10() +
  scale_y_log10()

4.3 The bind_tf_idf function

The idea of tf-idf is to find the important words for the content of each document by decreasing the weight for commonly used words and increasing the weight for words that are not used very much in a collection or corpus of documents, in this case, the group of Jane Austen’s novels as a whole. Calculating tf-idf attempts to find the words that are important (i.e., common) in a text, but not too common. Let’s do that now.

book_words <- book_words %>%
  bind_tf_idf(word, book, n)
book_words
## # A tibble: 40,379 × 7
##                 book  word     n  total         tf   idf tf_idf
##               <fctr> <chr> <int>  <int>      <dbl> <dbl>  <dbl>
## 1     Mansfield Park   the  6206 160460 0.03867631     0      0
## 2     Mansfield Park    to  5475 160460 0.03412065     0      0
## 3     Mansfield Park   and  5438 160460 0.03389007     0      0
## 4               Emma    to  5239 160996 0.03254118     0      0
## 5               Emma   the  5201 160996 0.03230515     0      0
## 6               Emma   and  4896 160996 0.03041069     0      0
## 7     Mansfield Park    of  4778 160460 0.02977689     0      0
## 8  Pride & Prejudice   the  4331 122204 0.03544074     0      0
## 9               Emma    of  4291 160996 0.02665284     0      0
## 10 Pride & Prejudice    to  4162 122204 0.03405780     0      0
## # ... with 40,369 more rows

Notice that idf and thus tf-idf are zero for these extremely common words. These are all words that appear in all six of Jane Austen’s novels, so the idf term (which will then be the natural log of 1) is zero. The inverse document frequency (and thus tf-idf) is very low (near zero) for words that occur in many of the documents in a collection; this is how this approach decreases the weight for common words. The inverse document frequency will be a higher number for words that occur in fewer of the documents in the collection. Let’s look at terms with high tf-idf in Jane Austen’s works.

book_words %>%
  select(-total) %>%
  arrange(desc(tf_idf))
## # A tibble: 40,379 × 6
##                   book      word     n          tf      idf      tf_idf
##                 <fctr>     <chr> <int>       <dbl>    <dbl>       <dbl>
## 1  Sense & Sensibility    elinor   623 0.005193528 1.791759 0.009305552
## 2  Sense & Sensibility  marianne   492 0.004101470 1.791759 0.007348847
## 3       Mansfield Park  crawford   493 0.003072417 1.791759 0.005505032
## 4    Pride & Prejudice     darcy   373 0.003052273 1.791759 0.005468939
## 5           Persuasion    elliot   254 0.003036207 1.791759 0.005440153
## 6                 Emma      emma   786 0.004882109 1.098612 0.005363545
## 7     Northanger Abbey    tilney   196 0.002519928 1.791759 0.004515105
## 8                 Emma    weston   389 0.002416209 1.791759 0.004329266
## 9    Pride & Prejudice    bennet   294 0.002405813 1.791759 0.004310639
## 10          Persuasion wentworth   191 0.002283132 1.791759 0.004090824
## # ... with 40,369 more rows

Here we see all proper nouns, names that are in fact important in these novels. None of them occur in all of novels, and they are important, characteristic words for each text. Some of the values for idf are the same for different terms because there are 6 documents in this corpus and we are seeing the numerical value for \(\ln(6/1)\), \(\ln(6/2)\), etc. Let’s look at a visualization for these high tf-idf words.

plot_austen <- book_words %>%
  arrange(desc(tf_idf)) %>%
  mutate(word = factor(word, levels = rev(unique(word))))

ggplot(plot_austen[1:20,], aes(word, tf_idf, fill = book)) +
  geom_bar(alpha = 0.8, stat = "identity") +
  labs(title = "Highest tf-idf words in Jane Austen's Novels",
       x = NULL, y = "tf-idf") +
  coord_flip()

Let’s look at the novels individually.

plot_austen <- plot_austen %>% 
  group_by(book) %>% 
  top_n(15) %>% 
  ungroup

ggplot(plot_austen, aes(word, tf_idf, fill = book)) +
  geom_bar(alpha = 0.8, stat = "identity", show.legend = FALSE) +
  labs(title = "Highest tf-idf words in Jane Austen's Novels",
       x = NULL, y = "tf-idf") +
  facet_wrap(~book, ncol = 2, scales = "free") +
  coord_flip()

Still all proper nouns! These words are, as measured by tf-idf, the most important to each novel and most readers would likely agree.

4.4 A corpus of physics texts

Let’s work with another corpus of documents, to see what terms are important in a different set of works. In fact, let’s leave the world of fiction and narrative entirely. Let’s download some classic physics texts from Project Gutenberg and see what terms are important in these works, as measured by tf-idf. Let’s download Discourse on Floating Bodies by Galileo Galilei, Treatise on Light by Christiaan Huygens, Experiments with Alternate Currents of High Potential and High Frequency by Nikola Tesla, and Relativity: The Special and General Theory by Albert Einstein.

This is a pretty diverse bunch. They may all be physics classics, but they were written across a 300-year timespan, and some of them were first written in other languages and then translated to English. Perfectly homogeneous these are not, but that doesn’t stop this from being an interesting exercise!

library(gutenbergr)
physics <- gutenberg_download(c(37729, 14725, 13476, 5001), 
                              meta_fields = "author")
physics_words <- physics %>%
  unnest_tokens(word, text) %>%
  count(author, word, sort = TRUE) %>%
  ungroup()

physics_words
## # A tibble: 12,592 × 3
##                 author  word     n
##                  <chr> <chr> <int>
## 1     Galilei, Galileo   the  3760
## 2        Tesla, Nikola   the  3604
## 3  Huygens, Christiaan   the  3553
## 4     Einstein, Albert   the  2994
## 5     Galilei, Galileo    of  2049
## 6     Einstein, Albert    of  2030
## 7        Tesla, Nikola    of  1737
## 8  Huygens, Christiaan    of  1708
## 9  Huygens, Christiaan    to  1207
## 10       Tesla, Nikola     a  1176
## # ... with 12,582 more rows

Here we see just the raw counts, and of course these documents are all different lengths. Let’s go ahead and calculate tf-idf.

physics_words <- physics_words %>%
  bind_tf_idf(word, author, n) 

plot_physics <- physics_words %>%
  arrange(desc(tf_idf)) %>%
  mutate(word = factor(word, levels = rev(unique(word)))) %>%
  mutate(author = factor(author, levels = c("Galilei, Galileo",
                                            "Huygens, Christiaan", 
                                            "Tesla, Nikola",
                                            "Einstein, Albert")))

ggplot(plot_physics[1:20,], aes(word, tf_idf, fill = author)) +
  geom_bar(alpha = 0.8, stat = "identity") +
  labs(title = "Highest tf-idf words in Classic Physics Texts",
       x = NULL, y = "tf-idf") +
  coord_flip()

Nice! Let’s look at each text individually.

plot_physics <- plot_physics %>% 
  group_by(author) %>% 
  top_n(15, tf_idf) %>% 
  mutate(word = reorder(word, tf_idf))

ggplot(plot_physics, aes(word, tf_idf, fill = author)) +
  geom_bar(alpha = 0.8, stat = "identity", show.legend = FALSE) +
  labs(title = "Highest tf-idf words in Classic Physics Texts",
       x = NULL, y = "tf-idf") +
  facet_wrap(~author, ncol = 2, scales = "free") +
  coord_flip()

Very interesting indeed. One thing we see here is “gif” in the Einstein text?!

grep("eq\\.", physics$text, value = TRUE)[1:10]
##  [1] "                         eq. 1: file eq01.gif"                  
##  [2] "                         eq. 2: file eq02.gif"                  
##  [3] "                         eq. 3: file eq03.gif"                  
##  [4] "                         eq. 4: file eq04.gif"                  
##  [5] "                       eq. 05a: file eq05a.gif"                 
##  [6] "                       eq. 05b: file eq05b.gif"                 
##  [7] "the distance between the points being eq. 06 ."                 
##  [8] "direction of its length with a velocity v is eq. 06 of a metre."
##  [9] "velocity v=c we should have eq. 06a ,"                          
## [10] "the rod as judged from K1 would have been eq. 06 ;"

Some cleaning up of the text might be in order. “K1” is the name of a coordinate system for Einstein:

grep("K1", physics$text, value = TRUE)[1]
## [1] "to a second co-ordinate system K1 provided that the latter is"

Maybe it makes sense to keep this one. Also notice that in this line we have “co-ordinate”, which explains why there are separate “co” and “ordinate” items in the high tf-idf words for the Einstein text.

“AB”, “RC”, and so forth are names of rays, circles, angles, and so forth for Huygens.

grep("AK", physics$text, value = TRUE)[1]
## [1] "Now let us assume that the ray has come from A to C along AK, KC; the"

Let’s remove some of these less meaningful words to make a better, more meaningful plot. Notice that we make a custom list of stop words and use anti_join to remove them.

mystopwords <- data_frame(word = c("eq", "co", "rc", "ac", "ak", "bn", 
                                   "fig", "file", "cg", "cb", "cm"))
physics_words <- anti_join(physics_words, mystopwords, by = "word")
plot_physics <- physics_words %>%
  arrange(desc(tf_idf)) %>%
  mutate(word = factor(word, levels = rev(unique(word)))) %>%
  group_by(author) %>% 
  top_n(15, tf_idf) %>%
  ungroup %>%
  mutate(author = factor(author, levels = c("Galilei, Galileo",
                                            "Huygens, Christiaan",
                                            "Tesla, Nikola",
                                            "Einstein, Albert")))

ggplot(plot_physics, aes(word, tf_idf, fill = author)) +
  geom_bar(alpha = 0.8, stat = "identity", show.legend = FALSE) +
  labs(title = "Highest tf-idf words in Classic Physics Texts",
       x = NULL, y = "tf-idf") +
  facet_wrap(~author, ncol = 2, scales = "free") +
  coord_flip()

We don’t hear enough about ramparts or things being ethereal in physics today.