# Term Frequency and Inverse Document Frequency (tf-idf) Using Tidy Data Principles

#### 2016-10-27

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. 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 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 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 the main vignette, 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.

Let’s look 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

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(show.legend = FALSE) +
xlim(NA, 0.0009) +
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. 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 specifically at Pride and Prejudice.

book_words %>%
filter(book == "Pride & Prejudice") %>%
select(-total) %>%
arrange(desc(tf_idf))
## # A tibble: 6,538 × 6
##                 book      word     n           tf       idf      tf_idf
##               <fctr>     <chr> <int>        <dbl>     <dbl>       <dbl>
## 1  Pride & Prejudice     darcy   373 0.0030522732 1.7917595 0.005468939
## 2  Pride & Prejudice    bennet   294 0.0024058132 1.7917595 0.004310639
## 3  Pride & Prejudice   bingley   257 0.0021030408 1.7917595 0.003768143
## 4  Pride & Prejudice elizabeth   597 0.0048852738 0.6931472 0.003386214
## 5  Pride & Prejudice   wickham   162 0.0013256522 1.7917595 0.002375250
## 6  Pride & Prejudice   collins   156 0.0012765540 1.7917595 0.002287278
## 7  Pride & Prejudice     lydia   133 0.0010883441 1.7917595 0.001950051
## 8  Pride & Prejudice     lizzy    95 0.0007773886 1.7917595 0.001392893
## 9  Pride & Prejudice longbourn    88 0.0007201074 1.7917595 0.001290259
## 10 Pride & Prejudice  gardiner    84 0.0006873752 1.7917595 0.001231611
## # ... with 6,528 more rows

These words are, as measured by tf-idf, the most important to Pride and Prejudice and most readers would likely agree.