…or your money back.
Setup
We've got a collection of documents:
doc1 <- "Stray cats are running all over the place. I see 10 a day!"
doc2 <- "Cats are killers. They kill billions of animals a year."
doc3 <- "The best food in Columbus, OH is the North Market."
doc4 <- "Brand A is the best tasting cat food around. Your cat will love it."
doc5 <- "Buy Brand C cat food for your cat. Brand C makes healthy and happy cats."
doc6 <- "The Arnold Classic came to town this weekend. It reminds us to be healthy."
doc7 <- "I have nothing to say. In summary, I have told you nothing."
doc.list <- list(doc1, doc2, doc3, doc4, doc5, doc6, doc7)
N.docs <- length(doc.list)
names(doc.list) <- paste0("doc", c(1:N.docs))
We also have an information need that is expressed via the following text query:
query <- "Healthy cat food"
We're going to use an old method that goes way back to the 1960's. Specifically, we'll implement the vector space model of information retrieval in R. In the process, you'll hopefully learn something about the tm package.
Text mining in R
Fundamentals of the tm package
If you have not installed the tm [1][2] and SnowballC [3] packages, please do so now.
install.packages("tm")
install.packages("SnowballC")
Load the tm package into memory.
library(tm)
In text mining and related fields, a corpus is a collection of texts, often with extensive manual annotation. Not surprisingly, the Corpus class is a fundamental data structure in tm.
my.docs <- VectorSource(c(doc.list, query))
my.docs$Names <- c(names(doc.list), "query")
my.corpus <- Corpus(my.docs)
my.corpus
<<VCorpus>>
Metadata: corpus specific: 0, document level (indexed): 0
Content: documents: 8
Above we treated the query like any other document. It is, after all, just another string of text. Queries are not typically known a priori, but in the processing steps that follow, we will pretend like we knew ours in advance to avoid repeating steps.
Standardizing
One of the nice things about the Corpus class is the tm_map function, which cleans and standardizes documents within a Corpus object. Below are some of the transformations.
getTransformations()
[1] "removeNumbers" "removePunctuation" "removeWords"
[4] "stemDocument" "stripWhitespace"
First, let's get rid of punctuation.
my.corpus <- tm_map(my.corpus, removePunctuation)
content(my.corpus[[1]])
## Stray cats are running all over the place I see 10 a day
Suppose we don't want to count “cats” and “cat” as two separate words. Then we will use the stemDocument transformation to implement the famous Porter Stemmer algorithm. To use this particular transformation, first load the SnowballC package.
library(SnowballC)
my.corpus <- tm_map(my.corpus, stemDocument)
content(my.corpus[[1]])
## Stray cat are run all over the place I see 10 a day
Finally, remove numbers and any extra white space.
my.corpus <- tm_map(my.corpus, removeNumbers)
my.corpus <- tm_map(my.corpus, content_transformer(tolower))
my.corpus <- tm_map(my.corpus, stripWhitespace)
content(my.corpus[[1]])
## stray cat are run all over the place i see a day
We applied all these standardization techniques without much thought. For instance, we sacrificed inflection in favor of fewer words. But at least the transformations make sense on a heuristic level, much like the similarity concepts to follow.
The vector space model
Document similarity
Here's a trick that's been around for a while: represent each document as a vector in \( \mathcal{R}^N \) (with \( N \) as the number of words) and use the angle \( \theta \) between the vectors as a similarity measure. Rank by the similarity of each document to the query and you have a search engine.
One of the simplest things we can do is to count words within documents. This naturally forms a two dimensional structure, the term document matrix, with rows corresponding to the words and the columns corresponding to the documents. As with any matrix, we may think of a term document matrix as a collection of column vectors existing in a space defined by the rows. The query lives in this space as well, though in practice we wouldn't know it beforehand.
term.doc.matrix.stm <- TermDocumentMatrix(my.corpus)
colnames(term.doc.matrix.stm) <- c(names(doc.list), "query")
inspect(term.doc.matrix.stm[0:14, ])
<<TermDocumentMatrix (terms: 14, documents: 8)>>
Non-/sparse entries: 21/91
Sparsity : 81%
Maximal term length: 8
Weighting : term frequency (tf)
Docs
Terms doc1 doc2 doc3 doc4 doc5 doc6 doc7 query
all 1 0 0 0 0 0 0 0
and 0 0 0 0 1 0 0 0
anim 0 1 0 0 0 0 0 0
are 1 1 0 0 0 0 0 0
arnold 0 0 0 0 0 1 0 0
around 0 0 0 1 0 0 0 0
best 0 0 1 1 0 0 0 0
billion 0 1 0 0 0 0 0 0
brand 0 0 0 1 2 0 0 0
buy 0 0 0 0 1 0 0 0
came 0 0 0 0 0 1 0 0
cat 1 1 0 2 3 0 0 1
classic 0 0 0 0 0 1 0 0
columbus 0 0 1 0 0 0 0 0
Sparsity and storage of the term document matrix
The matrices in tm are of type Simple Triplet Matrix where only the triples \( (i, j, value) \) are stored for non-zero values. To work directly with these objects, you may use install the slam [4] package. We bear some extra cost by making the matrix “dense” (i.e., storing all the zeros) below.
term.doc.matrix <- as.matrix(term.doc.matrix.stm)
cat("Dense matrix representation costs", object.size(term.doc.matrix), "bytes.\n",
"Simple triplet matrix representation costs", object.size(term.doc.matrix.stm),
"bytes.")
## Dense matrix representation costs 6688 bytes.
## Simple triplet matrix representation costs 5808 bytes.
Variations on a theme
In term.doc.matrix, the dimensions of the document space are simple term frequencies. This is fine, but other heuristics are available. For instance, rather than a linear increase in the term frequency \( tf \), perhaps \( \sqrt(tf) \) or \( \log(tf) \) would provide a more reasonable diminishing returns on word counts within documents.
Rare words can also get a boost. The word “healthy” appears in only one document, whereas “cat” appears in four. A word's document frequency \( df \) is the number of documents that contain it, and a natural choice is to weight words inversely proportional to their \( df \)s. As with term frequency, we may use logarithms or other transformations to achieve the desired effect.
The tm function weightTfIdf offers one variety of tfidf weighting, but below we build our own. Visit the Wikipedia page for the SMART Information Retrieval System for a brief history and a list of popular weighting choices.
Different weighting choices are often made for the query and the documents. For instance, Manning et al.'s worked example [5] uses \( idf \) weighting only for the query.
Choice and implementation
For both the document and query, we choose tfidf weights of \( (1 + \log_2(tf)) \times \log_2(N/df) \), which are defined to be \( 0 \) if \( tf = 0 \). Note that whenever a term does not occur in a specific document, or when it appears in every document, its weight is zero.
get.tf.idf.weights <- function(tf.vec) {
# Computes tfidf weights from term frequency vector
n.docs <- length(tf.vec)
doc.frequency <- length(tf.vec[tf.vec > 0])
weights <- rep(0, length(tf.vec))
weights[tf.vec > 0] <- (1 + log2(tf.vec[tf.vec > 0])) * log2(n.docs/doc.frequency)
return(weights)
}
# For a word appearing in 4 of 6 documents, occurring 1, 2, 3, and 6 times"
get.tf.idf.weights(c(1, 2, 3, 0, 0, 6))
[1] 0.5849625 1.1699250 1.5121061 0.0000000 0.0000000 2.0970686
Using apply, we run the tfidf weighting function on every row of the term document matrix. The document frequency is easily derived from each row by the counting the non-zero entries (not including the query).
tfidf.matrix <- t(apply(term.doc.matrix, 1,
FUN = function(row) {get.tf.idf.weights(row)}))
colnames(tfidf.matrix) <- colnames(term.doc.matrix)
tfidf.matrix[0:3, ]
Terms doc1 doc2 doc3 doc4 doc5 doc6 doc7 query
all 3 0 0 0 0 0 0 0
and 0 0 0 0 3 0 0 0
anim 0 3 0 0 0 0 0 0
Dot product geometry
A benefit of being in the vector space \( \mathcal{R}^N \) is the use of its dot product. For vectors \( a \) and \( b \), the geometric definition of the dot product is \( a \cdot b = \vert\vert a\vert\vert \, \vert\vert b \vert \vert \cos \theta \), where \( \vert\vert \cdot \vert \vert \) is the euclidean norm (the root sum of squares) and \( \theta \) is the angle between \( a \) and \( b \).
In fact, we can work directly with the cosine of \( \theta \). For \( \theta \) in the interval \( [-\pi, -\pi] \), the endpoints are orthogonality (totally unrelated documents) and the center, zero, is complete collinearity (maximally similar documents). We can see that the cosine decreases from its maximum value of \( 1.0 \) as the angle departs from zero in either direction.
angle <- seq(-pi, pi, by = pi/16)
plot(cos(angle) ~ angle, type = "b", xlab = "angle in radians",
main = "Cosine similarity by angle")
We may furthermore normalize each column vector in our tfidf matrix so that its norm is one. Now the dot product is \( \cos \theta \).
tfidf.matrix <- scale(tfidf.matrix, center = FALSE,
scale = sqrt(colSums(tfidf.matrix^2)))
tfidf.matrix[0:3, ]
Terms doc1 doc2 doc3 doc4 doc5 doc6 doc7 query
all 0.3625797 0.0000000 0 0 0.0000000 0 0 0
and 0.0000000 0.0000000 0 0 0.3558476 0 0 0
anim 0.0000000 0.3923672 0 0 0.0000000 0 0 0
Matrix multiplication: a dot product machine
Treating the query as just another document kept things simple for this article, though in a production system the query will effectively come from a different corpus (see @Lorien's comment below). Now it's time to split them up.
query.vector <- tfidf.matrix[, (N.docs + 1)]
tfidf.matrix <- tfidf.matrix[, 1:N.docs]
With the query vector and the set of document vectors in hand, it is time to go after the cosine similarities. These are simple dot products as our vectors have been normalized to unit length.
Recall that matrix multiplication is really just a sequence of vector dot products. The matrix operation below returns values of \( \cos \theta \) for each document vector and the query vector.
doc.scores <- t(query.vector) %*% tfidf.matrix
With scores in hand, rank the documents by their cosine similarities with the query vector.
results.df <- data.frame(doc = names(doc.list), score = t(doc.scores),
text = unlist(doc.list))
results.df <- results.df[order(results.df$score, decreasing = TRUE), ]
The results
How did our search engine do?
options(width = 2000)
print(results.df, row.names = FALSE, right = FALSE, digits = 2)
doc score text
doc5 0.267 Buy Brand C cat food for your cat. Brand C makes healthy and happy cats.
doc4 0.143 Brand A is the best tasting cat food around. Your cat will love it.
doc6 0.132 The Arnold Classic came to town this weekend. It reminds us to be healthy.
doc3 0.090 The best food in Columbus, OH is the North Market.
doc2 0.032 Cats are killers. They kill billions of animals a year.
doc1 0.030 Stray cats are running all over the place. I see 10 a day!
doc7 0.000 I have nothing to say. In summary, I have told you nothing.
Our “best” document, at least in an intuitive sense, comes out ahead with a score nearly twice as high as its nearest competitor. The second highest ranked document is still about cat food, and the profoundly uninformative document 7 has been ranked dead last.
Discussion
Though tfidf weighting and the vector space model may now be old school, its core concepts are still used in industrial search solutions built using Lucene. In more modern (and statistical) approaches based on probabilistic language modeling, documents are ranked by the probability that their underlying language model produced the query [6]. While there's nothing inherently statistical about the vector space model, a link to probabilistic language modeling has been demonstrated [7].
I hope you've enjoyed exploring the tm package and implementing classic ideas from the information retrieval.
Acknowledgements
The markdown [8] and knitr [9] packages, in conjunction with RStudio's IDE [10], were used to create this document. Thanks to Chris Nicholas and Shannon Terry for their comments and feedback. I first learned about information retrieval in Coursera's Stanford Natural Language Processing course taught by Dan Jurafsky and Christopher Manning. Keep up with ours and other great articles on R-Bloggers .
References
Ingo Feinerer and Kurt Hornik (2013). tm: Text Mining Package. R package version 0.5-8.3. http://CRAN.R-project.org/package=tm
Ingo Feinerer, Kurt Hornik, and David Meyer (2008). Text Mining Infrastructure in R. Journal of Statistical Software 25(5): 1-54. URL: http://www.jstatsoft.org/v25/i05/.
Kurt Hornik (2013). Snowball: Snowball Stemmers. R package version 0.0-8. http://CRAN.R-project.org/package=Snowball
Kurt Hornik, David Meyer and Christian Buchta (2013). slam: Sparse Lightweight Arrays and Matrices. R package version 0.1-28. http://CRAN.R-project.org/package=slam
Christopher D. Manning, Prabhakar Raghavan and Hinrich Schutze, Introduction to Information Retrieval, Cambridge University Press. 2008. URL: http://www-nlp.stanford.edu/IR-book/
Hugo Zaragoza, Djoerd Hiemstra, and Michael Tipping. “Bayesian extension to the language model for ad hoc information retrieval.” Proceedings of the 26th annual international ACM SIGIR conference on Research and development in information retrieval. ACM, 2003. URL
Thorsten Joachims. A Probabilistic Analysis of the Rocchio Algorithm with TFIDF for Text Categorization. No. CMU-CS-96-118. Carnegie-Mellon University of Pittsburgh, PA. Department of Computer Science, 1996.
JJ Allaire, Jeffrey Horner, Vicent Marti and Natacha Porte (2012). markdown: Markdown rendering for R. R package version 0.5.3. http://CRAN.R-project.org/package=markdown
Yihui Xie (2012). knitr: A general-purpose package for dynamic report generation in R. R package version 0.6. http://CRAN.R-project.org/package=knitr
RStudio IDE for Windows. URL http://www.rstudio.com/ide/
R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL: http://www.R-project.org/.
