Is this interpretation of sparsity accurate?

Question

According the documentation of the removeSparseTerms function from the tm package, this is what sparsity entails:

A term-document matrix where those terms from x are removed which have at least a sparse percentage of empty (i.e., terms occurring 0 times in a document) elements. I.e., the resulting matrix contains only terms with a sparse factor of less than sparse.

So, is a correct interpretation of this to say if sparse is equal to .99, then we are removing terms that only appear in at most 1% of the data?

Answer 1

Yes, although your confusion here is understandable, since the term "sparsity" is hard to define clearly in this context.

In the sense of the sparse argument to removeSparseTerms(), sparsity refers to the threshold of relative document frequency for a term, above which the term will be removed. Relative document frequency here means a proportion. As the help page for the command states (although not very clearly), sparsity is smaller as it approaches 1.0. (Note that sparsity cannot take values of 0 or 1.0, only values in between.)

So your interpretation is correct in that sparse = 0.99 will remove only terms that are more sparse than 0.99. The exact interpretation for sparse = 0.99 is that for term $j$, you will retain all terms for which $df_j > N * (1 - 0.99)$, where $N$ is the number of documents -- in this case probably all terms will be retained (see example below).

Near the other extreme, if sparse = .01, then only terms that appear in (nearly) every document will be retained. (Of course this depends on the number of terms and the number of documents, and in natural language, common words like "the" are likely to occur in every document and hence never be "sparse".)

An example of the sparsity threshold of 0.99, where a term that occurs at most in (first example) less than 0.01 documents, and (second example) just over 0.01 documents:

> # second term occurs in just 1 of 101 documents
> myTdm1 <- as.DocumentTermMatrix(slam::as.simple_triplet_matrix(matrix(c(rep(1, 101), rep(1,1), rep(0, 100)), ncol=2)), 
+                                weighting = weightTf)
> removeSparseTerms(myTdm1, .99)
<<DocumentTermMatrix (documents: 101, terms: 1)>>
Non-/sparse entries: 101/0
Sparsity           : 0%
Maximal term length: 2
Weighting          : term frequency (tf)
> 
> # second term occurs in 2 of 101 documents
> myTdm2 <- as.DocumentTermMatrix(slam::as.simple_triplet_matrix(matrix(c(rep(1, 101), rep(1,2), rep(0, 99)), ncol=2)), 
+                                weighting = weightTf)
> removeSparseTerms(myTdm2, .99)
<<DocumentTermMatrix (documents: 101, terms: 2)>>
Non-/sparse entries: 103/99
Sparsity           : 49%
Maximal term length: 2
Weighting          : term frequency (tf)

Here are a few additional examples with actual text and terms:

> myText <- c("the quick brown furry fox jumped over a second furry brown fox",
              "the sparse brown furry matrix",
              "the quick matrix")

> require(tm)
> myVCorpus <- VCorpus(VectorSource(myText))
> myTdm <- DocumentTermMatrix(myVCorpus)
> as.matrix(myTdm)
    Terms
Docs brown fox furry jumped matrix over quick second sparse the
   1     2   2     2      1      0    1     1      1      0   1
   2     1   0     1      0      1    0     0      0      1   1
   3     0   0     0      0      1    0     1      0      0   1
> as.matrix(removeSparseTerms(myTdm, .01))
    Terms
Docs the
   1   1
   2   1
   3   1
> as.matrix(removeSparseTerms(myTdm, .99))
    Terms
Docs brown fox furry jumped matrix over quick second sparse the
   1     2   2     2      1      0    1     1      1      0   1
   2     1   0     1      0      1    0     0      0      1   1
   3     0   0     0      0      1    0     1      0      0   1
> as.matrix(removeSparseTerms(myTdm, .5))
    Terms
Docs brown furry matrix quick the
   1     2     2      0     1   1
   2     1     1      1     0   1
   3     0     0      1     1   1

In the last example with sparse = 0.34, only terms occurring in two-thirds of the documents were retained.

An alternative approach for trimming terms from document-term matrixes based on a document frequency is the text analysis package quanteda. The same functionality here refers not to sparsity but rather directly to the document frequency of terms (as in tf-idf).

> require(quanteda)
> myDfm <- dfm(myText, verbose = FALSE)
> docfreq(myDfm)
     a  brown    fox  furry jumped matrix   over  quick second sparse    the 
     1      2      1      2      1      2      1      2      1      1      3 
> trim(myDfm, minDoc = 2)
Features occurring in fewer than 2 documents: 6 
Document-feature matrix of: 3 documents, 5 features.
3 x 5 sparse Matrix of class "dfmSparse"
       features
docs    brown furry the matrix quick
  text1     2     2   1      0     1
  text2     1     1   1      1     0
  text3     0     0   1      1     1

This usage seems much more straightforward to me.