Friday, March 30, 2012

NTA7:Dimensional Sparseness Or What Tokens To Use?

NTA7: Dimensional Sparseness Or What Tokens To Use?

Guide to all Numerical Text Analysis posts

 Word histogram and Zipf plot of corpus for identifying stop words

  • lowercasing,
  • depunctuating,
  • removing single atom words,
  • removing single occurrence tokens,
  • stemming and removing a few stop words (whether the 125 I have chosen or the ~ 450 in standard lists,
we are left with about 14000 tokens from a bit less than 100 documents.

Here is the Zipf plot:

Before going on to the Cosine Similarity business, let's think about Word-vector space. Each token represents a dimension, the frequency of occurrence is the coordinate. Each document represents a point in this space. So what we have is a 14,000 dimensional space for a 100 points. That's ridiculous! But let's consider we have a million documents, that is still only 70 points per dimension!

Now of course this is going to vary from case to case, e.g. the categorization problem I worked on had 25,000 points in 50,000 dimensions. Hardly a situation where standard clustering algorithms – most of which are designed for denser graphs (at least dimensionally speaking) can be expected to apply. My solution to this is on SlideShare, and I'll blob on it at length later.

My point, as was my point for selecting stop words, is that some [statistical or numerical] and purpose-based criteria have to be used to decide which set of tokens to use. If you look at the Zipf plot above, in my view there are clearly three families of tokens:
  • the high ranking ones that approximate a Zipf line with a coefficient marginally less than 1

  • the mid ranking tokens at the knee of the plot which fall off the Zipf line

  • the low ranking tokens whose power law deviates very strongly from 1

My choices for the cutoffs are somewhat arbitrary, but unimportant for now. Summarizing,

What do these mean? Consider the extremely rare ones that occur only once in the corpus, and hence in only one document. So they are extremely powerful for distinguishing one document from another and a reasonably small set of these plus a few that occur a few times (There may be a few documents which contain no singly occurring tokens.) will be sufficient to label every document uniquely. However, precisely because of their rarity, we can't use them for finding any similarities between texts. So these low ranking tokens lead to a very fine-graining of the set of documents. Rejecting these will have the advantage of reducing the dimensionality of word space by an order of magnitude.

What about the high-ranking ones? They suffer from the converse problem, they occur so often in so many texts that they are likely to not be able to distinguish well between texts at all, which is sort of the motivation for throwing out the stop words in the first place. (Though my suspicion is that stop words are, or should be, chosen based on the smallness of the standard deviation of their relative frequencies.) So short of calculating these frequency standard deviations for the tokens, which I hope to get to soon, we should reject these stochastic tokens, the ones that closely follow Zipf's law (or Mandelbrot's modification). Rejecting these will not reduce the dimensionality of word-vector space by much, but it will reduce the radial distance of the document-points from the origin, the de facto center for the Cosine Similarity.

So we can use only 1500 mid-ranking tokens (that is “only” 15 dimensions per point, which is better than 144 dimensions per point) for the next steps.

Last point: before calculating Cosine Similarities, a standard step is to scale all the token - frequencies by the log of the document frequency for the token. This is a flat metric on wordspace. This will have the effect of bringing down the weight at the high-ranking end, and since log(f) increases slower than f, not scaling down as much the low ranking end. So the overall weights of the tokens may show precisely the bump in the mid-range that we want. So another point for comparison.

So many choices, so little time.

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