Bigram Counts and Association Statistics

Description: In this exercise, we apply basic counts and some association statistics to a small corpus. We will:

Count unigrams
Count bigrams
Compute mutual information for bigrams
Compute likelihood ratios for bigrams

Turn in: written answers to all questions marked QUESTION.

Credits: The scripts used in this exercise, written by Philip Resnik, were originally derived from Ken Church's "NGRAMS" tutorial at ACL-1995. The likelihood ratio code was adapted from code written by Ted Dunning.

Prerequisites: This exercise assumes basic familiarity with typical Unix commands, and the ability to create text files (e.g. using a text editor such as vi or emacs). No programming is required.

Platforms:The executable files run under the Solaris operating system. C programs can generally be recompiled on different platforms (e.g. Linux) using cc -o filename filename.c. On most Unix platforms you can execute uname -sr to see what operating system you're running.

Notational Convention: The symbols <== will be used to identify a comment from the instructor, on lines where you're typing something in. So, for example, in

    
    %  cp file1.txt file2.txt   <== The "cp" is short for "copy"

what you're supposed to type at the prompt (identified by the percent sign, here) is

    cp file1.txt file2.txt

followed by a carriage return.

Getting the code

Log in to your course account. If this is your first time logging in, CHANGE YOUR PASSWORD:
```
  %  passwd
```
(Note that the prompt % may be different on your system; for example, another frequently seen prompt is [thismachine], where "thismachine" is the name of your workstation, or simply ">".)

Create a directory called stats and go into it.

  % mkdir stats               <== Create a subdirectory called "stats"
  % cd stats                  <== Go into that directory

Download and save ngrams.tar. Then extract the files by executing:
```
  % tar xvf ngrams.tar       <== Extract code from the file
  % rm ngrams.tar            <== Delete to conserve space
  % cd ngrams                                 <== Go into ngrams directory
  % chmod u+x *.pl            <== Make perl scripts executable
```
Generating Statistics for a Corpus
1. Take a look at file corpora/GEN.EN. You can do this as follows:
```
  %  more corpora/GEN.EN
```
  (Type spacebar for more pages, and "q" for "quit".) This contains an annotated version of the book of Genesis, King James version. It is a small corpus, by current standards -- somewhere on the order of 40,000 or 50,000 words. QUESTION: What words (unigrams) would you expect to have high frequency in this corpus? What bigrams do you think might be frequent?
2. Create a subdirectory called genesis to contain the files with statistics generated from this corpus:
```
  %  mkdir genesis
```
  Then run the Stats program to analyze the corpus. The program requires an input file, and a "prefix" to be used in creating output files. The input file will be corpora/GEN.EN, and the prefix will be genesis/out, so that output files will be created in the genesis subdirectory. That is, you should execute the following:
```
  %  Stats corpora/GEN.EN genesis/out
```
  The program will tell you what it's doing, as it counts unigrams, counts bigrams, computes mutual information, and computes likelihood ratio statistics. Depending on the machine you're working on, this may take differing amount of time to run, but it should be less than 5 minutes for most machines.
3. If you're on a slow machine, or don't care about doing your own computing, you can skip the above step (or hit control-C to cancel it), and download the results: ngram_out.tar. Then execute:
```
  % tar xvf ngram_out.tar
```
4. Whether you computed yourself or downloaded the results, you should now have a subdirectory called genesis containing a bunch of files that begin with out. (Note that the downloaded results are from an older version of the scripts and therefore the numbers will not be identical.)
Examining Unigram and Bigram Counts
1. Go into directory genesis.
```
  %  cd genesis
```
2. Look at file out.unigrams:
```
  %  more out.unigrams
```
  Seeing the vocabulary in alphabetical order isn't very useful, so let's sort the file by the unigram frequency, from highest to lowest:
```
  %  sort -nr out.unigrams > out.unigrams.sorted
  %  more out.unigrams.sorted
```
  Now examine out.unigrams.sorted. Note that v (verse), c (chapter), id, and GEN are part of the markup in file GEN.EN, for identifying verse boundaries. QUESTION: Other than those (which are a good example of why we need better pre-processing to handle markup), are the high frequency words what you would expect?
3. Analogously, look at the bigram counts out.bigrams:
```
  %  sort -nr out.bigrams > out.bigrams.sorted
  %  more out.bigrams.sorted
```
  - QUESTION: Markup aside, again, are the high frequency bigrams what you would expect?
Getting Quantitative with Mutual Information
1. Now let's look at mutual information. File out.mi contains bigrams sorted by mutual information value. Each line contains:
  1. I(wordX,wordY)
  2. freq(wordX)
  3. freq(wordY)
  4. freq(wordX,wordY)
  5. wordX
  6. wordY
  Low-frequency bigrams (bigram count less than 5) were excluded.
  As an exercise, compute mutual information by hand for the first bigram on the list, "savoury meat". Recall that
```
   I(x,y)  =  log2 [p(x,y)/(p(x)p(y))]
```
  and that the simplest estimates of probabilities, the maximum likelihood estimates, are given by
```
  p(x)   = freq(x)/N
  p(y)   = freq(y)/N
  p(x,y) = freq(x,y)/N
```
  where N is the number of observed words in the corpus. You can find out this value by counting the words in file out.words; it should match what you should get by summing the frequencies in either out.unigrams or out.bigrams.
```
  % wc -l out.words
```
  You can get a calculator on your screen on some systems (at least sunos and solaris) by executing:
```
  %  xcalc &
```
  Here's a sequence you can use to do the calculation:
```
  Compute p(savoury)         = freq(savoury)/N
  Compute p(meat)            = freq(meat)/N
  Compute p(savoury meat)    = freq(savoury,meat)/N
  Compute p(savoury)p(meat)  = p(savoury) * p(meat)
  Divide p(savoury,meat) by this value
  Take the log of the result (which in xcalc is log to the base 10)
  Convert that result to log base-2 by dividing by 0.30103 
    This uses the fact that for all M, N: logM(x) = logN(x)/logN(M).
```
  At some point, the calculator may give you scientific notation for a number. If you need to enter a number in scientific notation, you use EE:
```
   EE         Used for entering exponential numbers.  For example
              to get "-2.3E-4" you'd enter "2 . 3 +/- EE 4 +/-".   
```
  The number you some up with should be close to the mutual information reported in out.mi. It may be slightly different because your calculation used different precision than the program's.
2. As you've just seen, probabilities can be very low numbers. All the more so when using n-grams for n=3 or above! Underflow can be a problem in these sorts of calculations: when the probabilities are too low, they exceed the representational capacity of the computer. For this reason it's very common to do such calculations using the logs of the probability values (often called "log probabilities" or "logprobs"), using the following handy identities:
```
  log(a * b) = log(a) + log(b)
  log(a / b) = log(a) - log(b)
```
  Try converting the formula for mutual information using these identities so that probabilities are never multiplied or divided, before reading further.
  Solution: log[p(x,y)/p(x)p(y)] = log p(x,y) - log p(x) - log p(y)
  To really get a feel for things, first first substitute the maximum likelihood estimates in and then convert to using log probabilities, i.e.
  log[ (freq(x,y)/N)/(freq(x)/N)(freq(y)/N) ]
Examining the Results
1. Look at out.mi and the bigrams selected by mutual information as being strongly associated. QUESTION: What do you think of them? Notice how very many of them are low-frequency bigrams: it's well known that mutual information has overly high values for bigrams of low frequency, i.e. it reports word pairs as associated when they probably are not really that strongly associated after all.
2. Compare this to out.lr, where the leftmost column is the likelihood ratio. There are a lot of common words of English in there, so try filtering those out using the filter_stopwords program. First, access the program so it's easy to run in this directory:
```
  %  ln -s ../filter_stopwords      <== Creates a symbolic link
  %  ln -s ../stop.wrd              <== Creates a symbolic link
```
  Then run it:
```
  %  filter_stopwords stop.wrd < out.lr > out.lr.filtered
```
  If you see a lot of instances of GEN, you might want to also filter it out, e.g.
```
  %  filter_stopwords stop.wrd < out.lr | fgrep -v GEN > out.lr.filtered
```
  QUESTION: How does out.lr.filtered look as a file containing bigrams that are characteristic of this corpus?
Time for Fun
1. One thing you may have noticed is that there's more data sparseness because uppercase and lowercase are distinct, e.g. "Door" is treated as a different word from "door". In the corpora directory, you can create an all- lowercase version of GEN.EN by doing this:
```
  %   cat GEN.EN | tr "A-Z" "a-z" > GEN.EN.lc
```
  To save disk space, assuming you're done with GEN.EN, delete the original:
```
  %   rm GEN.EN
```
  QUESTION: Try re-doing the exercise with this version. What, if anything, changes?
2. Ok, perhaps that last one wasn't exactly fun. But this probably will be. Go into your corpora subdirectory. Then download one or more of adventures.dyl, hound.dyl, study.dyl. These are all Sherlock Holmes stories: A Study in Scarlet, The Hound of the Baskervilles, or Adventures of Sherlock Holmes, the last of which contains 12 different stories.
  Now get back into your stats directory, create an output directory, say, holmes1, and run the Stats program for the file of interest, e.g.:
```
  %   cd ..
  %   mkdir holmes1
  %   Stats corpora/study.dyl holmes1/out
  %   cd holmes1
```
  Or perhaps convert to lowercase before running Stats:
```
  %   cd corpora
  %   cat study.dyl | tr "A-Z" "a-z" > study.lc
  %   rm study.dyl
  %   cat hound.dyl | tr "A-Z" "a-z" > hound.lc
  %   rm hound.dyl
  %   cat adventures.dyl | tr "A-Z" "a-z" > adventures.lc
  %   rm adventures.dyl
  %   cd ..
```
  Look at out.lr, etc. for this corpus. Now go through the same process again, but creating a directory holmes2 and using a different file. QUESTION: Same author, same main character, same genre... how do the high-association bigrams compare between the two cases? If you use filter_stopwords, how do the results look -- what kinds of bigrams are you getting? What natural language processing problems might this be useful for?
When You are Done, We Need to Recover Disk Space
Corpora and data take up a lot of disk space. When you are done, PLEASE delete the output directories you have created, and even the corpus directory itself if you no longer need it. For example, if you are in your stats directory, you can type:
% /bin/rm -rf corpora genesis holmes1 holmes2 holmes3
to delete the entire directories. Your housekeeping will be much appreciated.

Bigram Counts and Association Statistics

Getting the code

Generating Statistics for a Corpus

Examining Unigram and Bigram Counts

Getting Quantitative with Mutual Information

Examining the Results

Time for Fun

When You are Done, We Need to Recover Disk Space