Ted Pedersen

combig.pl

SYNOPSIS

Combines (sums) the frequency counts of bigrams made up of the same pair of words in either possible order. It will count the number of time two words occur together in a bigram regardless of which one comes first.

DESCRIPTION

USAGE

`` combig.pl [OPTIONS] BIGRAM``

INPUT PARAMETERS

• BIGRAM

Specify a file of bigram counts created by NSP programs count.pl. The entries in BIGRAM will be formatted as follows:

``        word1<>word2<>n11 n1p np1  ``

Here, word1 is followed by word2 n11 times. word1 occurs as the 1st word in total n1p bigrams and word2 occurs as the 2nd word in np1 bigrams.

• OPTIONS

`` --help``

Displays this message.

`` --version``

Displays the version information.

OUTPUT

combig.pl produces a count of the number of times two words make up a bigram in either order, whereas count.pl produces counts for a single fixed ordering. In other words, combig.pl combines the counts of bigrams that are composed of the same words but in reverse order. While the BIGRAM shows pairs of words forming bigrams, output of combig will show the pairs of words that are co-occurrences or that co-occur irrespective of their order.

e.g. if bigrams

``````        word1<>word2<>n11 n1p np1
and
word2<>word1<>m11 m1p mp1 ``````

are found in BIGRAM file, then combig.pl treats these as a single unordered bigram

``        word1<>word2<>n11+m11 n1p+mp1 np1+m1p``

where the new bigram will show a combined contingency table in which the order of words doesn't matter.

``````                        word2           ~word2
___________________________________________________________
word1  |       n11+m11         n12+m21     | n1p+mp1
|                                   |
~word1  |       n21+m12         n22+m22-n   | n2p+mp2-n
___________________________________________________
np1+m1p         np2+m2p-n   |  n
here the entry ``````
• (word1,word2)=n11+m11

shows the number of bigrams having both word1 and word2 in either order

i.e. word1<>word2 + word2<>word1

• (word1,~word2)=n12+m21

shows the number of bigrams having word1 but not word2 at either position

i.e. word1<>~word2 + ~word2<>word1

• (~word1,word2)=n21+m12

shows the number of bigrams having word2 but not word1 at either position

i.e. ~word1<>word2 + word2<>~word1

• (~word1,~word2)=n22+m22

shows the number of bigrams not having word1 nor word2 at either position

i.e. ~word1<>~word2 + ~word2<>~word1 - n

where n=total number of bigrams

The mathematical proof of how the cell counts in the above contingency table are counted is explained in section Proof.

When a bigram appears in only one order i.e.

word1<>word2<>n11 n1p np1

appears but

word2<>word1<>m11 m1p mp1

does not, then the combined bigram will be same as the original bigram that appears. Or in other words,

word1<>word2<>n11 n1p np1

is displayed as it is.

PROOF OF CORRECTNESS

A bigram word1<>word2<>n11 n1p np1 represents a contingency table

``````                  word2         ~word2
--------------------------------------
word1   n11     |       n12     |       n1p
|               |
~word1  n21     |       n22     |       n2p
--------------------------------------
np1     |       np2     |       n``````

while a bigram word2<>word1<>m11 m1p mp1 represents a contingency table

``````                  word1         ~word1
--------------------------------------
word2   m11     |       m12     |       m1p
|               |
~word2  m21     |       m22     |       m2p
--------------------------------------
mp1     |       mp2     |       n
``````

Here,

`` n11+n12+n21+n22 = n ``

Also,

`` m11+m12+m21+m22 = n ``

combig.pl combines bigram counts into a single order independant word pair

`` word1<>word2<>n11+m11 n12+m21 n21+m12 ``

And the corresponding contingency table will be shown as

``````                        word2           ~word2
-----------------------------------------
word1   n11+m11   |     n12+m21   |     n1p+mp1
|               |
~word1  n21+m12   |     n22+m22-n |     n2p+mp2
-----------------------------------------
np1+m1p   |     np2+m2p   |     n
``````

The first cell (n11+m11) shows the #bigrams having word1 and word2 (irrespective of their positions) i.e. word1<>word2 or word2<>word1 which is n11+m11.

The second cell (n12+m21) shows the #bigrams having word1 but not word2 at any position i.e. word1<>~word2 or ~word2<>word1 which is n12+m21.

The third cell (n21+m12) shows the #bigrams having word2 but not word1 at any position i.e. ~word1<>word2 or word2<>~word1 which is n21+m12.

The fourth cell (m22+n22-n) shows the #bigrams not having word1 nor word2 at any position which

`````` = n - (n11+m11) - (n12+m21) - (n21+m12)

= n - (n11+n12+n21) - (m11+m12+m21)

= n - (n-n22) - (n-m22)

= n22 + m22 - n ``````

Alternative proof -

`````` n22 = m11 + m12 + m21 + X      (a)

m22 = n11 + n12 + n21 + X      (b)``````

where X = #bigrams not having either word1 or word2.

as both n22 and m22 have some terms in common which show the bigrams not having either word1 or word2. But,

`` m11+m12+m21 = n - m22 ``

Substituting this in eqn (a)

`` n22 = n - m22 + X ``

Or

`` X = n22 + m22 - n ``

Or add (a) and (b) to get

`` n22+m22 = (n11+m11) + (n12+m21) + (n21+m12) + 2X ``

rearranging terms,

`` n22+m22 = (n11+n12+n21) + (m11+m12+m21) + 2X``

but

`````` n11+n12+n21 = n - n22 and

m11+m12+m21 = n - m22 ``````

Hence,

`````` n22+m22 = (n-n22) + (n-m22) + 2X

2(n22+m22-n) = 2X``````

Or

`` (n22+m22-n) = X``

which is the fourth cell count.

Viewing Bigrams as Graphs

In bigrams, the order of words is important. Bigram word1<>word2 shows that word2 follows word1. Bigrams can be viewed as a directed graph where a bigram word1<>word2 will represent a directed edge e from initial vertex word1 to terminal vertex word2(word1->word2).

In this case,

n11, which is the number of times bigram word1<>word2 occurs, becomes the weight of the directed edge word1->word2.

n1p, which is the number of bigrams having word1 at 1st position, becomes the out degree of vertex word1

and

np1, which is the number of bigrams having word2 at 2nd position, becomes the in degree of vertex word2

combig.pl creates a new list of word pairs from these bigrams such that the order of words can be ignored. Viewed another way, it converts the directed graph of given bigrams to an undirected graph showing new word pairs.

A pair say

``        word1<>word2<>n11 n1p np1 ``

shown in the output of combig can be viewed as an undirected edge joining word1 and word2 having weight n11. If we count the degree of vertex word1 it will be n1p and degree of vertex word2 will be np1.

AUTHORS

`````` Amruta Purandare, pura0010@d.umn.edu
Ted Pedersen, tpederse@d.umn.edu

This work has been partially supported by a National Science Foundation Faculty Early CAREER Development award (#0092784).