opencog-learn / learn-lang-diary / learn-lang-diary-part-three.lyx

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\begin_layout Title
Language Learning Diary - Part Three
\end_layout

\begin_layout Date
Sept 2021
\end_layout

\begin_layout Author
Linas Vepstas
\end_layout

\begin_layout Abstract
The language-learning effort involves research and software development
 to implement the ideas concerning unsupervised learning of grammar, syntax
 and semantics from corpora.
 This document contains supplementary notes and a loosely-organized semi-chronol
ogical diary of results.
 The notes here might not always makes sense; they are a short-hand for
 my own benefit, rather than aimed at you, dear reader!
\end_layout

\begin_layout Section*
Introduction
\end_layout

\begin_layout Standard
Part Three of the diary on the language-learning effort continues work on
 the English dataset.
 This part is concluded, the next part is Part Four.
\end_layout

\begin_layout Section*
Summary Conclusions
\end_layout

\begin_layout Standard
A short summary of what is found here:
\end_layout

\begin_layout Itemize
Trimming the word-disjunct dataset to remove words or disjuncts or word-disjunct
 pairs observed only once is a good thing, and reduces dataset size by 90%
 or more.
 Trimming more than this is a bad thing.
\end_layout

\begin_layout Itemize
Part of the above result may be due to a bug in the word-segmentation code:
 leading/trailing double-quotes are not split from words.
 The processing code then tries to escape these with backslashes, and repeated
 processing creates a shower of backslashes.
 This appears to pointless inflate the number of words in a dataset.
 This bug needs to be fixed.
\end_layout

\begin_layout Itemize
It is not known if trimming word-pair files to remove low-count word-pairs
 is a good idea or not.
 This will require new (and time-consuming) experiments.
 Some trimming is surely good, but how much?
\end_layout

\begin_layout Itemize
The mutual information between word-vectors is given by a Gaussian distribution.
 This was seen before, but is confirmed again here.
 Recall that a word-vector consists of a word, and the vector basis elements
 are disjuncts.
 These word-vectors are reminiscent of SkipGrams, but are constructed differentl
y.
 I don't know of any theoretical work that could explain this experimental
 result (or any kind of result, for that matter).
\end_layout

\begin_layout Itemize
The self-MI of a word-vector against itself appears to be given by a weak
 log-normal distribution.
 It is almost but not quite uniformly distributed, ranging from about 2
 and gradually tapering to 30 for the dataset explored here.
 An easy theoretical argument can be made that the self-MI must be positive,
 but that's all.
 I don't know of any theoretical work that could explain this experimental
 result.
\end_layout

\begin_layout Itemize
Other ways of comparing word vectors, including basic Jaccard (vector overlap)
 distance, and a conditional-Jaccard distance do correlate in general with
 the MI.
 An exploration of these alternate ways of exploring word similarities was
 driven by work reported in the Diary part Two.
 Here, nothing unusual stands out.
 The distribution of these appears to be given by a log-normal curve, or
 possible a log logistic curve, but the data is not refined enough to tell.
 There is no theoretical basis to expect any of this.
 
\end_layout

\begin_layout Itemize
Examining the top-100 most similar word pairs by hand (by gut feel) suggests
 that all but one or two word-pair recommendations are excellent.
 This visual exam also seems to suggest that maybe MI provides better recommenda
tions than the other metrics (contradicting earlier results).
 Certainly, computing MI is much faster.
\end_layout

\begin_layout Itemize
Both of the Jaccard measures have a self-distance score of exactly zero.
 By contrast, the self-MI is not zero, and it suggests that there it can
 be used as a natural scale to help determine when words should be classified
 together.
 That is, a low self-MI word is similar to everything, and maybe should
 not be merged with high self-MI words.
 Or perhaps it is an indicator of a word with many distinct meanings.
 I don't understand this yet, and it might be a fruitful topic to examine.
\end_layout

\begin_layout Itemize
In order to make a determination to merge a pair of words, one must look
 at both their similarity, and their frequency of occurrence.
 One wishes to merge words that are similar and are also frequently-occurring.
 To this end, the text proposes that the similarity should be combined with
 the average log-frequency (log-probability) of the two words.
 Combining this idea with the formula for MI gives the explicit formula
\begin_inset Formula 
\[
\mbox{common}MI\left(w,u\right)=\log_{2}\frac{p\left(w,u\right)}{\sqrt{p\left(w\right)p\left(u\right)}}
\]

\end_inset

The square-root arises from the factor of 1/2 that arises from averaging
 two things together.
 The square-root makes this formula look unusual, but it seems to provide
 an excellent way of ranking word-pairs for merging.
 The distribution again follows the normal Gaussian Bell curve.
 As mentioned, the top 100 suggestions for merge candidates look very healthy.
 
\end_layout

\begin_layout Standard
The diary Part Four will examine a merge in gruesome detail.
\end_layout

\begin_layout Section*
September 2021
\end_layout

\begin_layout Standard
After the previous setback with grammatical clustering, it seems like more
 exploratory work is in order.
 Part of the exploratory effort is to discover faster ways of finding similar
 words, and to find algorithms that can do incremental or continuous learning.
 Some questions along this path:
\end_layout

\begin_layout Itemize
What is the distribution of disjuncts? (Yes, we've done this before, but
 lets do it again for the latest 
\begin_inset Quotes eld
\end_inset

clean
\begin_inset Quotes erd
\end_inset

 dataset.) Create a graph showing rank vs number of times a disjunct has
 been observed.
\end_layout

\begin_layout Itemize
How are disjuncts shared? Create a graph showing the rank vs.
 the number of words that share this disjunct.
\end_layout

\begin_layout Itemize
Scatterplots: ...
\end_layout

\begin_layout Subsection*
Expt-4 – Trimming – Sept 2021
\end_layout

\begin_layout Standard
The above questions will be aimed at a specific dataset.
 The initial choice of dataset was `
\family typewriter
run-1-en_mpg-tranche-123.rdb
\family default
`, as described earlier in the diary.
 The intent is to explore two versions: the full dataset, and a lightly
 trimmed dataset.
\end_layout

\begin_layout Standard
What is explored here (in expt-4) is how the size of the dataset varies
 with the amount of trimming.
 Trimming is desirable, since it can be used to manage dataset sizes.
 Computations run faster on smaller datasets.
\end_layout

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The outcome of trimming proves to be surprising: even 
\begin_inset Quotes eld
\end_inset

mild
\begin_inset Quotes erd
\end_inset

 trimming removes large parts of the word–disjunct datasets.
 It removes much more than one might naively guess from a Zipfian distribution.
 The underlying reason for this is confusing: it turns out that a bug in
 tokenization may be ruining the distribution.
 The tokenizer does not split double-quotes from the beginning/ending of
 words, and later stages add backslashes to escape the quotes.
 A waterfall of quotes results, polluting the dataset with oodles of backslashes.
 These are observed infrequently, and how they interact with the quality
 of the datasets is unclear.
\end_layout

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Coupled with the earlier observation that even mild trimming damages the
 data (as seen in expt-3, at the end of diary Part Two), this suggests that
 trimming should be kept to a minimum.
 By the conclusion, it emerges that perhaps the ideal trim is one that removes
 all words, disjuncts and word–disjunct pairs that are observed only once.
\end_layout

\begin_layout Standard
\begin_inset Quotes eld
\end_inset

Trimming
\begin_inset Quotes erd
\end_inset

 means removing all words, disjuncts and word–disjunct pairs with observation
 counts below some given thresholds.
 The thresholds for these three counts can be set independently.
 These are the three numbers below: e.g.
 10-4-2 means 
\begin_inset Quotes eld
\end_inset

discard all words with an observation count of 10 or less, all disjuncts
 with a count of 4 or less, and all word–disjunct pairs with a count of
 2 or less.
\begin_inset Quotes erd
\end_inset


\end_layout

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The trimming process is funny: trimming (as implemented) is not idempotent
 (but it should be).
 Basically, the linkage filter, which guarantees that connectors have correspond
ing word entries, knocks out disjuncts and so pushes words below the cut-off.
 Repeating this process then knocks out more connectors.
 It needs to be run until its stable.
 This is CPU consuming.
\end_layout

\begin_layout Standard
The table below shows the result of repeated trimming, interleaved with
 a linkage filter to remove disjuncts with connectors in them that do not
 have words any more.
 Each was written to a distinct file.
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So that took an alarmingly large number of iterations to stabilize.
 It also trimmed the dataset extremely sharply.
 Alarmingly sharply.
 Lets try again.
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Same idea, modified algo: here, instead of filtering, just delete outright.
 This will shrink the AtomSpace as we go along, avoiding some of the hassle
 with filtering.
 Hopefully it will run faster, too.
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xxx-trim-1-1-1-pass-1
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r4-trim-1-1-1-pass-1
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r4-trim-1-1-1-pass-3
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r4-trim-1-1-1-pass-6
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Notes:
\end_layout

\begin_layout Itemize
The `
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
xxx-trim-1-1-1-pass-1` version is not a file, but is rather the number of
 WordNodes, ConnectorSeqs and Sections in the AtomSpace, after removing
 those with only one observation count.
 This is not the same as the matrix dimensions, because the matrix is built
 by finding all Sections, and then taking the Words and ConnectorSeqs in
 those Sections.
 There are fewer of those, because some disjuncts have connectors to words
 that have been removed.
\end_layout

\begin_layout Itemize
The `
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
r4-1-1-1-pass-1` version removes all words, disjuncts and sections that
 have one observation count.
 This is followed by removing all disjuncts and sections that do not have
 words as the germs of the sections.
 This results in words that don't appear anywhere; these are removed.
\end_layout

\begin_layout Itemize
The `
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
r4-1-1-1-pass-n` versions repeat this process, until it terminates.
 Iterations until termination are the same as for the earlier cut.
\end_layout

\begin_layout Standard
It is quite remarkable how much the dataset shrinks, simply by removing
 sections and words with just one count! This suggests several activities:
\end_layout

\begin_layout Itemize
Plot the distribution of words, disjuncts and sections and verify that they
 are Zipfian.
 I thought that the Zipfian distribution implied that half of the dataset
 would have a count of one, and not 5/6th or 9/10th's of it.
\end_layout

\begin_layout Itemize
Take a look at how trimming affects the other datasets.
\end_layout

\begin_layout Itemize
Introduce a periodic trim step into the processing pipeline.
 The hope is that this could dramatically shrink the size of the datasets,
 with very little loss of information.
\end_layout

\begin_layout Standard
Let's look at the previous graphs of the Zipfian distributions.
 These can be found in `
\family typewriter
connector-sets-revised.pdf
\family default
` page 9ff.
 For words, the graph on page 9, left hand side shows that the count drops
 precipitously as a function of the rank.
 The logarithmic scale is deceiving; it looks like more than half of words
 have a count of only 1.
 The right-hand graph suggests the same for disjuncts.
 Likewise for the graphs on page 11 and 12.
 Thus, the present case is not new: this was seen in earlier datasets.
\end_layout

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Lets try some deeper trimming.
 What happens then? Shown in table below.
\end_layout

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r4-trim-1-1-1-pass-6
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r4-trim-5-2-2
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r4-trim-10-4-2
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There is a very dramatic drop-off in the number of vocabulary words, as
 trimming is increased.
 Each of the later trims was created by starting with the earlier version,
 and trimming that.
 This history dependence should not matter.
 Looks like the earlier r4-trim-10-4-2 result is reproducible.
\end_layout

\begin_layout Subsection*
Single-count trim
\end_layout

\begin_layout Standard
Based on general results, it seems that a trim of removing all words, disjuncts
 and word–disjunct pairs with an observation count of one is a good thing.
 Based on this, some 
\begin_inset Quotes eld
\end_inset

master
\begin_inset Quotes erd
\end_inset

 trimmed files are prepared, stored under 
\begin_inset Quotes eld
\end_inset

run-1
\begin_inset Quotes erd
\end_inset

.
 These were carefully processed.
 monitored and double-checked.
 They contain MMT values for the word–disjunct pairs.
 (They also contain word-pairs, and updated MI states for the word-pairs.)
 
\end_layout

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Results of trimming are shown in the table below.
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r4-trim-1-1-1-pass-6
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\begin_layout Standard
The shrinkage is dramatic, in each case.
 Overall effects in each case are quite similar.
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\begin_layout Standard
I wonder what would happen, if MPG parsing was performed a second time,
 with the trimmed word-pairs...
\end_layout

\begin_layout Subsection*
Support Trim
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\begin_layout Standard
The single-count trim above was the right idea, but it missed a spot: we
 also need to trim away all words/disjuncts that have a support of just
 one! So trimming is repeated, and summarized below.
 Trim steps alternate between support-trim and count trim.
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Holy Smokes! That is one heck of a trim! That is one heck of a trim! Word
 counts reduced by factors of 3 to 6, disjunct counts reduced by factors
 of 8 to 12, total pairs reduced by a factor of 3 to 4.
 And that's compared to the trimmed files! 
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Compared to the untrimmed: 
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Those are just huge numbers! Oddly reassuring though.
 I wonder how much of this is due to the bad tokenization and bad quote-escaping
 in the MPG parse pipeline...
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Expt-5 – Trimming pairs – Sept 2021
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What happens if we trimmed the pair files? Scripts in 
\family typewriter
experiments/run-5
\family default
.
 Table below.
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Summary of results: the effect of trimming on quality is ambiguous.
 Because there are 24 observations made of each sentence, most word-pairs
 have observation counts of 24 or higher, and thus trimming less than this
 has almost no effect on the word-pair dataset size.
 Whether or not trimming more than this is a good idea is unknown.
 Conclusion: don't trim the word-pair dataset, at least not without further
 experimentation and measurements.
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So...
 trimming has a very different effect, here.
 Trimming to thresh of one does not have much of an effect.
 Trimming 10 raises the MI to an all-time high.
 Presumably it is cutting out the low-MI pairs.
 Don't we have a graph of MI vs.
 count, somewhere? Trimming to 10 also raises the sparsity to an all-time
 high.
 Is it a good idea to trim pairs? Maybe.
\end_layout

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Its possible that the low-MI pairs never show up in sections anyway, to
 the trimming might not affect the computation of the sections, except maybe
 to speed up MPG parsing slightly, by offering fewer edges.
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Word pairs in disjunct files
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So ...
 it appears that the disjunct files all have the word-pairs in them.
 If the disjuncts are trimmed, then, as a side effect, many word-pairs are
 also trimmed.
 How does this work out? Lets take a look.
 The table below reports the word-pair subset of the indicated data files,
 after trimming and after recomputing the MI for the word-pairs.
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\begin_inset Text

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run-1-marg-tranche-1234
\end_layout

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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\begin_layout Plain Layout
run-1-t1234-trim-1-1-1
\end_layout

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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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\begin_inset Text

\begin_layout Plain Layout
26173771
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.0263
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
51.439
\end_layout

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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
17.934
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1.2433
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Note that there are fewer words in the word-word pairs, than there are words
 in the word-disjunct pairs; about half as many (15350 avg vs.
 23865).
 Note that MI dropped by 
\begin_inset Quotes eld
\end_inset

a lot
\begin_inset Quotes erd
\end_inset

; i.e.
 by much more than the variation between files.
 The avg word-pair MI of the untrimmed tranches ranges from 1.55 to 1.44.
\end_layout

\begin_layout Standard
The explanation for this might be as follows: there are lots of unique words
 in the file, arising from poor cleanup of marked up text.
 Examples include "state,
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
" "shot.
\backslash

\backslash
" "
\backslash

\backslash

\backslash
"Always" "party!
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
" "Sit.
\backslash

\backslash
" "hide,
\backslash

\backslash
" "
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
"Prince" "
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
"Tick" "
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
"A
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
" "
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
"Exactly,
\backslash

\backslash

\backslash

\backslash

\backslash

\backslash
" ...
 There's actually a vast number of these, indicating that our cleanup scripts
 are broken.
 (XXX TBD Fix these.
 It appears that they have something to do with quotations.) Anyway, such
 words will be observed only a few times, in fixed combinations, and thus
 have a naturally high MI.
 Yet they are effectively garbage.
 When parsed via MPG, they appear only once, and thus become singletons,
 and get trimmed away.
 Untrimmed, they inflate the MI.
\end_layout

\begin_layout Standard
After trimming, these words with these backslashes are mostly absent.
\end_layout

\begin_layout Subsection*
Expt-6 – Similarity – Sept 2021
\end_layout

\begin_layout Standard
OK, so, finally onto similarity.
 The `
\family typewriter
experiments/run-6
\family default
` directory contains scripts to compute similarity, although the actual
 scripts are in `
\family typewriter
learn-lang-diary/utils/similarity-jacc.scm
\family default
`.
 Three similarity measures are computed: conventional MI, as before, overlap,
 and conditional-Jaccard.
 See diary Part Two for the definitions of these.
\end_layout

\begin_layout Standard
Even the most casual peek at the MI data shows that its pretty darned good,
 and its not obvious that overlap and cond-jacc are actually better.
 Interestingly, the self-MI of words are all over the place, and sometimes
 quite high, in the 7 to 11 range for the top most frequent 300 words, or
 even extremely high, in the 20-32 range for the least-frequently observed
 words.
 Many (most) infrequently-observed words have an MI of 31.809.
 This might be due to the fact that these words appear to have only one
 disjunct, grand-total?! This needs to be checked, and it suggests that
 perhaps a further trim step should be taken: get rid of words with only
 one disjunct on them! One the other hand, they might be idioms or set phrases.
 Are they? Lets look.
 But first, a characterization of the basic dataset.
\end_layout

\begin_layout Subsubsection*
Dataset generics
\end_layout

\begin_layout Standard
We are using the `
\family typewriter
run-1-t1234-trim-1-1-1.rdb
\family default
` dataset.
 It has 75431 words in it.
 Datasets generated with code in `
\family typewriter
utils/similarity-graphs.scm
\family default
`.
 First up: rank vs observation count.
 This is the classic Zipf graph.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/rank-count.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
The above graphs the classic Zipf distribution graph, although the distribution
 is not quite classic: it seems to have two regimes: the first 80 or 90
 words follow a 
\begin_inset Formula $rank^{-1.1}$
\end_inset

 distribution, which is more-or-less the classic slope, while later words
 follow a 
\begin_inset Formula $rank^{-1.4}$
\end_inset

 slope, which is considerably steeper than conventional Zipf distributions.
 I don't think I've seen this before.
 Is this due to trimming?
\end_layout

\begin_layout Subsubsection*
Self-MI
\end_layout

\begin_layout Standard
Lets look at the structure of Self-MI.
 Self-MI is defined as the MI between a word and itself.
\end_layout

\begin_layout Standard
Here is a scatterplot of the rank vs self-MI.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/rank-mi-scatter.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
It was noted that some words have a very high self-MI value, and it was
 hypothesized that these are words with a low number of disjuncts.
 The scatterplot show that there's no particularly clear self-MI vs observation-
count-rank correlation.
 The above can be re-graphed with a ranking taken from the word-vector support
 size; it does not change substantially.
 Looks nearly exactly the same.
\end_layout

\begin_layout Standard
Lets explore further.
 Support vs.
 rank-by-observation-count.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/rank-support-scatter.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
The 
\begin_inset Quotes eld
\end_inset

support
\begin_inset Quotes erd
\end_inset

 of a word-vector is the number of disjuncts it has on it.
 This does have the expected shape: the fewer observation counts on a word,
 the fewer disjuncts on it, too.
 It even seems to follow the classic Zipf slope.
\end_layout

\begin_layout Standard
So, lets take a look at some specific cases, those with a low count, but
 with very high or very low MI.
 Here's a table of words with an MI of greater than 25 and a rank of greater
 than 3000:
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
rank
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
count
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
support
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
word
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
self-MI
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3011
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
170
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
85
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
units
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25.399
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3154
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
160
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
80
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
aircraft
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25.398
\end_layout

\end_inset
</cell>
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<row>
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\begin_inset Text

\begin_layout Plain Layout
3185
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
158
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
73
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Seigneur
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25.216
\end_layout

\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3312
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
150
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
73
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
antenna
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25.433
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3349
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
148
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
69
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Battalion
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25.416
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
And again, with a rank greater than 12000 and MI greater than 28:
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
rank
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
count
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
support
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
word
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
self-MI
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12127
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
11
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Sauch
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28.349
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12129
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
11
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
trails
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28.349
\end_layout

\end_inset
</cell>
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12132
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
11
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
contractors
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28.349
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12134
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
11
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
fishermen
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28.349
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12139
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
11
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
villagers
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
28.349
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
For comparison, words with a very low MI, in the same range:
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
rank
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
count
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
support
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
word
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
self-MI
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3020
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
170
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
25
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Sch
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9911
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3043
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
168
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
13
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3s
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9844
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3058
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
167
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Scribner
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9647
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3141
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
161
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
57
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
destruction
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.8399
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
3178
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
159
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
51
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
wisdom
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9809
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
And again, with a rank greater than 12000:
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="5">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
rank
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
count
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
support
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
word
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
self-MI
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12122
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
23
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
810w
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9878
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12563
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
22
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
CONCLUSION
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.8572
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12848
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
21
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1140w
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9987
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12859
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
21
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Wessels
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9627
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12861
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
21
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
1
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Leipzig
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
2.9627
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Comparing the tables, we see that, for the same rank (same observation count),
 the low-MI words have a much smaller support than the high-MI words.
 A low support/count ratio suggests idioms or set phrases: they may have
 been counted a lot, but in only a few contexts.
 Yet, this result is counter-intuitive: shouldn't idiomatic, set phrases
 have a high self-MI, as the contexts are not shared?
\end_layout

\begin_layout Standard
The below is a scatterplot of MI vs.
 the ratio of support to observation count.
 There is no obvious correlation, other than a cap on the highest possible
 MI for a fixed ratio, and some sampling under-observations when the support-cou
nt ratio approaches 1.0.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/ratio-mi-scatter.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
The lack of correlations in the above graph seems to make it hard to say
 anything interesting about set phrases.
 Lets look at some details.
\end_layout

\begin_layout Itemize
Both Leipzig and Wessels appear to be in single-word sentences (a connector
 to LEFT-WALL to left, and a period on the right).
 So, in some list or index or table.
 Likewise for 810w and 1140w.
\end_layout

\begin_layout Itemize
The word CONCLUSION appears in four sentences:
\end_layout

\begin_deeper
\begin_layout Itemize
CONCLUSION.
\end_layout

\begin_layout Itemize
CHAPTER CONCLUSION.
\end_layout

\begin_layout Itemize
RECAPITULATION AND CONCLUSION.
\end_layout

\begin_layout Itemize
CONCLUSION (without the period at the end)
\end_layout

\end_deeper
\begin_layout Itemize
The word Scribner appears by itself, and as 
\begin_inset Quotes eld
\end_inset

New York: Scribner, Armstrong Co.
\begin_inset Quotes erd
\end_inset

 So, clearly a set phrase.
\end_layout

\begin_layout Standard
But how about 
\begin_inset Quotes eld
\end_inset

destruction
\begin_inset Quotes erd
\end_inset

 which has a very low self-MI but a support of 57, and 
\begin_inset Quotes eld
\end_inset

fishermen
\begin_inset Quotes erd
\end_inset

 which has a very high MI but low support?
\end_layout

\begin_layout Itemize
Most of the disjuncts on the word 
\begin_inset Quotes eld
\end_inset

destruction
\begin_inset Quotes erd
\end_inset

 are quite large: 4 or 5 or 6 connectors.
 Many of the connectors are prepositions.
 Almost all connector sequences have a determiner in them.
 Almost all sections (disjuncts) have a count of two.
 We conclude: no particular grammatical structure seems to be emerging.
 This word is not mergeable with others.
\end_layout

\begin_layout Itemize
The disjuncts on the word 
\begin_inset Quotes eld
\end_inset

fishermen
\begin_inset Quotes erd
\end_inset

 are crazy-large: the lengths are (as sampled): 9 5 7 7 10 7 10 6 3 3 3.
 All of these sections were sampled twice.
\end_layout

\begin_layout Standard
Comparing 
\begin_inset Quotes eld
\end_inset

destruction
\begin_inset Quotes erd
\end_inset

 to 
\begin_inset Quotes eld
\end_inset

fishermen
\begin_inset Quotes erd
\end_inset

 now reveals the origin for the extremely large self-MI value: the long
 connector sequences mean that the contexts are extremely specific, and
 effectively no other words will appear in these contexts.
\end_layout

\begin_layout Standard
Is this correct? Lets see...
\end_layout

\begin_layout Itemize
\begin_inset Quotes eld
\end_inset

fishermen
\begin_inset Quotes erd
\end_inset

 – all of the connector sequences on 
\begin_inset Quotes eld
\end_inset

fishermen
\begin_inset Quotes erd
\end_inset

 have a support of one!
\end_layout

\begin_layout Itemize
\begin_inset Quotes eld
\end_inset

destruction
\begin_inset Quotes erd
\end_inset

 – of the 57 connector sequences on 
\begin_inset Quotes eld
\end_inset

destruction
\begin_inset Quotes erd
\end_inset

, one has a support of 2385, and 13 have a support of more than ten.
 There are 38 connector sequences with a support of just one.
\end_layout

\begin_layout Standard
Yikes! The above reveals that the trimming was too conservative.
 The trim should remove all connector sequences with a support of one! Before
 we get too far, maybe we should do this.
 But first, some additional characterizations of the existing dataset, so
 that we can see how trimming affects it.
\end_layout

\begin_layout Standard
What is the scatterplot of the MI vs the average connector sequence length?
 How about the weighted avg sequence length? Shown below.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/conlen-mi-scatter.eps
	width 49col%

\end_inset


\begin_inset Graphics
	filename en-sims-p3/conlenw-mi-scatter.eps
	width 49col%

\end_inset


\end_layout

\begin_layout Standard
Given a fixed word 
\begin_inset Formula $w$
\end_inset

, the average connector sequence length is give by 
\begin_inset Formula 
\[
\left\langle \mbox{len}d\right\rangle =\frac{\sum_{d}I\left(w,d\right)\mbox{len}d}{\sum_{d}I\left(w,d\right)}
\]

\end_inset

where 
\begin_inset Formula $I\left(w,d\right)$
\end_inset

 is the indicator function, equal to 1 if the word-disjunct pair 
\begin_inset Formula $\left(w,d\right)$
\end_inset

 exists, else equal to zero.
 The 
\begin_inset Formula $\mbox{len d}$
\end_inset

 is the number of connectors in the connector sequence (aka disjunct) 
\begin_inset Formula $d$
\end_inset

.
 The weighted average is 
\begin_inset Formula 
\[
\left\langle \mbox{len}d\right\rangle _{\mbox{weighted}}=\frac{\sum_{d}I\left(w,d\right)\mbox{len}d\sum_{v}I\left(v,d\right)}{\sum_{d}I\left(w,d\right)\sum_{v}I\left(v,d\right)}
\]

\end_inset

so that the weighting is the total number of words in which a disjunct 
\begin_inset Formula $d$
\end_inset

appears in.
\end_layout

\begin_layout Standard
This seems to suggest that the commonly used connector sequences, those
 having a large support, have a small number of connectors in them.
 The long connector sequences appear to have a small support.
 Again, this points at trimming away connector sequences that have a small
 support.
\end_layout

\begin_layout Standard
How is the self-MI distributed? Shown below.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/self-mi-hist.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
The above shows the probability distribution of the self-MI of the 75K words
 in the dataset.
 As can be seen, almost all of these have a high MI.
 Based on the earlier scatterplots, almost all of these have low observation
 counts.
 The high MI is almost surely due to the untrimmed connector-seq support.
 Ignoring these shows that the self-MI distribution is otherwise rather
 noisy and quasi-uniform.
\end_layout

\begin_layout Subsubsection*
Pair distributions
\end_layout

\begin_layout Standard
Lets take a quick look at the distributions of the MI, log overlap and log
 conditional Jaccard of word pairs.
 The similarity between pairs of the top-ranked 1200 words.
 In principle, there are 
\begin_inset Formula $N\left(N+1\right)/2=720600$
\end_inset

 such pairs.
 In practice, 386380 pairs are observed; the remaining pairs have no overlap!
 (and thus a 
\begin_inset Formula $-\infty$
\end_inset

 for the three measures).
 
\end_layout

\begin_layout Standard
Here's is the MI distribution:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/mi-dist.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
The distribution appears to be an almost perfect Gaussian, centered at 
\begin_inset Formula $\mu=-1.85$
\end_inset

 and stddev 
\begin_inset Formula $\sigma=3.7$
\end_inset

.
 That is, the smooth parabola superimposed on the data is 
\begin_inset Formula 
\[
G\left(x;\mu,\sigma\right)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{\left(x-\mu\right)^{2}}{2\sigma^{2}}\right)
\]

\end_inset

The two parameters are an eyeballed fit.
 The fact that the MI distributions follow a Gaussian has been noted before.
\end_layout

\begin_layout Standard
Distribution of overlaps is shown below.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/overlap-dist.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
Two eyeballed fits are shown.
 One is to the log normal distribution
\begin_inset Formula 
\[
N\left(x;\mu,\sigma\right)=\frac{1}{x\sigma\sqrt{2\pi}}\exp\left(-\frac{\left(\ln x-\mu\right)^{2}}{2\sigma^{2}}\right)
\]

\end_inset

with 
\begin_inset Formula $\mu=2.15$
\end_inset

 and 
\begin_inset Formula $\sigma=0.25$
\end_inset

.
 Note that the normal peak occurs at and overlap of 
\begin_inset Formula $\exp\mu=8.6$
\end_inset

.
 The other curve is an eyeballed fit to the log logistic distribution
\begin_inset Formula 
\[
F\left(x;\alpha,\beta\right)=\frac{\beta\left(x/\alpha\right)^{\beta-1}}{\alpha\left(1+\left(x/\alpha\right)^{\beta}\right)^{2}}
\]

\end_inset

with 
\begin_inset Formula $\alpha=8$
\end_inset

 and 
\begin_inset Formula $\beta=8.8$
\end_inset

.
 Both curves seem to provide reasonable descriptions of the data, and I
 know of no theoretical reason to prefer one or the other.
\end_layout

\begin_layout Standard
Distribution of the Conditional Jaccard is shown below.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/condjacc-dist.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
Two fits are shown; this time, neither is very convincing, as they get either
 one end wrong, or the other, as parameters are adjusted.
 The log normal fit shown is for 
\begin_inset Formula $\mu=2$
\end_inset

 and 
\begin_inset Formula $\sigma=0.32$
\end_inset

 while the log logistic curve is for 
\begin_inset Formula $\alpha=6$
\end_inset

 and 
\begin_inset Formula $\beta=8$
\end_inset

.
\end_layout

\begin_layout Subsubsection*
MI vs.
 Overlap vs.
 CondJacc scatterplots
\end_layout

\begin_layout Standard
These should be correlated.
 What do they look like? Below.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/mi-ov-scatter.eps
	lyxscale 80
	width 32col%

\end_inset


\begin_inset Graphics
	filename en-sims-p3/mi-cj-scatter.eps
	lyxscale 80
	width 32col%

\end_inset


\begin_inset Graphics
	filename en-sims-p3/ov-cj-scatter.eps
	lyxscale 80
	width 32col%

\end_inset


\end_layout

\begin_layout Standard
Well, yes, all three are roughly correlated.
 Roughly; the clouds are quite fat.
 They're not very good proxies for each other.
 Note that the projections of these distributions to the x and y axes give
 the three distributions given in the subsection immediately previous.
\end_layout

\begin_layout Standard
Note that MI below 5, and overlap, condjacc below -4 mean that the two words
 of the word-pair are very different from one-another.
 They're junk, for the purposes of merging.
\end_layout

\begin_layout Subsubsection*
MI Inversion
\end_layout

\begin_layout Standard
There are some words whose self-MI is lower than the MI to another word!
 Out of the list of the 700 top-ranked words, there are 135 words whose
 self-MI is less than the MI to some other word.
 Out of the 1200 top-ranked words, there are 286 such instances.
 That's a lot.
 
\end_layout

\begin_layout Standard
Why this matters: There are several merge strategies: one is to merge a
 pair of words, whenever their MI is above some threshold.
 A second merge strategy is to compare the pair MI to the two self-MI's,
 and merge whenever it is close enough.
 That is, we should compare the MI of the word-pair to the average of the
 two self-MI's, and accept for merge if that is high enough.
\end_layout

\begin_layout Standard
Lets look at a few cases.
\end_layout

\begin_layout Standard
Here, the self-MI of 
\begin_inset Quotes eld
\end_inset

I
\begin_inset Quotes erd
\end_inset

 is lower than the MI of 
\begin_inset Quotes eld
\end_inset

I
\begin_inset Quotes erd
\end_inset

 to four other words.
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
MI
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
MI
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.0460
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
13.395
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.6618
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.3167
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.5361
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.0321
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3938
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12.304
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3380
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Note all of these are capitalized.
 Note the single-quote is a UTF8 tilted quote.
 The characteristic here is that the self-MI of 
\begin_inset Quotes eld
\end_inset

I
\begin_inset Quotes erd
\end_inset

 is fairly low, and that it has an MI that is higher to four other words,
 all of which have a high self-MI.
 Its as if these other words pull up the MI.
\end_layout

\begin_layout Standard
The above table is awkward.
 Lets try a matrix.
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
13.395
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.4164
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.3167
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2146
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.7200
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.0321
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.9883
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.8425
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.3143
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12.304
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.0460
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.6618
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.5361
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3938
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3380
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
This matrix shows that in this cluster, it is only the word 
\begin_inset Quotes eld
\end_inset

I
\begin_inset Quotes erd
\end_inset

 that has the low self-MI.
 None of the others do.
\end_layout

\begin_layout Standard
A few more.
 Here, 
\begin_inset Quotes eld
\end_inset

will
\begin_inset Quotes erd
\end_inset

 has a low self-MI.
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="8" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
MI
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
MI
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
cannot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.3204
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
cannot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
cannot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.3846
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
may
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.2399
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
may
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
may
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2852
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
must
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.2347
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
must
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
must
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2428
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
would
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.0957
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
would
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
would
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.0066
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
can
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.9624
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
can
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
can
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.6600
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.8143
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.8750
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.7761
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Let's do that again, in matrix form:
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="8" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
cannot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
may
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
must
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
would
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
can
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.3846
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
may
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.9085
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2852
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
must
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.0241
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.9907
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2428
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
6.5425
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.8959
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.9021
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.0066
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2619
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.2926
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.4708
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.0848
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.6600
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.2076
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.9178
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.5754
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.5990
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.5074
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.8750
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.3204
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.2399
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.2347
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.0957
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.9624
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.8143
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.7761
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Of all the word-pairs above, only 
\begin_inset Quotes eld
\end_inset

will
\begin_inset Quotes erd
\end_inset

 has the self-MI inversion.
\end_layout

\begin_layout Standard
Here's another one: The inversion sequence is 
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
straight, down, forth, forward, up, out, off, back.
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="9" columns="9">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
straight
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
down
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
forth
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
forward
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
up
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
out
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
off
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
back
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
straight
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
10.18
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
down
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.102
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
9.431
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
forth
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.127
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.732
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\end_inset
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7.922
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7.433
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\begin_inset Text

\begin_layout Plain Layout

\end_layout

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<row>
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\begin_inset Text

\begin_layout Plain Layout
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\begin_inset Text

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7.556
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\begin_layout Plain Layout
7.246
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\end_inset
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\begin_inset Text

\begin_layout Plain Layout
8.251
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\begin_inset Text

\begin_layout Plain Layout
7.382
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\begin_inset Text

\begin_layout Plain Layout
9.257
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\begin_layout Plain Layout

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\begin_inset Text

\begin_layout Plain Layout

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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\begin_inset Text

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8.209
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\begin_inset Text

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\begin_inset Text

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8.069
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\begin_inset Text

\begin_layout Plain Layout
7.210
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\begin_inset Text

\begin_layout Plain Layout
7.213
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.842
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
9.852
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
back
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.701
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\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.689
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.662
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.388
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.274
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.099
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.943
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
What should be done here?
\end_layout

\begin_layout Standard
Let's fill in these tables with the difference, the delta:
\begin_inset Formula 
\[
\Delta\left(w,u\right)=MI\left(w,u\right)-\frac{1}{2}MI\left(w,w\right)-\frac{1}{2}MI\left(u,u\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Why the average, instead of the geometric mean? Well, MI has a logarithm
 in it, and it gives the geometric mean after consolidation:
\begin_inset Formula 
\begin{align*}
\Delta\left(w,u\right)= & \log_{2}\frac{f\left(w,u\right)f\left(*,*\right)}{f\left(w\right)f\left(u\right)}-\frac{1}{2}\log_{2}\frac{f\left(w,w\right)f\left(*,*\right)}{f\left(w\right)f\left(w\right)}-\frac{1}{2}\log_{2}\frac{f\left(u,u\right)f\left(*,*\right)}{f\left(u\right)f\left(u\right)}\\
= & \log_{2}\frac{f\left(w,u\right)f\left(*,*\right)}{f\left(w\right)f\left(u\right)}-\frac{1}{2}\log_{2}\frac{f\left(w,w\right)f\left(*,*\right)}{f\left(w\right)f\left(w\right)}\frac{f\left(u,u\right)f\left(*,*\right)}{f\left(u\right)f\left(u\right)}\\
= & \log_{2}\frac{f\left(w,u\right)f\left(*,*\right)}{f\left(w\right)f\left(u\right)}-\log_{2}\frac{f\left(*,*\right)\sqrt{f\left(w,w\right)f\left(u,u\right)}}{f\left(w\right)f\left(u\right)}\\
= & \log_{2}\frac{f\left(w,u\right)}{\sqrt{f\left(w,w\right)f\left(u,u\right)}}
\end{align*}

\end_inset

Well, that's ..
 interesting.
 Just like the regular MI, the 
\begin_inset Formula $\Delta$
\end_inset

 is 
\begin_inset Quotes eld
\end_inset

projective
\begin_inset Quotes erd
\end_inset

; it is invariant under the rescaling of 
\begin_inset Formula $f$
\end_inset

.
 So, fill it in.
 BTW, caution with the definitions of the above.
 As before, the joint probabilities are sums over the normalized word-disjunct
 probabilities:
\begin_inset Formula 
\[
f\left(w,u\right)=\sum_{d}P\left(w,d\right)P\left(u,d\right)
\]

\end_inset

and that, in the above, 
\begin_inset Formula 
\[
f\left(w\right)=\sum_{d}P\left(w,d\right)P\left(*,d\right)=f\left(w,*\right)
\]

\end_inset

 
\end_layout

\begin_layout Standard
Below the diagonal, as before.
 Above the diagonal is the delta.
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="6" columns="6">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ll
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
13.395
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-3.440
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-3.499
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-6.861
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-3.821
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
You
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.4164
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.3167
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-0.454
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-5.468
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1.665
\end_layout

\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
We
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.2146
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
7.7200
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
8.0321
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-3.854
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-1.649
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I’ve
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.9883
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.8425
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
6.3143
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
12.304
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
-3.927
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
5.0460
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.6618
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.5361
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3938
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
4.3380
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Hmm.
 So anything with 
\begin_inset Quotes eld
\end_inset

I’ll
\begin_inset Quotes erd
\end_inset

 is deeply negative, because the self-MI of 
\begin_inset Quotes eld
\end_inset

I’ll
\begin_inset Quotes erd
\end_inset

 is so high.
 Should we use 
\begin_inset Formula $\Delta$
\end_inset

 as the merge-recommendation value?
\end_layout

\begin_layout Standard
Another 
\end_layout

\begin_layout Standard
\align center
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="8" columns="8">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout

\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
cannot
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
may
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
must
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
would
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
can
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
can’t
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
will
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
cannot
\end_layout

\end_inset
</cell>
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</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
These are pretty mildly behaved.
\end_layout

\begin_layout Subsubsection*
Distribution of Delta-MI
\end_layout

\begin_layout Standard
Here's the distribution of Delta-MI:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/delta-mi-dist.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
Compared to the distribution of the MI, given further up, this has exactly
 the same width, just a different offset.
 The distribution is smoother: the little prong is missing.
 There are no 
\begin_inset Formula $\Delta$
\end_inset

 values that are positive, thus, although the above was fit with a Gaussian
 normal distribution, this cannot be the theoretically-correct distribution,
 because of that cutoff.
 The theoretical reasons and implications of normal distributions for MI
 remain unknown to me.
\end_layout

\begin_layout Standard
How does this compare to MI? A scatterplot:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/mi-delta-scatter.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
Well ...
 we see that it performs a kind of clamping.
 High MI values seem to be clamped to a maximum of 0.0.
 Many, but not all of the 
\begin_inset Formula $\Delta=0$
\end_inset

 word-pairs are self-pairs (a word against itself).
\end_layout

\begin_layout Subsubsection*
Word-pair Rankings
\end_layout

\begin_layout Standard
Lets look at some top-ten lists of word pairs.
 Oh no! Inhabiting the top of the pops are word-pairs where the words have
 low observation counts.
 That's not nice, or, at least, not very informative.
 So lets re-rank taking into account the average log probability of seeing
 each word.
 Well, that leads to a curious situation.
 Let 
\begin_inset Formula 
\[
\mbox{common}sim\left(w,u\right)=sim\left(w,u\right)+\frac{1}{2}\log_{2}f\left(w\right)+\frac{1}{2}\log_{2}f\left(u\right)
\]

\end_inset

For the 
\begin_inset Formula $MI$
\end_inset

 this leads to
\begin_inset Formula 
\[
\mbox{common}MI\left(w,u\right)=\log_{2}\frac{f\left(w,u\right)f\left(*,*\right)}{\sqrt{f\left(w\right)f\left(u\right)}}
\]

\end_inset

which is ..
 also curious.
 Oh, hang on...
 what do we mean by 
\begin_inset Formula $f\left(w\right)$
\end_inset

? Because this can be interpreted two different ways:
\begin_inset Formula 
\[
f\left(w\right)=f\left(w,*\right)=\sum_{d}P\left(w,d\right)P\left(*,d\right)
\]

\end_inset

which is appropriate for MI.
 But there is also the plain-old marginal
\begin_inset Formula 
\[
p\left(w\right)=\sum_{d}P\left(w,d\right)
\]

\end_inset

which is ...
 a direct count of how often a word is observed.
 From here on out, we'll try to use lower-case 
\begin_inset Formula $p$
\end_inset

 for the word-disjunct marginal.
 (This is OK, since it's normalized: 
\begin_inset Formula $p\left(*\right)=1$
\end_inset

.)
\end_layout

\begin_layout Standard
This raises yet more questions.
 Define 
\begin_inset Formula 
\[
\mbox{marg}sim\left(w,u\right)=sim\left(w,u\right)+\frac{1}{2}\log_{2}p\left(w\right)+\frac{1}{2}\log_{2}p\left(u\right)
\]

\end_inset


\end_layout

\begin_layout Standard
Lets take some samples.
 The top-ten word pairs:
\end_layout

\begin_layout Standard
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
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\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
Looks pretty healthy, and there's some commonality.
 Some distributions:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/common-mi-dist.eps
	width 49col%

\end_inset


\begin_inset Graphics
	filename en-sims-p3/marg-mi-dist.eps
	width 49col%

\end_inset


\end_layout

\begin_layout Standard
Both look approximately normal.
 Common-MI is curiously flat-topped.
 Marg-MI retains a bit of the horn that the MI distribution has.
 Perhaps that horn is what makes Common-MI flat-topped.
\end_layout

\begin_layout Standard
Anyway, either of these seems like a good way to chose which words to rank.
 Marg-MI might be slightly preferable, as it seems to give slightly higher
 priority to frequent words (?).
\end_layout

\begin_layout Standard
How about the other two?
\end_layout

\begin_layout Standard
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="11" columns="4">
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<column alignment="center" valignment="top">
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_inset Text

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\end_layout

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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
he
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
he
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
her
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
his
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
her
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
his
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
to
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
,
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
She
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
He
\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
This does not seem as high-quality as either of the MI pair lists.
 the of-comma and the to-comma pairing is disturbing.
 The the-his is also disturbing.
 And these are in the top-ten! Yow! What's up with that? 
\end_layout

\begin_layout Standard
Lets look at the top-100 from common-MI and see how that works out.
\end_layout

\begin_layout Standard
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="21" columns="9">
<features tabularvalignment="middle">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<row>
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common-MI
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\end_inset
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\end_layout

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\begin_inset Text

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\begin_inset Text

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\end_layout

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\begin_inset Text

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\begin_inset Text

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common-MI
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\end_inset
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\begin_inset Text

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<cell multicolumn="2" alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

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</cell>
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<row>
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\begin_inset Text

\begin_layout Plain Layout

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
1
\end_layout

\end_inset
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21
\end_layout

\end_inset
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\begin_inset Text

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They
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\begin_inset Text

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41
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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25
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26
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
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</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
There's not much to object to, here.
 Maybe 39 and 40.
 How about marg-MI?
\end_layout

\begin_layout Standard
\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
\align center
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<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
<column alignment="center" valignment="top">
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<row>
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\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\end_layout

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\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
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42
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Her
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Their
\end_layout

\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
3
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<row>
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\begin_layout Plain Layout
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\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

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\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
24
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
more
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
less
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
44
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset
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<row>
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\begin_layout Plain Layout
5
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
?
\end_layout

\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
45
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<row>
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\begin_layout Plain Layout
6
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\end_inset
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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26
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\end_inset
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<row>
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\end_inset
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<row>
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\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
him
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
50
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
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\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
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<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
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<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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\begin_layout Plain Layout
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\end_inset
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<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
12
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
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\begin_layout Plain Layout
His
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\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
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</lyxtabular>

\end_inset


\begin_inset VSpace defskip
\end_inset


\end_layout

\begin_layout Standard
There's one screwy item in this list: United-several at position 46.
 It also notably elevates common Project Gutenberg markup: lower-case roman,
 FIG-Fig, Chapter-Footnote but otherwise looks pretty healthy.
\end_layout

\begin_layout Standard
Looking at the top-100 marg-overlap entries, these look pretty healthy,
 except for five more questionable comma-preposition suggestions! The top-100
 cond-jacc looks reasonable, too, except for some oddball determiner-pronoun
 similarities.
\end_layout

\begin_layout Standard
Overall, common-MI really looks best overall! I might be biased.
 It's hard to be scientific here.
\end_layout

\begin_layout Subsubsection*
TODO Items
\end_layout

\begin_layout Standard
Other things: 
\end_layout

\begin_layout Itemize
Explore the word 
\begin_inset Quotes eld
\end_inset

idea
\begin_inset Quotes erd
\end_inset

 and everything it is similar to!!! Wow!
\end_layout

\begin_layout Itemize
Graph of similarity cutoff vs number of things at are similar at that cutoff
 - ranked.
\end_layout

\begin_layout Itemize
Scatterplot: rank of word vs.
 number of disjuncts on it 
\end_layout

\begin_layout Itemize
Histogram of length of disjuncts.
\end_layout

\begin_layout Subsubsection*
Important Hypothesis
\end_layout

\begin_layout Standard
One important task: Compute the marginal MI of individual words, and rank
 those.
 The hypothesis is that words with low marginal MI are 
\begin_inset Quotes eld
\end_inset

filler words
\begin_inset Quotes erd
\end_inset

, not 
\begin_inset Quotes eld
\end_inset

function words
\begin_inset Quotes erd
\end_inset

: they are syntactic place-holders but are secondary in the transmission
 of meaning.
 The function words presumably carry most of the meaning.
 Is this true? Can we look?
\end_layout

\begin_layout Subsubsection*
Correctly trimmed similarity
\end_layout

\begin_layout Standard
Lets redo the similarity calculations, after trimming away all disjuncts,
 words with a support of one.
 As illustrated in table way above, this is a *huge* trim, removing most
 words and disjuncts.
 So copy `
\family typewriter
run-1-t1234-tsup-1-1-1.rdb
\family default
` to `
\family typewriter
r6-similarity-tsup.rdb
\family default
` and compute MI pairs for top-ranked words.
 Computation is a 
\emph on
lot
\emph default
 faster.
 There are 15083 words in this set ...
 still too many for computing all possible pairs ...
 but far more manageable.
 A skim of these reveals that a large fraction contain backslashes, 
\emph on
i.e.

\emph default
 are still polluted by bad tokenization.
\end_layout

\begin_layout Standard
One of the 
\begin_inset Quotes eld
\end_inset

bad charts
\begin_inset Quotes erd
\end_inset

 above was the self-MI.
 Words with only a single disjunct on them had a huge self-MI.
 Upon trimming, this goes away, as shown below:
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/self-mi-hist-tsup.eps
	width 80col%

\end_inset


\end_layout

\begin_layout Standard
This time, there is a clear downward trend.
 It can be fit with a log-normal distribution, the formula given previously
 above, with 
\begin_inset Formula $\mu=2.3$
\end_inset

 and 
\begin_inset Formula $\sigma=0.55$
\end_inset

.
 Recall that the 
\begin_inset Quotes eld
\end_inset

N
\begin_inset Quotes erd
\end_inset

 label in the graph is a shorthand for the log-normal distribution, the
 formula for which is given previously.
\end_layout

\begin_layout Standard
Of the 
\begin_inset Formula $N\left(N+1\right)/2=720600$
\end_inset

 possible pairs for the top 
\begin_inset Formula $N=1200$
\end_inset

 words, a total of 398431 are observed.
 This is a bit more than the earlier dataset.
 The pair distribution is the same as before, and only the high-MI excess
 has been removed.
 Three data curves are shown: 
\begin_inset Quotes eld
\end_inset

now
\begin_inset Quotes erd
\end_inset

, 
\begin_inset Quotes eld
\end_inset

before
\begin_inset Quotes erd
\end_inset

 and 
\begin_inset Quotes eld
\end_inset

uniq
\begin_inset Quotes erd
\end_inset

.
 The 
\begin_inset Quotes eld
\end_inset

before
\begin_inset Quotes erd
\end_inset

 curve is as before, and the 
\begin_inset Quotes eld
\end_inset

now
\begin_inset Quotes erd
\end_inset

 curve is for the current dataset.
 The 
\begin_inset Quotes eld
\end_inset

uniq
\begin_inset Quotes erd
\end_inset

 curve repeats the the current dataset, but with all self-MI pairs removed.
 This makes it especially clear that the goings-on at the high MI end of
 the curve is entirely due to the high-self-MI of some words.
\end_layout

\begin_layout Standard
\align center
\begin_inset Graphics
	filename en-sims-p3/mi-dist-tsup.eps
	width 49text%

\end_inset


\begin_inset Graphics
	filename en-sims-p3/common-mi-tsup.eps
	width 49text%

\end_inset


\end_layout

\begin_layout Standard
The Common-MI graph from before is nearly identical to the one from before.
 No comment.
\end_layout

\begin_layout Subsection*
Expt-7 – Merging – Sept 2021
\end_layout

\begin_layout Standard
Time to open a new chapter.
 We are ready to merge word-pairs, with a mostly-trustworthy dataset.
 The goal here will be to merge just one word at a time, by hand, and see
 what happens.
 We're doing this because of the disastrously bad July 2021 merge attempt,
 which generated complete garbage, quite in contrast to the merges from
 previous years.
 
\end_layout

\begin_layout Standard
Part of the reason for the failure of the July merge was that the merge
 threshold was set much too low.
 Part of the problem is that the datasets had too much grunge in them; they
 needed to be trimmed.
 Part of the problem is that the default merge used a proxy for the common-MI
 value, based on ranking.
 Probably most of the failure can be ascribed to the first part: the merge
 threshold was set too low.
 
\end_layout

\begin_layout Standard
We've never studied the merges carefully, before.
 Mostly, they seemed to 
\begin_inset Quotes eld
\end_inset

just work
\begin_inset Quotes erd
\end_inset

, and we unleashed the machinery, and let it go.
 The first time, it worked, the second time, it failed.
 Anyway, now that we've got a better dataset, its worth looking at these
 in detail.
\end_layout

\begin_layout Standard
This work resumes in the Diary Part Four.
\end_layout

\begin_layout Section*
The End
\end_layout

\begin_layout Standard
This is the end of the diary.
 The next part is Part Four.
\end_layout

\end_body
\end_document