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I show how the Burrows-Wheeler Transform can be used to compute the similarity between two strings. We submitted results from this method (along with results from the Context-Chain metric developed by my colleagues Frank Schilder and Ravi Kondadadi) for the Automatically Evaluating the Summaries of Peers (AESOP) task of the TAC 2009 conference.

The task was to produce an automatic metric to evaluate machine generated summaries (i.e., system summaries) against human generated summaries for the TAC ‘09 Update Summarization Task. Clearly the automatic metric is just some function that produces a similarity score between the system summary and the human generated (the so-called model) summary.

The  proposed metrics were evaluated by comparing their rankings of the system summaries from different peers to that of the ranking produced by human judges.

We use an estimate of the conditional “compressibility” of the model summary given the system summary as the similarity metric. The conditional compressibility is defined as the increase in the compressibility of the model summary when the system summary has been observed.

In order to judge the similarity of the system summary , to the model summary , we propose to use the difference in compressibility of when is not seen to when is given. This metric basically

captures the reduction in the uncertainty in when is known.

We define the compressibility of any string by

and the conditional compressibility of string over an alphabet given another string over the same alphabet as

where is the concatenation of the strings and , is the entropy of string , and is the length of the string .

The fractional increase in compressibility of given can then measured by

.

We use as the similarity metric to measure the similarity of a system summary to the model summary .

Our metric is similar to the one proposed by Li and Vitanyi and is theoretically well-justified from the perspective of algorithmic information theory. One peculiarity is that our similarity is asymmetric.

The only thing that is needed to implement the above similarity metric is an estimate of the entropy for a string . We use the BWT for this estimate.

BWT-based String Entropy Estimate

We use the Move-To-Front (MTF) entropy of the Burrows-Wheeler transform of a given string as an estimate for its entropy $H(S)$.

In this paper the MTF coding is used to define the MTF entropy (which the authors also call local entropy) of a string as

where is the  symbol of the MTF coding of the string .

Now we define , the entropy of string as

where is the BWT of string .

Since the Burrows-Wheeler transform involves just the construction of a suffix array, the computation of our compression based evaluation metric is linear in time and space in the length of the model and system summary strings.

Some Technical Details

Results

The results on the TAC-AESOP task (above) show that the BWT based metric (FraCC in the table) is reasonable for summarization evaluation, especially because there are not very many knobs to tune. I obtained these results from Frank (who will present them at TAC next week). The “best metric” is the AESOP submission that seemed to have high scores across several measures.

The Burrows-Wheeler transform (BWT), which is the main step in the bzip2 compression algorithm, is a permutation transform on a string over an ordered alphabet. It is a clever idea and can be useful for some string processing for natural language processing.  I will present one such use.

BWT massages the original string into being more amenable to compression. Of course the transform doesn’t alter the compressibility (entropy rate) of the original string. All it does is make the string more compressible by algorithms we know.

The reason string permutation by BWT (as opposed to say sorting the string, which makes it really compressible) is useful is that the reverse transform (undoing the permutation) can be done with very little additional information. Mark Nelson wrote a nice introduction to the transform.  Moreover, the BWT essentially involves the construction of the suffix array for the string, and therefore can be done in time and space linear in the length of the string.

Here is an example of the Burrows-Wheeler tranformation of the first stanza of Yeats’ Sailing to Byzantium. I added some newlines to the transformed string, and the underscores represent spaces in the original string. Notice the long runs of characters in the transformed string.

Original string

THAT is no country for old men. The young In one another’s arms, birds in the trees – Those dying generations – at their song, The salmon-falls, the mackerel-crowded seas, Fish, flesh, or fowl, commend all summer long Whatever is begotten, born, and dies. Caught in that sensual music all neglect Monuments of unageing intellect.

BWTransformed string

rsgnsnlhhs__lntsnH__T__.A____ss.,gt,.-gcd,es s,,,ode,yrgtsgrTredllssrn,edtrln,ntefemnu__fs___eh_hrC___ia__-eennlew_r_nshhhhslldrnbghrttmmgsmhvmnkielto-___nnnnna_ueesstWtTtTttTgsd__ye_teb__Fcweallolgfaaeaa_l

__mumoulr_reoeIiiueao_eouoii_aoeiueon__cm_sliM_

fbhngycrfeoeeoieiteaoctamleen’idit_o__ieu_n_cchaanta

____oa_nnosans_oomeoord_

A useful property

Effros et. al. showed that for a string generated by a finite-memory source, the BWT of the string is asymptotically (in the length of the string) indistinguishable from a piece-wise independent and identically distributed (i.i.d.) string. This is not surprising given that symbols with similar contexts appear sequentially in the BWT string, and for finite memory sources the current symbol is generated i.i.d. given a finite length context.

This property can be exploited to easily cluster words according to context by using BWT.

Word clustering

In this paper, among other things, Brown et.al. present a word clustering algorithm based on maximizing the average mutual information between the cluster ids of adjacent words. Some results are presented in Table 2 in the paper.

Such word clusters can be useful for feature engineering for sequence tagging tasks such as part-of-speech tagging or named-entity recognition. One of the most commonly used features for such tasks is one which checks if the current word is in a carefully constructed list of words.

Brown et. al. admit that, even after optimizations, their algorithm is slow and resort to approximations. (I realize that computers have gotten much faster since but still their algorithm is cubic in the size of the vocabulary.)

Word clustering based on BWT

We will cluster two words together if they appear independently given certain contexts (albeit with different probabilities). We first perform a BW transform on the input string of words (considering each word as a symbol, unlike in the example above) and measure whether the two words appear independently in an i.i.d. fragment.

Instead of actually trying to chop the BWT string into i.i.d. fragments before analysis, we adopt a proxy metric. We check if the number of times the two words are next to each other in the BWT string is large compared to what we would expect from their frequencies. We compute this as probability ratio with appropriate smoothing.

Another neat consequence of doing the clustering by BWT is that we only need to consider pairs of words that do appear next to each other in the BWT string. Therefore the selection of candidates for clustering is linear in the length of the string and not quadratic in the size of the vocabulary.

Some results

I ran this algorithm on about a month’s worth of New York Times and Wall Street Journal news data and these are the pairs of words with the highest scores.

january february 0.177721578886

january march 0.143172972502

march february 0.142398170589

englandgeoneng jerseyusanj 0.141412321852

news becdnews 0.135642386152

finala final 0.131901568726

finala finalb 0.122728309966

finala finalc 0.113085215849

cafd cea 0.107549686029

february april 0.100734422316

january april 0.0993752546848

has have 0.0967101802923

march april 0.0929933503714

did does 0.0854452561942

has had 0.0833642704346

will would 0.0827179598199

have had 0.0773517518078

…

I constructed a graph by joining all word pairs that have a score above a threshold and ran a greedy maximal clique algorithm. These are some of the resulting word clusters.

older young younger

announced today yesterday said reported

month today week yesterday

days month months decade year weeks years

decades months decade weeks years

com org www

writing write wrote

directed edited produced

should will probably could would may might can

worries worried concerns

work worked working works

wearing wear wore

win lost losing

man people men

against like to about that by for on in with from of at

under by on with into over from of

baton moulin khmer

daughter husband sister father wife mother son

red green blue black

ice sour whipped

time days months year years day

eastern coast southeastern

bergen orange nassau westchester

east ivory west

goes gone go going went

known seen well

travel review leisure weekly editorial

cultural financial foreign editorial national metropolitan

thursdays wednesdays fridays sundays tuesdays

thursday today monday sunday yesterday wednesday saturday friday tuesday

…

Discussion

1. For the above results, I only did the clustering based on right contexts. We can easily extend the word-pair score to take into account left contexts as well by concatenating the BWT of the reversed string to the BWT of the original string, and calculating the scores on this double length transformed string.

2. The word clustering algorithm of Brown et. al. proceeds by iteratively merging the best pair of words and replacing the two words in the alphabet (and the string) by a merged word. We can imagine doing something similar with our approach, except, because BWT uses the order on the alphabet, we need to decide where to insert the merged word.

3. One thing that I should have done but didn’t for the above results is to order the alphabet (of words) lexicographically. Instead I assign positive integers to the words based on their first appearance in the string, which is the order BWT uses to sort. Fixing this should improve the results.

Finally we gots some new clothes!

Here am I in mono comme ca and Brigitte in Hollywood. Doesn’t she have super long legs?!

And then, we gots some new outfits from Volks Who’s That Girl collection!

I decided to get a job so I can buy more clothes and shoes. Since I already had the uniform I applied to work at McDowell’s. It didn’t go so well…

Me: So, how do I look?

Prince Akeem: Hmmm.. not too shabby…

Me: Oh yeah! I look good!

Prince Akeem: Good enough to mop up the floor!

Me: Hold on, let me pose for you in my cute uniform!

Semmi: Pose here next to the mop!

Me: Oh wait! I hear my fans calling!

Prince Akeem: That’s not your fans, it’s the mop calling!

Me: I quit! Bye!

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Brigitte and I went to our usual cafe for some cupcakes. And we were surprised by a topless waiter! I thought it was funny but Brigitte was a bit stunned!

Ta dah! Yummy cupcake!

The cupcake was so delicious! It was a rich chocolate cake topped with a dreamy cream cheese frosting  with coconut flakes!

Hee Hee! Brigitte said I looked like a cupcake in my Blythism dress!

Hi! It’s Brigitte and Jacqui reporting live at the 81st Annual Academy Awards! (Okay, we’re a little late, not really live, and we weren’t really there.. but let’s just pretend cuz that’s what Blythes do best!) As you can see behind us all the stars have arrived on the red carpet! Everyone looks so beautiful!

These ladies were our favorites on the red carpet:

Taraji Henson in Roberto Cavalli and Penelope Cruz in vintage Balmain Couture.

And of course, a huge Congratulations to Kate Winslet! Woo Hoo! Go Kate!

She does look quite the Hollywood Royalty in Yves Saint Laurent… the golden statue helps too!

Our outfits:

Brigitte: dress: VM stock

Me: dress: Blythism, necklace: BWT stock, gloves: BWT stock

We still haven’t gone anywhere so Brigitte and I had to entertain ourselves. Today we played rockstars with me on the turntable and B on the guitar. At first B didn’t want to play but eventually she started to really get into her role!

Rock on dudes! Rock on! Woo Hoo!

Our outfits:

Me: shirt and pants: Bratz Boyz, shoes: BWT stock

Brigitte: shirt: Rement, Hoodie: nava kiwi, skirt: TIB shop, fishnets: Onionblythe, socks: TIB shop, shoes: BHC

I haven’t posted in awhile cuz my human didn’t take me anywhere. So today Brigitte and I decided to put on our stock outfits so my human will remember how special we are and take us out more!

Today Brigitte and I went to a Parisian cafe…. well, kinda! Our human set us up a mock cafe in Paris all with the Eiffel Tower in the background. We had a lot of yummy pastries. I think I had too much pink champagne though, the room started to spin a bit!

Today my little sister and I got to go to the playground. Even though we live in Southern California, it’s still chilly so we wore our cute pixie hats our human made for us. (You can get the pattern by clicking here. It’s by Creative Blythe. So cute!) But don’t even get me started on B’s outfit! She insisted on dressing herself. She looks so silly! But don’t tell her I said that, she’ll start crying.

Me: Come over here B!

Brigitte: ummm.. no thanks, you go ahead I’ll just stay here where I can’t fall through the cracks.

Me: See, it’s not scary at all. Let’s go on the slide next!

Brigitte: ummm… we’re up kinda high….

Me: Yeah! It’s gonna be so fun!

Brigitte: I don’t think I wanna go down the slide anymore. I don’t want to dirty my clothes.

Me: Whatever scaredy cat!

Weeeeeeeeee!!!!

Woo Hoo!

hmmmph!

Our outfits:

Me – hat: handmade, sweater: Rement, jeans: Bratz Boyz, shoes: BWT stock

Brigitte – hat: handmade, dress: Blythism, bloomers and shoes: YNH stock, socks: TIB store

Happy New Year! On New Year’s Day Brigitte and I went to my human’s mom’s (I call her madame) for lunch. She served us a traditional Korean New Year meal which was quite an adventure for me and B.

Brigitte: Jacqui, that’s a lot of rice! Do you think we can finish all of it? I wish it was a bowl of snow instead. *sigh*

This is the New Year’s soup called dduk gook, which is basically rice dumpling soup. The dumplings are really chewy. It’s actually yummy and kinda fun to eat and keeps your tummy warm! B had trouble eating it at first. LOL!

Brigitte: Jacqui, how long do you chew it for?

Me: Just swallow it, okay?!

Brigitte: But, what if it’s like gum? Ooooh I’m going to blow a bubble!

After lunch we had to bow to madame to wish her a prosperous and healthy new year and in exchange she gives us some shinies! It’s another korean tradition. We’re suppossed to wear hanboks, which are the traditional korean attire, but we don’t have any so we stole Mulan’s dresses! We thought it was close enough! Then my human took some pretty pictures of us in madame’s backyard. I had a lot of fun, but B preferred having tea indoors.

We ended the day watching the Rose Bowl. Fight on USC!  I’m going to ask my human to make us USC Songleader outfits.

So this was my biggest Christmas surprise – a little sister! Her name is Brigitte but I just call her B. She calls me Jacqui. She’s so cute, but a little airheaded. She thinks she’s a real princess! She’s always asking where the snow is and insists on skiing with her dress and coat. I told her there’s really not much snow here (actually none) and that we’ll have to take a trip to the mountains to go skiing. She started crying. arghhhhh! Little sisters are a handful!

I’m going to have so much to teach her about fashion and life and stuff! You just can’t go around wearing a princess lolita dress around here. yikes! I think I hear B crying again!

my outfit:

dress: Blythism, belt: Bratz Boyz, socks, BWT stock, shoes: BHC

Brigitte’s outfit:  all YNH stock

I decided that “Eat, Shop, Drink” will by my motto!

Whose shoes do you like better? Mine or my human’s? I choose mine!

Oh! I’m so hungry! Are we eating here?

Boo.. my human is not into fancy eateries so we went to Mitsuwa instead. We were all hungry so everything was delicious!

my outfit:

sweater: rement, skirt: BWT stock, socks: TIB shop, shoes: BHC