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In Part 2 we argued that most relevance ranking algorithms used for ranking text documents are based on three fundamental features:

1. Document Frequency: the number of documents containing a query term.
2. Term Frequency: the number of times a query term occurs in the document.
3. Document Length: the number of words in the document.

This post discusses document length normalization.

Document length normalization is related to term frequency.  Without length normalization, long documents would tend to get ranked above shorter documents even if the term in question is important in the short document but incidental to the topic of the long document.

 In the example below, the 32 page book Well Water Testing Guide has 200 occurrences of the word “water.”   If we rank documents based on term frequency, we will use some function of the number of occurrences of query terms.  For the query [water], without length normalization, long documents such as dictionaries and other reference books would be ranked higher than this book because they have more occurrences (higher term frequency) of the word “water”.[i]  In the chart below you can see that the Engineering Index, which contains around 3, 000 occurrences of the word “water”, and the American Encyclopaedic Dictionary, which contains about 2,000 occurrences of the word “water”, would be ranked much higher than the short document Well Water Testing Guide. This is because they have many more occurrences of the word “water” than the  Well Water Testing Guide.

One simple length normalization formula is to divide the number of occurrences by the length of the document.  For example, we can measure the length in pages and divide the number of occurrences (term frequency) by the number of pages as seen in Column 4 above.  Dividing the number of occurrences by the number of pages increases the score of the 32 page document relative to the dictionary and index because it has many more occurrences per page of the word “water.”

In practice, relevance ranking algorithms use more complex length normalization formulas, and there are more complex considerations regarding length normalization.  In order to understand the details, it will be useful to discuss the history of length normalization in information retrieval.

History of Document Length Normalization

Early experiments with the SMART system in the 1970s and 1980s

In the early 1970’s and 1980’s Gerald Salton’s group at Cornell did extensive experiments with the SMART information retrieval system.   The SMART system is an implementation of the vector space model designed for experimentation.   The SMART system was designed so that particular choices for handling term frequency, document frequency, and length normalization could be easily changed so that researchers could study and compare different combinations of these components. In a series of experiments based on twenty years of prior experimentation, Salton and Buckley (Salton and Buckley 1988) tested several different formulations of these three components.  For example for the document frequency component ( called collection frequency in the table below) they tried three different formulations:

,

and

  1 (no idf/collection frequency component)

The table below shows the components and formulas used in their 1988 experiments[ii].  Note that inverse document frequency, idf, is called “collection frequency” in this early paper and in the chart below:

Each particular formulation of a component is designated by a letter.  This allows the experimenter to specify a particular combination of components using a series of letters.  Since each of the three different components could be used for weighting documents or for weighting queries,  each particular configuration of document and query weighting can be described by listing six letters such as “tfc*nfx.” Salton and Buckley estimated that there were 1800 possible combinations of which 287 were unique.  They tested these against six standard test collections in order to determine which combination was best.

The test collections and documents they used were both very small: 25-50 word abstracts with bibliographic data, bundled into collections of  between 1,000 and 12,000 documents. The queries, however, were very long by modern web standards, averaging 10-20 words.^[iii] 

Salton and Buckley found that the best combination of components varied with the collection and types of queries. They attempted to describe how one could decide which combinations to use based on these characteristics.  For example, some combinations worked better with very long queries. [iv]

At the time, operational information retrieval systems were several orders of magnitude larger than these test collections and various researchers suggested that a larger test collection was needed.[v]

The Text REtrieval Conference (TREC) in the 1990s

In 1992 the National Institute of Standards and Technology started the Text REtrieval Conference (TREC) with an initial collection of 2GB of text. At the time, this was a big challenge for existing systems.[vi]   Document sizes in the TREC collections averaged around 300 words.  However, a collection of relatively long (average 1,500 words) documents from the Federal Register was added to the TREC collections to serve as “noise” documents[vii]  Collections were on the order of 500,000 documents. When the Text Retrieval Conference started up, systems had difficulty dealing with the huge (for the time) collections and the relatively large documents.  As a result, over the first few years of the TREC conferences, various techniques were developed to deal with long documents.[viii]

Okapi at TREC and BM25

The Okapi information retrieval system of City University London did not do well with the "large" documents and "large" collections in TREC 1.  As a result, between TREC 1 and TREC 3, the City University London team did a number of experiments with their relevance ranking algorithm (with at least 25 different versions of their ranking algorithm.) [ix]  The eventual outcome of these experiments was that the core algorithm for the Okapi system (which was previously based only on idf) was modified to take term frequency and document length into account. [x]  One of the findings of the experiments was that adding a tunable parameter to control the amount of length normalization improved retrieval results. In TREC 3 this parameter was added, and the version of the algorithm , which is known as BM25 (Best Match Algorithm Number 25), was first used.  This algorithm has been widely used as a baseline algorithm in information retrieval experiments for the last two decades.[xi]

In adapting the Okapi algorithms to deal with long documents, Stephen Robertson articulated the “scope vs. verbosity” hypothesis:

We may postulate at least two reasons why documents might vary in length.  Some documents may simply cover more material than others; an extreme version would have a long document consisting of a number of unrelated short documents concatenated together (the ‘scope hypothesis”).  An opposite view would have long documents like short documents but longer; in other words, a long document covers a similar scope to a short document, but simply uses more words (the “verbosity hypothesis”).

Robertson et al. (1994a ,1994b p 235)

Robertson argued documents that are long because they are verbose should be normalized and documents that are long because they cover a wide scope should not be normalized. However,  most documents are long because of a mix of scope and verbosity: “The verbosity hypothesis suggests that we should simply normalise any observed tfs by dividing by document length.  The scope hypothesis, on the other hand, at least in its extreme version, suggests the opposite.  In a real collection of documents we will observe variations in length, which might be due to either effect, or a combination.” Robertson and Zaragoza (2009: p 359)

Since most documents are long because of a mix of scope and verbosity, Robinson’s practical solution to this issue was to create a mixture model with a parameter “b” to control the mix of normalization vs no normalization.[xii]  Below is the part of the BM25 formula that deals with length normalization:

• w = document score before length normalization
• dl = document length.
• avdl = average document length.
• “b” is the tuning parameter that controls the mix of normalization/non-normalization to compensate for scope vs verbosity.
• “b” is a number between 0 and 1.   Think of it as percentage normalization

If b is set to 1, then the equation becomes:

This is 100% normalization and favors the verbosity hypothesis.  If average document length were 100 words and you had a document with 200 words then the score would be multiplied by ½.

If b is set to 0, the equation just multiplies the weight by 1 so there is no normalization.

As implemented, the BM25 algorithm does not attempt to determine whether a particular document being ranked is long because it covers a wide scope or because it is verbose or any specific mix between the two.  The same value of “b” is used for every document in a collection.

The “b” parameter of BM25 is tuned to general collection characteristics by testing it against a training collection of documents, queries, and relevance judgments that are similar to the collection to be used in production.  Parameters are tweaked to produce the best results on some measure of relevance.  In tests on various TREC collections and other IR research collections this strategy of tuning the algorithm has proven to be effective.

SMART catches up

From the 1960s until TREC started in 1992, the SMART information retrieval system from Cornell was one of the leading systems and was generally used as the baseline system for testing new systems and algorithms.  By the third TREC, the SMART system was not doing nearly as well as the Okapi system from City University.[xiii]  Amit Singhal, a graduate student in Salton’s group at Cornell  (now senior VP of Google and the head of the Google core relevance ranking team ) investigated why SMART was doing so much worse than Okapi.  Singhal determined that the comparatively poor performance of the SMART algorithms was due to Okapi’s  superior document length normalization. ^[xiv]  

In his analysis, Singhal sorted documents that were judged relevant in the TREC collections by size and then put them into equal size bins. Using the median document size for each bin, he then graphed the probability of relevance.  He discovered that the SMART system was doing too much length normalization (i.e. reducing the score based on the length of the document.)    Thus as a result the SMART system was biased against long documents.  The system was retrieving shorter documents with a probability greater than their probability of relevance and longer documents with a probability smaller than their probability of relevance. 

In the figures below, the graph on the left shows a smoothed representation of the probability of relevance and probability of retrieval using SMART’s default (cosine) term frequency  normalization.[xv]   In order to move the probability of retrieval closer to the probability of relevance, scores for documents smaller than a “pivot point” need to be reduced, and scores for documents longer than the pivot point need to be increased.  

Singhal explained that the probability of retrieval is inversely proportional to the normalization factor.[xvi]  The figure on the right shows how the normalization factor needs to be modified and demonstrates the “pivot.  For documents shorter than the pivot point the normalization factor needs to be increased (which will decrease the scores of short documents), and for documents longer than the pivot point the normalization factor needs to be decreased (which will increase the scores of long documents.)   Note that the slope of the line is controlled by a tunable parameter (alpha in the chart on the right).

Singhal did his doctoral thesis on length  normalization (Singhal 1997).  By using his "pivoted document weight normalization", the Cornell team got results comparable with the Okapi team.[xvii]  Singhal’s methodology has been replicated in a number of subsequent studies, and most groups using the vector space model have adopted Singhal’s pivoted normalization method.

We suspect that the default Solr/Lucene ranking algorithm, which is loosely based on the vector space model, suffers from the same problem of ranking short documents too high and long documents too low.  Robert Muir  contributed a patch that implements “pivoted document length normalization” for Lucene. (https://issues.apache.org/jira/browse/LUCENE-2187)  However, the patch has not been incorporated into Lucene, probably because the implementation of newer algorithms such as BM25, Language Models, Informaton-Based Models  and Divergence from Randomness were considered a higher development priority.

Tuning Parameters

Both BM25 and “Pivoted Document Length Normalization” have parameters that need to be tuned to the characteristics of a document collection.   More recent algorithms--such as the Divergence from Randomness (DFR) and Information-Based algorithms--also have tunable length normalization parameters.[xviii]

The values for these parameters are determined empirically by using a test collection of documents, queries, and relevance judgments, and doing a parameter sweep to optimize a particular retrieval effectiveness metric, such as mean average precision (MAP).[xix]   There are several important points here:

1. The optimal parameter settings differ for different collections and may be impacted by characteristics of documents, queries, and relevance judgments[xx]
2. The values for the parameters are not determined by a theoretically justified method, nor are they determined based on some particular quantifiable property of the documents, queries or relevance judgments.[xxi]
3. Because the parameter settings are collection-specific, for optimal results they have to be empirically determined for each collection.  In practice a test collection that is similar to the collection used in production is generally used.[xxii]

The empirical approach to tuning length normalization parameters using test collections has proven to be effective in tests on various TREC collections and other IR research collections.  However, there remain many unanswered questions for a practitioner working with production systems:

1. Given a large production collection, such as the 11 million volume HathiTrust corpus, what criteria should be used for creating a representative test collection? [xxiii]
   1. How large does a test collection need to be compared to the target (production) collection?  The INEX Book Track test collection contains about 50,000 books.  Would results of experiments on the INEX Book Track test collection apply to a collection such as HathiTrust that is two orders of magnitude larger? 
   2. What are the important properties of documents, queries, and relevance judgments that need to be similar between the production collection and the test collection?
2. Given limited resources, how can the relevance judgments needed for a test collection be collected at a reasonable cost?[xxiv]
3. How should length normalization be applied in different types of collections or different user tasks?  Are there particular properties of documents, queries, or relevance judgments that might call for a different type of length normalization?

The TREC ad hoc collections used in almost all of the studies of length normalization (including both Robertson’s Okapi work and Singhal’s SMART work) make some specific assumptions about user needs and relevance that may not apply to different tasks or collections.  The next post will discuss the assumptions used in the TREC ad hoc collections and their implications for relevance ranking of book-length documents.

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