Outline

• More on Vector Space Model
• Proximity Ranking
• Quiz
• Boolean Retrieval

Introduction

• Last week, we gave the vector space model as one model for selecting which documents to return for a given term query and how to rank them.
• We viewed the query and the document as vectors with coordinates for each word in the corpus. The value in a given coordinate is some meassure of the importance of that term to the query or document (we'll say more on this).
• The similarity between a query and document is then defined in terms of the normalized dot-product between the two vectors. This value is `cos (theta)` where `theta` is the angle between the two vectors, so is sometimes called cosine similarity.
• Today, we begin by using this measure to come up with an algorithm for ranked retrieval.
• We then look at two other models: proximity ranking and boolean retrieval.
• Finally, we consider different techniques for evaluating query results.

TF-IDF

• Using our vector space model, the dimensionality of the vectors in question might be in the millions -- there could be a lot of distinct words in our corpus. So at first blush it might seem for even a small corpus that computing this cosine similarity would be costly.
• The important thing to note is that the actual runtime depends on only the number of non-zero entries in the vector with fewer non-zero entries (usually the query), and this is typically small.
• We still have to figure out what real values to use for the components of our query and document vectors.
• The most common way to do this is to use TF-IDF weights.
• Here TF means Term Frequency and is some function which measures how common the term is in the given document, and IDF is inverse document frequency which is typically relates the document frequency to the total number of documents in the corpus.
• Over the years, several different functions have been proposed for TF and for IDF. For this class, we will define `IDF = log(frac(N)(N_t))` and we will define `TF = log (f_(t,d)) + 1` if `f_(t,d) > 0` and `0` otherwise.
• We use log's (base 2) for TF since this models the number of bits of information we are getting by knowing the exact frequency of the number of occurrences of the term in the document.
• Notice if a term appears in close to every document, it doesn't tell you much to say the document has the term. The IDF in this case is small.
• For example, if we have a collection of five documents and the word "sir" appears in four, then `IDF(sir) = log(5/4) = 0.32`. If in document 2 the term "sir" occurs twice, then its TF would be `log(f_(t,d)) + 1 = 2`. So the TF-IDF("doc 2","sir"), which is the product of these two values, would be 0.64.
• This would represent the weight we would use for the "sir" component of doc 2's document vector. We could have one such component for each vocabulary term.

Algorithm to Compute Cosine Rank

rankCosine(t[1],...t[n], k) 
// t is an array of query terms
// k is the number of documents we want to return
{
    j := 1
    d := min_(1 <= i <= n) nextDoc(t[i], - infty)
    //we only need to consider docs containing at least one term
    while d < infty do
        Result[j].docid := d;
        Result[j].score := sim(vec d, vec t)
        j++;
        d := min_(1 <= i <= n, nextDoc(t[i], d))
    sort Result by score;
    return Result[1 .. k];
} 

Proximity Ranking

• Results and ranking in the VSM depend only on term frequency, TF, and Inverse Document Frequency, IDF.
• The proximity ranking method in contrast depends explicitly on how close the terms in the term vector occur, but only implicitly depends of TF and IDF plays no role at all.
• Given a term vector `langle t_1, ... t_n rangle`, we define a cover for this vector to be an interval `[u,v]` in the collection that contains all the terms in the vector, such that no smaller interval `[u', v']` contained in `[u,v]` also has a match to all the terms in the vector.
• For example, if the document 1 in our collection began as "To be or not to be", and the term vector was "to be" then the intervals `[1:1, 1:2]` and `[1:5, 1:6]` would be covers, but `[1:1, 1:3]` would not be as it contains `[1:1, 1:2]`.
• Covers are allowed to overlap. If "meow meow" was the term vector and document 2 began "meow meow meow meow meow meow ..." then the intervals `[1:1, 1:2]` and `[1:2, 1:3]` would both be covers.
• A given token though can appear in at most `n cdot l` covers, where `l` is the length of the shortest posting list.

Algorithm for Finding Covers

nextCover(t[1],.., t[n], position) 
{
    v:= max_(1≤ i ≤ n)(next(t[i], position));
    if(v == infty)
        return [ infty, infty];
    u := min_(1≤ i ≤ n)(prev(t[i], v+1))
    if(docid(u) == docid(v) ) then 
        return [u,v]; 
       // covers need to be in the same document
    else
        return nextCover(t[1],.., t[n], u);
}

Ranking Covers

• We now want to come up with a formula that scores documents according to the covers it contains for the term query.
• We would like that smaller covers are worth more than larger covers.
• We would also like more covers in a document to count more than fewer covers in a document.
• Keeping this in mind, suppose document `d` has covers `[u_1, v_1]`, `[u_2, v_2]`, ...
• Then we define the score for `d` to be:
  `mbox(score)(d) = sum(frac(1)(v_i - u_i + 1))`.

Ranking Algorithm with Proximity Scores

rankProximity(t[1],.., t[n], k)
// t[] term vector
// k number of results to return 
{
    u := - infty;
    [u,v] := nextCover(t[1],.., t[n], u);
    d := docid(u);
    score := 0;
    j := 0;
    while( u < infty) do
        if(d < docid(u) ) then
        // if docid changes record info about last docid
            j := j + 1;
            Result[j].docid := d;
            Result[j].score := score;
            d := docid(u);
            score := 0;
        score := score + 1/(v - u +1);
        [u, v] := nextCover(t[1],.., t[n], u);
    if(d < infty) then
        // record last score if not recorded
        j := j + 1;
        Result[j].docid := d;
        Result[j].score := score;
    sort Result[1..j] by score;
    return Result[1..k];   
}
  

Using an analysis similar to that used for galloping search in the book, you can prove this algorithm has running time:
`O(n^2 l cdot log(L/l))`.

Quiz

Which of the following is true?

1. The posting list next() operation should always be implemented using a binary search between first() and last().
2. Using a frequency index we can determine the number of times a term occurred in a document.
3. The vector space model uses sine similarity to determine how close a query is to a document.

Boolean Retrieval

• Besides being used as implicit filters in some web search engines, explicit support for Boolean queries is important for some specific applications such as digital libraries and in the legal domain.
• For example, when we do a patent search, we want an exhaustive list of patents returned which match the term query, not just the top 10.
• In the Boolean retrieval model we thus return sets of results rather than ranked lists.
• The standard boolean operators AND, OR, and NOT are used to construct queries.
• Here "A AND B" means the intersection of sets A and B; "A OR B" means the union of the sets A and B; and NOT A means those documents in the collection not contained in A.
• A term `t` corresponds in a query to the collection of documents containing `t`.
• So ("quarrel" OR "sir") AND "you" represents the set of documents that contain "you" and also contain at least one of the two terms "quarrel" and "sir".
• An algorithm to solve a Boolean query locates candidate solutions to the query, where each candidate solution represents a range of documents that together satisfy the query.

Extending Our ADT for Boolean Retrieval

• To simplify our Boolean search algorithm, we define two functions that operate over Boolean queries:

  docRight(Q, u)

  end point of the first candidate solution to `Q` starting after document `u`

  docLeft(Q, v)

  start point of the last candidate solution to `Q` ending before document `v`

• For terms we define docRight(t, u) := nextDoc(t, u) and docLeft(t,v) := prevDoc(t,v).
• For AND and OR operators we define:

  docRight(A AND B, u) := max(docRight(A, u), docRight(B,u))
  docLeft(A AND B, v) := min(docLeft(A,v), docLeft(B,v))
  docRight(A OR B, u) := min(docRight(A, u), docRight(B,u))
  docLeft(A OR B, v) := max(docLeft(A,v), docLeft(B,v))
  

• Notice the above rules give us a recursive algorithm given a Boolean query `Q` to compute docRight, docLeft.

Algorithm to Return the Next Solution to a Positive Boolean Query (No NOT's).

nextSolution(Q, position)
{
    v := docRight(Q, position);
    if v = infty then
        return infty;
    u := docLeft(Q, v+1);
    if(u == v) then
        return u;
    else
        return nextSolution(Q, v);
}

Algorithm to Return All Solutions to a Positive Boolean Query

u :=  -infty
while u < infty do
    u := nextSolution(Q, u);
    if(u < infty) then
        report docid(u);

If we implement nextDoc, prevDoc with galloping search, the complexity of this algorithm is `O(n cdot l cdot log(L/l))`

Handling Queries with NOT

• To handle negations we first use De Morgan's rules to push negations so that they are only on terms.
• i.e., NOT (A AND B ) = NOT A OR NOT B and NOT (A OR B) = NOT A AND NOT B
• Then we enhance our methods nextDoc and prevDoc, so that they can directly handle negations. i.e., So nextDoc(NOT t, u) and prevDoc(NOT t, u) make sense.
• Finally, we set docRight(NOT t, u) := nextDoc(NOT t, u) and docLeft(NOT t,v) := prevDoc(NOT t,v).