Decision Trees and Recommender Systems - Data Mining with Decision Trees: Theory and Applications - page 265

Table 16.1 Illustration of 4 4 judgment matrix

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corresponding to ratings

r A =5, r B =1, r C =3,

r D = 2 using Equation (16.7).

A B C D

A 1 5/1 5/3 5/2

B 1/5 1 1/3 1/2

C 3/5 3/1 1 3/2

D 2/5 2/1 2/3 1

Based on Equation (16.7) we can prepare a square judgment matrix,
where every element in the matrix refers to a single, pairwise comparison.
Table 16.1 presents the corresponding table.
Once the matrix is obtained, the original rating of the items can be
reconstructed by calculating the right principal eigenvector of the judgment
matrix. It follows from the fact that, for any completely consistent matrix,
any column is essentially (i.e. to within multiplication by a constant) the
dominant right eigenvector.
It should be noted that there are other methods for converting a
judgment matrix into an anity vector, including, for example, the least
squares method, logarithmic least squares method, weighted least squares
method, logarithmic least absolute values method and singular value
decomposition. Each of these methods has advantages and weaknesses. In
this chapter, we use the eigenvector method due to its popularity. The
eigenvector can be approximated by using the geometric mean of each row.
That is, the elements in each row are multiplied with each other and then
the k th root is taken (where k is the number of items).
Recall that the user selects a linguistic phrase such as “I much prefer
item A to item B” or “I equally like item A and item B”. In order to map it
into a five stars rating scale, we first need to quantify the linguistic phrase
by using a scale. Such a scale is a one-to-one mapping between the set of
discrete linguistic choices and a discrete set of numbers representing the
importance or weight of the choice. Satty suggests matching the linguistic
phrases to the set of integers k =1 ,..., 9 values for representing the degree
to which item A is preferred over item B. Here, the value 1 indicates that
both of the items are equally preferred. The value 2 shows that item A is
slightly preferred over item B, etc.
The value 3 indicates that item A is preferred over item B. The value
4 signifies that item A is much more preferred over item B and so on.