Decision Trees and Recommender Systems - Data Mining with Decision Trees: Theory and Applications - page 271

Fig. 16.6 An illustration of the pairwise decision tree.

Since each tree node is associated with a set of users, we can create a new
user profile vector for each node by taking the mean of the vectors of the
corresponding users. When a user decides to terminate the profile initializa-
tion, we generate her profile vector p u based on the node she has reached
on the tree. Then Equation (16.5) can be used to predict ratings. Using
this vector we can generate a recommendation list by sorting the items
according to Equation (16.5). We could also use Equation (16.5) without
actually knowing the user bias b u because the user bias does not affect the
way the items are sorted. Sometime sorting the items is not sucient and
one is interested in the actual rating, for example, when it is necessary to
calculate the RMSE evaluation measure. For predicting the user rating, we
have to take it into account. But for that we need to know the user bias. To
do this, we first reconstruct the user's supposed individual ratings ˜ r u,i .We

distinguish supposed ratings from original ones by using the tilde notation.
Supposed ratings are pairwise responses that are converted to ratings using
the right dominant eigenvector mapping procedure presented above.
In order to estimate b u , we solve the least squares problem presented
in Equation (16.6) by replacing r u,i with

r u,i . Note that in this case the
parameters p u , q i , b i and µ are fixed. Thus, we need to minimize an
univariate function with respect to b u over the domain [

˜

5 , +5] (when
ratings range from 1 to 5). Minimization is performed by the golden section
search.

−