Database Reference
In-Depth Information
Table 16.2
4
judgment matrix corresponding to
ratings
Illustration of 4
×
r C =3,
r D = 2 using Equation (16.8).
r A =5,
r B =1,
ABCD
A1
9
2
4
B
1/9
1
1/5
1/3
C
1/2
5
1
2
D
1/4
3
1/2
1
Similarly, the inverse numbers (1 /k ) are used to represent the degree to
which item B is preferred over item A. In order to fit the five stars ratings
to Satty's pairwise scores, we need to adjust Equation (16.7) to:
round 2 r ui
1
r uj
if r ui
r uj
C uij =
.
(16.8)
round 2 r uj
1
1
\
r ui
if r ui <r uj
This will generate the judgment matrix presented in Table 16.2. The
dominant right eigenvector of the matrix in Table 16.2 is (3 . 71; 0 . 37; 1 . 90; 1).
After rounding, we obtain the vector of (4 , 0 , 2 , 1). After scaling we
successfully restore the original ratings, i.e.: (5 , 1 , 3 , 2). The rounding of
the vector's components is used to show that it is possible to restore the
original rating values of the user. However, for obtaining a prediction for a
rating, rounding is not required and therefore it is not used from here on.
Note that because we are working in a SVD setting, instead of using the
original rating provided by the user, we first subtract the baseline predictors
(i.e. r ui
b u ) and then scale it to the selected rating scale.
Our last example assumed that all pairwise comparisons should be
performed before the judgment matrix can be converted into the regular
rating scale. However, it is possible to approximate the rating using the
incomplete pairwise comparison (IPC) algorithm. In fact, if each item is
compared at least once, the approximation is quite good.
µ
b i
16.3.5
Profile Representation
Since we are working in a SVD setting, the profile of the newcomer should
be represented as a vector p v in the latent factor space, as is the case with
existing users. Nevertheless, we still need to fit the p v of the newcomer to her
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