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Table 9.1 Comparison of
prediction rates (absolute):
adaptive SVD with
variable rank
k
p 1
10
26
50
105
100
132
200
142
300
145
400
140
500
152
600
148
700
148
800
145
The products correspond to the first, and the sessions to the second dimension.
We deploy the adaptive SVD Algorithm 8.1 and compare the (absolute) prediction
rates (i.e., numbers of correctly predicted products) with respect to the test set. To
do so, we establish the vector a over all products, whereupon we assign the value
1 to the hitherto visited products and 0 to the remaining ones. By virtue of the
projection procedure ( 8.21 ) , we compute the vector a k and recommend the product
with the largest value therein. Upon termination of each session, we carry out an
incremental SVD step, i.e., compute the new rank- k SVD.
The result is displayed in Table 9.1 .
Rank 500 yields the highest prediction rate, namely,
152
5000 100 % 3 % .
Example 9.4 Let us now turn to the three-dimensional case. The selected product
may, for instance, act as the first dimension, the time of transaction as dimension
2, and, again, the session as dimension 3. A new slice thus corresponds to the matrix
of the hitherto occurred rewards for the selected products and the times of their being
invoked. Then we may write ( 9.5 ) as follows (provided that one insists on using it
despite its lack of correctness):
B t ¼ U 1 U 1 BU 2 U 2 :
Thus, or by virtue of Algorithm 9.2, respectively, we obtain a matrix of scores
over products and times. For the moment of the next prediction, we insert its time
and recommend in the corresponding column again the products with the entries of
the highest scores.
In the following, we consider the experimental results for the data set from the
previous Example 9.3. For technical reasons, though, we use product variations rather
than times as a new dimension.
We shall explain this in more detail. The retailer organizes the product by a
master-variation scheme. Thereupon, the master describes the product and the varia-
tion its varieties, which, in this case, are given by colors. A possible master might, e.g.,
be “Nike T-Shirt Air Jordan T-56” and the available colors “white,” “blue,” and “red.”
Hence, the master does not exist physically, but only the pairs (master, variation).
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