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Table 8.2 Comparison of
prediction qualities: adaptive
SVD with variable rank
Rank
Clicks (%)
Baskets (%)
Buys (%)
5
6.3
11.14
12.07
10
7.59
12.1
14.22
14
8.07
13.72
15.35
15
8.09
13.37
15.39
16
7.98
13.72
15.44
18
8.38
13.78
15.33
20
7.67
13.06
15.28
25
7.78
12.63
14.77
¼
(
U
k
S
k
)(
U
k
S
k
)
T
¼ L
k
L
k
. Again, normalizing
L
k
along its columns leads to
L
k
,and
we can calculate
S
k
very easy without requiring the right singular vectors
V
k
:
S
k
¼ L
k
L
T
k
:
Here, for the full-rank SVD, we arrive at the original ITI-CF similarity matrix
S
.
The efficient computation of maximum values of a low-rank matrix is described in
Sect.
8.6
.
Example 8.5
For the assessment of the prediction quality, we use the methodology
of Sect.
4.4
with one difference: a product is counted as correctly predicted not only
when it directly follows a recommendation but also if it appears in the remaining
course of the session. This follows the logic of the prediction method because for
transactions within a session, their sequential order is ignored.
The transaction log file used contains 695,154 transactions. For the test all
sessions with less than 3 transactions have been removed, and only products of
the core shop have been considered (because the products of the remaining assort-
ment only rarely occur in the transactions). This resulted in 23,461 remaining
sessions. The number of different products is 558. We use the following mapping:
click
1,
basket
10,
order
20
for a reward function.
In Table
8.2
the results of the simulation with adaptive singular value decom-
position of Algorithm 8.1 for variable ranks on the described data of one day are
summarized.
From Table
8.2
it follows that rank 16 delivers the best prediction quality
concerning the orders, whereas baskets and clicks are slightly better predicted by
a model with rank 18.
Figure
8.3
, which is a graphical interpretation of Table
8.2
, clearly shows the
expected graphs of under- and over-fitting. If the rank is too small, the model does
not approximate the data good enough, and this results in a lower prediction rate
(under-fitting); a rank that is too high approximates the noise too exactly and also
reduces the prediction rate (over-fitting).
The results may suggest good chances for the application of the method in REs
although we will see later that reality is more complex.
■
