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In-Depth Information
Table 8.3 Comparison of prediction qualities and error norms of SVD, LVP, and NMF with
variable rank
SVD
LVP
NMF
k
p
1
p
3
e
F
t
p
1
p
3
e
F
t
1
p
3
e
F
t
2
1.98
4.47
4.64
41
0.77
2.59
4.73
0
1.88
4.28
4.64
1
5
2.51
5.61
4.50
48
2.43
5.31
4.59
0
2.22
6.37
4.51
2
50
5.35
9.55
3.44
199
5.14
9.54
3.74
1
5.14
8.56
3.52
26
100
5.75
10.06
2.73
378
5.69
10.00
3.13
2
5.50
9.22
2.98
64
200
6.27
10.69
1.88
757
6.12
10.49
2.19
7
6.09
10.31
2.36
185
500
6.35
10.32
0.74
5168
6.35
10.35
0.75
38
6.35
10.51
2.53
752
558 (full)
6.37
10.33
0
8.4.4 Experimental Results
In the following, we shall apply the above-described factorization procedures to
predict product transitions and evaluate the results. To be able to employ the
nonadaptive procedures from this chapter, we divide each of the data sets into a
training and a test set.
We shall first address factorization of the transition probabilities
P
. To this end,
we first compute the matrix
P
from the training data. Then, we factorize it and
subsequently use the factorized matrix
P
k
to predict the considered products of the
test set. To do so, we traverse all sessions of the test set step by step and recommend
for each considered product
s
the product with the highest transition probability in
P
k
, i.e., arg max
s
0
pð
ss
0
, which we compare with the immediate successor product.
Correspondingly, we also recommend the three strongest products to inquire the
case of multiple recommendations. This so corresponds to the Markov approach
from the previous chapters.
Example 8.6
We consider the shop from Example 8.5, while taking all transactions
and its entire product assortment (2671 products) into account. The training and test
set both consist of 134,832 sessions.
For the factorization, we use each of the three methods presented in this section:
the truncated SVD (
8.14
), computed by means of the Lanczos Algorithm 8.2 with
Ritz projections; the Lanczos vector projection (LVP), i.e., Algorithm 8.2 with left-
projection approximation (
8.25
); and the NMF according to the ALS Algorithm 8.3.
The latter has always been carried out with 100 iterations.
The result is displayed in Table
8.3
.Here,
k
denotes the rank,
p
1
and
p
3
are
the rates of correct prediction for 1 and 3 recommendations, respectively,
e
F
:
¼
kP P
k
k
F
the Frobenius norm of the approximation error, and
t
the computing time
(in seconds). The last row contains the results for usage of the complete matrix
P
.
The result is rather disappointing and may be summarized as follows: none of
the three approaches to factorization of transition probabilities is really sensible.
A ridiculously high rank is necessary to achieve a quality of prediction that is
comparable with that attained when using the nonfactorized matrix
P
.
