Database Reference
In-Depth Information
Table 7.3 Results for the Ripley data set
Tenfold
Best
Level
Tenfold testing (%)
λ
On test data (%)
λ
Testing (%)
1
84.8
0.01005
89.8
0.00370
90.3
2
85.2
0.000001
90.4
0.00041
90.9
3
88.4
0.00166
90.6
0.00370
91.2
4
87.6
0.00248
90.6
0.01500
91.2
5
87.6
0.01005
90.9
0.00673
91.1
6
86.4
0.00673
90.8
0.00673
90.8
7
86.4
0.00075
88.5
0.00673
91.0
8
88.0
0.00166
89.7
0.00673
91.0
9
88.4
0.00203
90.9
0.00823
91.0
10
88.4
0.00166
90.6
0.00452
91.1
first two columns of Table
7.3
. With this
, we then compute the sparse grid
classifier from the 250 training data. Column 3 of Table
7.3
gives the result of
this classifier on the (previously unknown) test data set. We see that our method
works well. Already level 5 is sufficient to obtain results of 90.0 %. We also see
that there is not much need to use any higher levels. The reason is surely the
relative simplicity of the data. Just a few hyperplanes should be enough to
separate the classes quite properly. This is achieved with the sparse grid already
for a small number
n.
Additionally, we give in Table
7.3
the testing correctness which is achieved for
the best possible
λ
. To this end we compute for
all
(discrete) values of
λ
the sparse
grid classifier from the 250 data points and evaluate them on the test set. We then
pick the best result. We clearly see that there is not much of a difference. This
indicates that our approach to determine the value of
λ
from the training set by
cross-validation works well. Note that a testing correctness of 90.6 % was achieved
with neural networks in [Rip94].
λ
■
7.3.2 High-Dimensional Problems
Example 7.4
The 10-dimensional data set
ndcHArd
consists of two million
instances synthetically generated and was first used in [MM01]. Here, the main
observations concern the run time.
In Table
7.4
we give the results using the combination technique with simplicial
basis functions as described in Sect.
7.2.6
. More than 50 % of the run time is spent
for the assembly of the data matrix. The time needed for the data matrix scales
linearly with the number of data points. The total run time seems to scale even
better than linearly. Already at level 1, we get 84.9 % testing correctness, and no
improvement with level 2 is achieved. Notice that with support vector machines,
correctness rates of 69.5 % were reported in [FM01].
