Database Reference
In-Depth Information
Fig. 7.3 Spiral data set, sparse grid with levels 5 (
left
) and 7 (
right
)
Table 7.2 Leave-one-out cross-validation results for the spiral data set
Level
λ
Training correctness (%)
Testing correctness (%)
4
0.00001
95.31
87.63
5
0.001
94.36
87.11
6
0.00075
100.00
89.69
7
0.00075
100.00
88.14
8
0.0005
100.00
87.63
applications. However, it serves as a hard test case for new data mining algorithms.
It is known that neural networks can have severe problems with this data set and
some neural networks cannot separate the two spirals at all. In Table
7.2
we give the
correctness rate achieved with the leave-one-out cross-validation method, i.e., a
194-fold cross-validation. For the sparse grids, use the tensor-product basis func-
tions as described in this chapter.
The best testing correctness was achieved on level 6 with 89.69 % in comparison
to 77.20 % in [Sin98].
In Fig.
7.3
we show the corresponding results obtained with our sparse grid
combination method for levels 5 and 7. With level 7 the two spirals are clearly
detected and resolved. Note that here 1,281 grid points are contained in the
sparse grid.
■
Example 7.3
This data set Ripley, taken from [Rip94], consists of 250 training data
and 1,000 test points. It is shown in Fig.
6.4a
. The data set was generated synthet-
ically and is known to exhibit 8 % error. Thus no better testing correctness than
92 % can be expected. As before, we use tensor-product basis functions.
Since we now have training and test data, we proceed as follows: first, we use
the training set to determine the best regularization parameter
λ
. The best test
correctness rate and the corresponding
λ
are given for different levels
n
in the
