Information Technology Reference
In-Depth Information
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.2
0.2
0.2
0.2
0
0
0
0
-0.2
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
-0.4
-0.6
-0.6
-0.6
-0.6
0.5
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0
0
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0
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-0.5
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0
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-0.5
I
RD
=1.1816
I
RD
=1.3822
I
RD
=1.6223
I
RD
=1.7903
Fig. 3.7
A class with three sources altered by different grades of residual dependencies
Monte Carlo datasets were prepared, adapting the MLP to yield different grades
of nonlinearities in the data. Thus, depending on the nonlinearity mapping, there
were data with different degrees of adjustment to the source pdf's of the ICA model.
In order to test the correction of residual dependencies, 30 % of the observation
vectors with the lower values of source pdf, p
ð
s
ð
n
Þ
k
Þ
k
¼
1...Kn
¼
1...N
;
were
selected for the testing stage. The remaining 70 % of the observation vectors were
selected for the training stage. The parameters of the ICA mixture were estimated
using the data that best fit into the ICA model, and the classification was made on
the data that supposed were to be distorted with residual dependencies. We denote
any training observation vector as x
training
;
and we denote any testing observation
vector as x
testing
;
with p
ð
C
k
=
x
training
Þ
and p
ð
C
k
=
x
testing
Þ
being their respective pos-
terior probabilities. An index that measures the residual dependencies (I
RD
) can be
defined as the ratio between the mean of the posterior probability estimated in
training (p
ð
C
k
=
x
training
Þ
k
¼
1...K) and the mean of the posterior probability
estimated in testing (p
ð
C
k
=
x
testing
Þ
k
¼
1...K),
I
RD
¼
p
ð
C
k
=
x
training
Þ
p
ð
C
k
=
x
testing
Þ
k
¼
1...K
ð
3
:
14
Þ
I
RD
tends to 1 in the case of low or null nonlinearities, i.e., all the data are well-
adjusted to the ICA model. Conversely, if the data for testing do not fit the linear
ICA model, I
RD
increases since the posterior probability for testing data decreases.
Figure
3.7
shows some examples of generated datasets altered by nonlinearities
for a class in a dimensional space defined by three sources. The points in grey
represent the linear part of the data, and the points in black represent the points that
are most altered by nonlinearities. Note the different degrees of nonlinearities and
the corresponding index of residual dependencies I
RD
estimated after classification
for each of the graphs in Fig.
3.7
.
The results are summarized in Fig.
3.8
. The improvement in classification by
the correction is presented through different values of the residual dependency
index and for different numbers of sources (D
¼
2
;
...
;
5). Multi-dimensional
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