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where
N
is the number of rules of the system. The fourth layer has adaptive nodes:
a
i
(
x
)
·
f
i
(
x
)
=
a
i
(
x
)
·
(
g
0
i
+
g
1
i
x
1
+···+
g
ni
x
n
)
(3.5)
Finally, the fifth layer is the defuzzyfication node. For TS systems, the output
will be:
i
=
1
ω
i
(
N
x
)
·
f
i
(
x
)
a
i
(
x
)
·
f
i
(
x
)
=
(3.6)
i
=
1
ω
i
(
x
)
i
=
1
consequent functions. The
output of the Fuzzy complete model may be described by
We will call
a
i
(
x
)
antecedent functions and
f
i
(
x
)
N
)
g
0
j
+
g
nj
x
n
y
(
x
)
=
a
j
(
x
g
1
j
x
1
+···+
(3.7)
j
=
1
The idea to reduce complexity is to obtain a new description of the complete
model with less number of terms
R
:
R
)
h
0
j
+
h
nj
x
n
y
(
x
)
=
1
ξ
j
(
x
h
1
j
x
1
+···+
(3.8)
j
=
We can see the Fuzzy model as a function which is a lineal combination of
functions. Then, doing an analysis of principal components, we could obtain the
subspace where the total function is described. In the following section, we will
study this analysis.
3.3 Functional Principal Component Analysis
Principal Components Analysis (PCA) is well known in the field of multivariate
statistical analysis. Using PCA, we managed to reduce the dimensionality of the
space variables. The idea behind PCA is to find the subspace where the data have
a high covariance. Supposing
r
variables and
N
real samples, a real data set is
represented by
x
11
+
x
12
+··· +
x
1
r
e
1
e
2
e
r
x
21
+
x
22
+··· +
x
2
r
e
1
e
2
e
r
(3.9)
.
···
x
N
1
e
1
+
x
N
2
e
2
+··· +
x
Nr
e
r
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