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3
VARIANTS OF THE MCA
EXIN NEURON
*
3.1 HIGH-ORDER MCA NEURONS
High-order units have been advocated in [65]. The MCA neurons can be gener-
alized into high-order neurons for fitting nonlinear hypersurfaces, which can be
expressed as
w
1
f
1
(
x
)
+
w
2
f
2
(
x
)
+···+
w
n
f
n
(
x
)
+
w
0
=
0
(3.1)
where
f
i
(
x
)
is a function of
x
[e.g., for polynomial hypersurfaces
f
i
(
x
)
=
x
r
1
x
r
2
···
x
r
n
n
with integers
r
j
≥
0]. A generalized high-order neuron is just like
a linear neuron, but each weight is now regarded as a high-order connection,
and each input is not the component of the data vector
x
but its high-order
combination
f
i
(
2
. From this point of view, the neuron can be considered as a
functional link
neuron [148]. Hence, the same reasoning of the hyperplane fitting
can be made: Defining
f
x
)
=
f
1
)
T
, calculate
e
f
(
x
)
,
f
2
(
x
)
,
...
,
f
n
(
x
=
E
(
f
)
;if
it is different from zero, preprocess the data, and so on.
Remark 80 (MCA Statistical Limit)
If the observation datum x has Gaussian
noise, the noise in the transformed input f
is generally non-Gaussian
[
145
]
.It
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