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currents are given by Eq. (
5
), where i
a
, i
b
and i
c
denote the three stator currents, I
max
their maximum value, f their frequency,
their phase angle and t the time. It is
known that each stator current is given by the combination of the others:
/
8
<
i
a
ð
Þ
¼
I
max
sin
ð
p
/Þ
t
2
ft
i
b
ð
Þ
¼
I
max
sin
ð
p
p=
/Þ
ð
5
Þ
t
2
ft
2
3
:
i
c
ð
t
Þ
¼
I
max
sin
ð
2
p
ft
4
p=
3
/Þ:
The PCA transform (
4
), applied to the signals in Eq. (
5
), makes the smallest
singular value equal to zero. This implies that the information of the principal
component, captured by the smallest singular value is null, then the last principal
component could be deleted and the original space reduced from three to two
without losing information. This is justi
ed by the fact that in Eq. (
5
), each stator
current is perfectly correlated to the sum of the others. Adding Gaussian white
noise, with standard deviation
, to the stator current signals (Eq.
5
), the smallest
singular value will not be equal to zero, but it will depend by the ratio between I
max
and
r
.
r
2.1.2 Kernel Density Estimation
Given N independent
and identically distributed (i.i.d.)
random vectors
X
¼
½
X
1
; ...;
X
N
, where X
i
¼
½
X
i1
; ...;
X
id
, whose distribution function F
ð
x
Þ
¼
P
½
X
x
is absolutely continuous with unknown PDF f
ð
x
Þ
. The estimated density
at x is given by Parzen (
1962
):
!
:
N
X
N
1
1
x
X
i
f
ð
x
Þ
¼
d
K
ð
6
Þ
d
i¼1
j
H
j
j
H
j
In the present study a two-dimensional Gaussian kernel function is used so d is 2
and a further simpli
cation, which follows from the restriction of kernel bandwidth
, leads to the single bandwidth estimator so the estimated den-
H
¼
h
2
I
:
h
0
[
sity f
ð
x
Þ
becomes:
N
X
N
2
1
1
kk
x
X
i
f
ð
x
Þ
¼
2
e
:
ð
7
Þ
2h
2
1
=
i¼1
ð
2
p
h
2
Þ
d
whose size in
grid
is the points number in which the PDF is estimated,
accordingly to Wand and Jones (
1994a
). It is well known that the value of the
bandwidth h and the shape of the kernel function are of critical importance as stated
in Mugdadi and Ahmad (
2004
). In many computational-intelligence methods that
employ KDE, the issue is to
where x
2 R
find the appropriate bandwidth h [see for example
Comaniciu (
2003
), Mugdadi and Ahmad (
2004
), Sheather (
2004
)]. In the present
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