Information Technology Reference
In-Depth Information
T
n
(
i
b
can be obtained
)
For any
x
, we have
p
(
x
)
=
c
y
(
x
)
. Each normal vector
n
from the derivatives of
p
. Consider the derivative of
p
n
(
x
)
as follows,
∂
p
(
x
)
∂
n
n
∏
∑
∏
n
T
(
i
)
(
i
)
T
(
j
)
∇
p
(
x
)
=
=
x
b
=
b
x
b
.
n
∂
x
∂
x
i
i
j
≠
i
∏
≠
i
T
(
l
)
(
j
)
(
l
)
For a point
x
∈
S
,
b
x
=
0
for
l
≠
i
. It can be noted that there is only
l
j
∏
≠
i
T
one non-zero term in
p
n
(
x
)
(
l
)
(
i
)
(
j
)
(
l
)
for
l
=
i
. Then,
∇
, i.e.
∇
p
(
x
)
=
b
b
x
≠
0
n
j
the normal vector of
S
is yielded as,
(
l
)
∇
p
(
x
)
b
(
i
)
=
n
.
(A2)
∇
p
(
x
(
l
)
)
n
In order to get a set of points lying on each hyperplane respectively, so as to de-
termine the corresponding normal vectors
(
i
b
, we can choose a point in the given
X
close to one of the hyperplanes as follows,
)
px
()
n
i
=
arg
min
, where
x
∈
X
.
(A3)
∇
px
()
∇≠
px
()0
n
n
(
i
)
After given the normal vectors
{
b
}
, we can classify the whole point set
X
into
n
hyperplanes in
R
as follows,
(
)
T
(
j
)
label
=
arg
min
x
b
,
j
=
1
...
n
.
(A4)
j
This algorithm is called GPCA-PDA Alg. in [12,13].
Polynomial Segmentation Algorithm
Consider a special case of piecewise constant data. Given
N
data points
x
œ
R
, we
hope to segment them into an unknown number of groups
n
. This implies that
there exist
n
unknown cluster centers
μ
≠≠
...
μ
, so that,
1
n
(
x
=
μ
)
∪∪
...
(
x
=
μ
)
,
1
n
which can be described in a polynomial form as follows,
n
n
∏
∑
.
k
px x
( )
=
(
−
μ
)
=
cx
=
0
(A5)
n
i
k
i
=
1
k
=
0
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