Anak Agung Ngurah Gunawan 26
with ,
, ,
,
1
∑
=
=
t r
y y
y r
q q
m
d y
y H
d y
H 3.6
, ,
, ,
1
∑
=
=
t q
y y
y r
q r
m
d y
y H
d y
H 3.7
∑
=
=
t q
y y
y q
m q
d y
H y
M Mean
1
, ,
3.8
, ,
,
1 1
2
∑ ∑
= =
⎥⎦ ⎤
⎢⎣ ⎡
− =
t q
t p
y y
y q
m y
y y
p m
p q
d y
H d
y H
y y
D Deviation
3.9
∑ ∑
= −
= =
=
t r
q q
t r
y y
y y
y y
y r
q diff
d y
y H
d i
H
1
, ,
, ,
1
3.10
Entropy of ,
, log
,
1
∑
=
− =
t
i i
i diff
diff diff
d i
H d
i H
EHD H
3.11 ASM of
[ ]
, ,
,
1
2
∑
=
=
t
i i
i diff
diff
d i
H ASMHD
d i
H 3.12
Mean of
∑
=
=
t
i i
i diff
diff
d i
iH MHD
H
1
, ,
3.13 where
d y
y
r q
, ,
are the gray-level pixel value of unity, the value of the second pixel gray-level and the distance between the two pixels with pixels
unity, respectively. d
y y
H
r q
, ,
is a second-order histogram that describes the distribution of probability of occurrence of a pair of gray-level.
4. Logistic Regression Mapping Function
Review the following probability function: Y
P
r
and ,
X f
Y =
where the dependent variable that is bound to free variables
{ }
,
i
X and
i
X are linearly independent with
j
X that is
A Novel Model Determination of Breast Cancer Stage … 27
∑
≠
j j
j i
X a
X , where Y is output category, e.g.,
, =
y stage 1 category,
, 1
= y
stage 2 category and so on, ,
k y
= particular category.
This form is multinomial or multiple linear rate. Review of the logistic function logic as follows [24]:
{ }
⎭ ⎬
⎫ ⎩
⎨ ⎧
| =
− |
= =
| =
X Y
P X
Y P
log X
Y P
logic
r r
r
1 1
1 1
. 1
1 1
ln Y
X Y
P X
Y P
r r
= ⎭
⎬ ⎫
⎩ ⎨
⎧ |
= −
| =
≅ 4.1
Further,
{ }
4 3
2 1
, ,
, :
Z Z
Z Z
Y with
{ }
. :
entropies of
all X
For example, the category
k
Z Y
= .
1 1
1 ln
k r
r
Z X
Y P
X Y
P =
⎭ ⎬
⎫ ⎩
⎨ ⎧
| =
− |
= 4.2
Note.
Use of functional in natural logarithm related to qualitative mapping entropy to qualitative stage types of breast cancer, which does
not satisfy the normal Gaussian, statistically,
k
Z r
r
e X
Y P
X Y
P =
| =
− |
= 1
1 1
or
k
Z r
r
e X
Y P
X Y
P
−
= |
= |
= −
1 1
1 to obtain:
, 1
1 1
k
Z r
r
e X
Y P
X Y
P
−
| =
= |
= −
{ }
, 1
1 1
= +
| =
−
k
Z r
e X
Y P
{ }
, 1
1 1
k
Z r
e X
Y P
−
+ =
| =
4.3 and
X Y
P
r
| = 1
as a multinomial logistic regression of statistical model. For example
{ }
,
2 ,
1 =
=
k k
Z Y
it will be found in all categories
∑
=
= |
=
2 1
, 1
1
k k
r
X Z
P to
Anak Agung Ngurah Gunawan 28
{ }
, ,
3 ,
1 ,
2 ,
1 ,
1 ,
1 1
3 2
1
⎪⎭ ⎪
⎬ ⎫
⎪⎩ ⎪
⎨ ⎧
= =
= =
detected stage
Z detected
stage Z
detected stage
Z Z
k
{ }
, 1
1 1
1
1 Z
r
e X
Z P
−
+ =
| =
4.4
{ }
, 1
1 1
2
2 Z
r
e X
Z P
−
+ =
| =
4.5
{ }
3
1 1
1
3 Z
r
e X
Z P
−
+ =
| =
4.6 and because the fulfillment of all categoriesstages into force,
, 1
1
4 1
∑
=
= |
=
k k
r
X Z
P
, 1
1 1
1
3 2
1
= |
= +
| =
+ |
= X
Z P
X Z
P X
Z P
r r
r
until ,
1 1
1 1
3 2
1
X Z
P X
Z P
X Z
P
r r
r
= −
| =
− =
| =
{ }
{ }
. 1
1 1
1 1
1
3 2
1
⎥ ⎦
⎤ ⎢
⎣ ⎡
+ −
⎥ ⎦
⎤ ⎢
⎣ ⎡
+ −
= |
=
− −
Z Z
r
e e
X Z
P 4.7
5. Linear Regression Multinomial Function as an Outcome of the Stage Type
Review the following linear regression [24]:
MeanHd10, .
Bkn
1 ,
+ ⎭
⎬ ⎫
⎩ ⎨
⎧ +
=
∑
= n
j k
j kj
k k
Entr B
Z Z
5.1
k
Z is the outcomeimpact of a number of
{ }
,
j
Entr
k
Z is the intersectionintersection of the axis or the initial value of outcome,
.
k k
Z Z
=
A Novel Model Determination of Breast Cancer Stage … 29
For the tribe
0, MeanHd10
. Bkn
1 ,
= +
⎭ ⎬
⎫ ⎩
⎨ ⎧
∑
= n
j k
j kj
Entr B
∑
= n
j k
j kj
Entr B
1 ,
is a nuisance parametervariable-free number
{ }
j
Entr
the rank of 1 one or linear,
Bkn.MeanHd10 is the correction factor by the number of outcome
{ }
.
j
Entr
For example: MeanHd10,
. B2n
1 ,
2 2
20 2
+ ⎭
⎬ ⎫
⎩ ⎨
⎧ +
=
∑
= n
j j
j
Entr B
Z Z
5.2
MeanHd10. .
B3n
1 ,
3 3
30 3
+ ⎭
⎬ ⎫
⎩ ⎨
⎧ +
=
∑
= n
j j
j
Entr B
Z Z
5.3 These are illustrated in Figure 1 as follows:
Figure 1. Linear regression model and logistic regression model.
6. Results and Discussion
