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CC B
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SA
li
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se
.
C
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s
p
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A
uth
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r
:
Vijay
Ku
m
ar
Dep
ar
tm
en
t o
f
Ap
p
lied
Scien
c
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Sch
o
o
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E
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in
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T
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Ma
n
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ter
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I
n
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titu
te
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d
Stu
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Far
id
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I
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E
m
ail: v
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k
u
m
ar
.
s
et@
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iu
.
ed
u
.
in
1.
I
NT
RO
D
UCT
I
O
N
E
n
tr
o
p
y
esti
m
atio
n
p
lay
s
a
s
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g
n
if
ican
t
r
o
le
ac
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o
s
s
th
e
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is
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o
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s
cien
ce
an
d
tech
n
o
lo
g
y
s
u
ch
as
en
g
in
ee
r
in
g
[
1
]
–
[
3
]
,
b
i
o
l
o
g
y
[
4
]
,
an
d
p
h
y
s
ics
[
5
]
.
Ma
th
em
atica
l
g
en
er
aliza
tio
n
o
f
n
o
n
-
p
ar
am
et
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tr
o
p
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ter
m
s
o
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tin
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o
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s
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v
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as
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ee
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r
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s
ed
b
y
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an
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Un
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ltiv
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p
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b
ab
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ed
in
[
6
]
,
[
7
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.
I
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s
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ates
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d
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a
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W
eg
m
an
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Dav
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[
8
]
in
tr
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d
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ce
d
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r
s
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e
d
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s
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esti
m
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to
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d
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th
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y
[
9
]
b
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m
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tatio
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ally
f
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a
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s
im
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to
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tan
d
[
1
0
]
.
2.
P
L
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-
I
N
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ST
I
M
AT
O
RS
Plu
g
-
in
esti
m
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esti
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f
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p
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.
Dif
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r
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s
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−
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esti
m
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ap
p
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f
in
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in
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.
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al.
Dm
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d
T
ar
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o
[
1
1
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p
r
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a
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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&
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I
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N:
2088
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8
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tima
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p
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.
I
n
m
u
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ca
s
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J
o
e
[
1
2
]
ap
p
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Sh
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p
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−
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I
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1
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7
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[
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o
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1
4
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–
[
1
6
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.
−
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ate
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1
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3.
USE
F
U
L
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M
A
T
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A
s
am
p
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d
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a
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ate
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ap
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x
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ir
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is
tr
ib
u
ti
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Oth
er
th
an
p
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g
-
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esti
m
ates,
m
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ates
h
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d
is
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s
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in
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e,
p
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ch
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3
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1
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E
s
t
im
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s
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o
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m
ple sp
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ased
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cted
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s
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t tr
iv
ial
[
1
8
]
–
[
2
0
]
.
3
.
2
.
E
s
t
im
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t
es ba
s
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o
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re
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esti
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eig
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d
is
tan
ce
s
w
ith
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co
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s
tan
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I
n
[
2
1
]
–
[
2
3
]
,
th
e
p
r
o
p
o
s
ed
n
ea
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n
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h
b
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esti
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ates
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m
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is
ten
cy
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n
d
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tain
co
n
d
itio
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s
.
3
.
3
.
E
ntr
o
py
estima
t
io
n f
o
r
lea
rning
T
h
e
liter
atu
r
e
s
h
o
ws
th
at
th
e
Sh
an
n
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m
ea
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u
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f
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tr
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p
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d
r
ew
a
lo
t
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f
atten
tio
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f
r
o
m
r
esear
ch
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s
to
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p
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s
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alg
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r
ith
m
s
f
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v
ar
io
u
s
lear
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in
g
s
y
s
tem
s
.
E
s
tim
at
in
g
Sh
an
n
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n
’
s
en
tr
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p
y
h
as
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lied
to
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y
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s
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tr
o
p
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g
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aliza
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o
f
Sh
an
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s
en
tr
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p
y
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h
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wo
r
k
p
r
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p
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s
es
s
im
p
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tr
o
p
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esti
m
ato
r
s
f
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tr
ain
in
g
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in
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en
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f
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ty
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d
d
if
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le
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ter
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s
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f
s
am
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T
h
e
m
ain
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o
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th
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wo
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k
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to
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s
tan
d
th
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m
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eh
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4.
P
RE
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M
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NAR
I
E
S
T
h
e
ter
m
s
lik
e
e
n
tr
o
p
y
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d
t
h
e
m
ea
s
u
r
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o
f
en
tr
o
p
y
ar
e
n
o
t
co
v
er
ed
as
t
h
ese
wer
e
d
i
s
cu
s
s
ed
b
y
m
an
y
r
esear
ch
e
r
s
.
So
m
e
o
f
th
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r
eq
u
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e
d
p
r
elim
in
a
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ies ar
e
d
e
f
in
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as
f
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4
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1
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Wind
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Fo
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Hass
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2
4
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M
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(
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a
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5
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1
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T
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1
Giv
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P
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r
z
e
n
[
2
5
]
,
i
n
t
h
e
e
s
t
i
m
a
t
i
o
n
o
f
t
h
e
P
D
F
,
t
h
e
s
a
m
p
l
e
m
e
a
n
c
o
n
v
e
r
g
e
s
t
o
t
h
e
p
o
p
u
l
a
t
i
o
n
m
e
a
n
,
w
h
i
c
h
is
t
h
e
d
i
r
e
ct
i
m
p
l
ic
a
t
i
o
n
o
f
t
h
e
c
o
n
s
is
te
n
t
P
a
r
ze
n
W
i
n
d
o
w
es
t
i
m
a
t
o
r
(
1
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
E
s
tima
tio
n
o
f
k
ern
el
d
en
s
ity
fu
n
ctio
n
u
s
in
g
K
a
p
u
r
en
tr
o
p
y
(
L
ee
n
a
C
h
a
w
la
)
6019
5
.
2
.
P
ro
pert
y
5
.
2
E
n
tr
o
p
y
esti
m
ato
r
(
1
)
u
s
in
g
P
ar
ze
n
W
in
d
o
w
in
g
an
d
m
ea
n
ap
p
r
o
x
im
atio
n
is
th
e
lim
itin
g
ca
s
e
o
f
Sh
an
n
o
n
e
n
tr
o
p
y
f
o
r
(
,
→
1
)
b
o
th
in
c
o
n
tin
u
o
u
s
an
d
d
is
cr
ete
ca
s
e.
Pro
o
f
:
I
t
s
h
o
u
ld
b
e
n
o
ted
th
at
Kap
u
r
en
tr
o
p
y
o
f
o
r
d
er
an
d
ty
p
e
(
1
)
is
d
is
co
n
tin
u
o
u
s
at
=
.
Fo
r
(
,
→
1
)
,
s
h
an
n
o
n
m
ea
s
u
r
e
o
f
en
tr
o
p
y
is
o
b
tain
ed
.
I
n
co
n
tin
u
o
u
s
ca
s
e:
,
(
)
=
1
−
[
∫
(
)
∫
(
)
]
=
→
1
→
1
{
1
−
[
(
∫
(
)
∫
(
)
)
]
}
=
−
∫
(
)
(
)
−
∫
(
)
(
)
=
2
(
)
I
n
d
is
cr
ete
ca
s
e:
→
1
→
1
̂
,
(
)
=
→
1
→
1
1
−
[
1
∑
(
1
∑
(
−
)
)
−
1
1
∑
(
1
∑
(
−
)
)
−
1
]
=
→
1
−
1
∑
(
1
∑
(
−
)
)
−
→
1
1
∑
(
1
∑
(
−
)
)
=
−
2
̂
(
)
T
h
er
ef
o
r
e,
it is
p
r
o
v
ed
th
at
th
e
p
r
o
p
o
s
e
d
en
tr
o
p
y
m
ea
s
u
r
e
(
1
)
ap
p
r
o
ac
h
es
Sh
an
n
o
n
en
tr
o
p
y
.
5
.
3
.
P
r
o
pert
y
5
.
3
Scalin
g
p
r
o
p
e
r
ty
o
f
e
n
tr
o
p
y
e
s
tim
ato
r
(
1
)
.
L
et
{
1
,
.
.
.
.
,
}
b
e
th
e
s
am
p
les
o
f
r
an
d
o
m
v
ar
ia
b
le
,
esti
m
ated
with
k
er
n
el
s
ize
.
T
o
esti
m
ate
s
am
p
les
{
1
,
.
.
.
.
,
}
o
f
a
r
an
d
o
m
v
ar
iab
le
,
th
e
s
ize
o
f
th
e
k
er
n
el
will
b
e
s
ca
led
to
|
|
,
i.e
.
,
n
ew
k
er
n
el
s
ize
is
|
|
.
Pro
o
f
: Co
n
s
id
er
th
at
f
o
r
a
r
an
d
o
m
v
ar
iab
le
,
th
e
PDF
is
(
/
)
/
|
|
I
n
co
n
tin
u
o
u
s
ca
s
e:
,
(
)
=
1
−
[
∫
(
)
∫
(
)
]
=
1
−
[
∫
1
|
|
(
)
∞
−
∞
−
∫
1
|
|
(
)
∞
−
∞
]
=
1
−
[
(
1
−
)
(
)
−
(
1
−
)
(
)
]
+
|
|
⇒
,
(
)
=
{
(
)
+
|
|
,
=
1
(
)
+
|
|
,
=
1
}
I
n
d
is
cr
ete
ca
s
e:
̂
,
(
)
=
1
−
[
1
∑
(
∑
|
|
(
−
)
)
−
1
−
1
∑
(
∑
|
|
(
−
)
)
−
1
]
=
1
−
[
(
1
−
)
̂
(
)
−
(
1
−
)
̂
(
)
+
(
−
)
|
|
]
⇒
̂
,
(
)
=
{
̂
(
)
+
|
|
,
=
1
̂
(
)
+
|
|
,
=
1
}
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
1
4
,
No
.
5
,
Octo
b
e
r
2
0
2
4
:
6
0
1
6
-
6
0
2
2
6020
I
t is co
n
clu
d
ed
th
at
th
e
k
e
r
n
el
s
ize
af
f
ec
ts
th
e
wid
th
o
f
th
e
k
e
r
n
el
f
u
n
ctio
n
lin
ea
r
ly
.
R
em
ar
k
s
: T
h
e
p
r
o
p
er
ty
(
5
.
3
)
h
as a
p
p
licatio
n
s
wh
en
th
e
en
tr
o
p
y
co
s
t f
u
n
ctio
n
is
n
ea
r
th
e
g
lo
b
al
ex
tr
em
u
m
an
d
is
o
b
tain
ed
b
y
s
ca
lin
g
th
e
en
tr
o
p
y
co
s
t f
u
n
ctio
n
.
5
.
4
.
T
heo
re
m
5
.
4
Acc
o
r
d
in
g
to
C
h
awla
[
2
7
]
,
f
o
r
eq
u
al
s
am
p
les,
1
=
2
=
.
.
.
.
=
=
(
s
a
y
)
with
m
ax
im
u
m
v
alu
e
o
f
k
er
n
el
(
0
)
,
th
e
p
r
o
p
o
s
ed
en
tr
o
p
y
esti
m
ato
r
(
1
)
is
m
in
im
u
m
.
Pro
o
f
:
Sin
ce
,
1
=
2
=
.
.
.
.
=
=
(
s
a
y
)
,
th
e
en
tr
o
p
y
est
im
ato
r
(
1
)
o
b
tain
e
d
th
e
v
al
u
e
−
(
0
)
.
W
e
h
av
e
to
s
h
o
w
t
h
at
1
−
[
1
∑
(
∑
(
−
)
)
−
1
−
1
∑
(
∑
(
−
)
)
−
1
]
≥
−
(
0
)
⇒
{
1
∑
(
∑
(
−
)
)
−
1
1
∑
(
∑
(
−
)
)
−
1
}
≤
−
(
0
)
⇒
∑
(
∑
(
−
)
)
−
1
∑
(
∑
(
−
)
)
−
1
≤
−
−
(
0
)
r
ep
lacin
g
with
th
e
u
p
p
e
r
b
o
u
n
d
s
=
∑
(
∑
(
−
)
)
−
1
∑
(
∑
(
−
)
)
−
1
≤
(
∑
(
−
)
)
−
1
(
∑
(
−
)
)
−
1
≤
(
−
1
[
−
1
(
−
)
]
)
(
−
1
[
−
1
(
−
)
]
)
=
−
,
−
(
−
)
=
−
−
(
0
)
I
t
is
ev
id
e
n
t
th
at
en
tr
o
p
y
co
s
t
f
u
n
ctio
n
r
ea
ch
es
its
g
l
o
b
al
m
in
im
u
m
,
w
h
en
all
th
e
er
r
o
r
s
a
m
p
les
ar
e
ze
r
o
an
d
u
s
ed
to
tr
ain
k
er
n
els in
s
u
p
e
r
v
is
ed
lear
n
in
g
.
5
.
5
.
P
r
o
pert
y
5
.
5
Fo
r
jo
in
t
en
tr
o
p
y
esti
m
atio
n
o
f
a
r
an
d
o
m
v
a
r
iab
le,
all
o
r
th
o
n
o
r
m
al
m
atr
ices
,
s
atis
f
ies
th
e
co
n
d
itio
n
∑
(
)
=
∑
(
−
1
)
,
th
e
p
r
o
p
o
s
ed
e
n
tr
o
p
y
esti
m
a
to
r
(
1
)
is
in
v
ar
ian
t
u
n
d
e
r
r
o
tat
io
n
with
∑
(
)
as
th
e
m
u
lti
-
d
im
en
s
io
n
al
k
er
n
el
f
u
n
ctio
n
.
Pro
o
f
:
Fo
r
n
-
d
im
en
s
io
n
al
r
a
n
d
o
m
v
ec
to
r
s
an
d
,
∃
=
in
wh
ic
h
[
]
×
r
ea
l
o
r
th
o
n
o
r
m
al
m
atr
ix
.
I
n
co
n
tin
u
o
u
s
ca
s
e:
,
(
)
=
1
−
[
∫
(
)
∫
(
)
]
=
1
−
[
∫
1
|
|
(
−
1
)
∞
−
∞
−
∫
1
|
|
(
−
1
)
∞
−
∞
]
=
1
−
[
|
|
1
−
∫
(
)
∞
−
∞
−
|
|
1
−
∫
(
)
∞
−
∞
]
=
1
−
[
(
1
−
)
(
)
−
(
1
−
)
(
)
+
(
−
)
|
|
]
,
(
)
=
{
(
)
+
|
|
,
=
1
(
)
+
|
|
,
=
1
}
I
n
d
is
cr
ete
ca
s
e:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
E
s
tima
tio
n
o
f
k
ern
el
d
en
s
ity
fu
n
ctio
n
u
s
in
g
K
a
p
u
r
en
tr
o
p
y
(
L
ee
n
a
C
h
a
w
la
)
6021
̂
,
(
)
=
1
−
[
1
∑
(
∑
∑
(
−
)
)
−
1
1
∑
(
∑
∑
(
−
)
)
−
1
]
=
1
−
[
1
∑
(
∑
1
|
|
∑
(
−
1
(
−
)
)
)
−
1
−
1
∑
(
∑
1
|
|
∑
(
−
1
(
−
)
)
)
−
1
]
=
1
−
[
|
|
−
1
1
∑
(
∑
∑
(
−
)
)
−
1
−
|
|
−
1
1
∑
(
∑
∑
(
−
)
)
−
1
]
=
1
−
[
(
1
−
)
̂
(
)
−
(
1
−
)
̂
(
)
]
̂
,
(
)
=
{
̂
(
)
,
=
1
̂
(
)
,
=
1
}
6
.
CO
NCLU
SI
O
N
I
n
th
is
wo
r
k
,
g
en
er
al
p
r
o
p
er
ties
o
f
th
e
en
tr
o
p
y
esti
m
ato
r
(
1
)
th
at
h
av
e
ap
p
licatio
n
s
in
lear
n
in
g
s
y
s
tem
s
h
av
e
b
ee
n
ad
d
r
ess
ed
.
T
h
e
r
esu
lts
wer
e
o
b
tain
ed
in
m
ath
em
atica
l
v
alid
atio
n
s
,
wh
ich
ca
n
b
e
s
ee
n
in
th
eo
r
em
s
5
.
1
an
d
5
.
4
,
an
d
p
r
o
p
er
ties
5
.
2
,
5
.
3
,
5
.
5
u
n
d
e
r
ce
r
tain
co
n
d
itio
n
s
f
o
r
c
h
o
o
s
in
g
t
h
e
k
er
n
el
f
u
n
ctio
n
.
T
h
e
p
r
o
p
o
s
ed
e
n
tr
o
p
y
esti
m
ato
r
co
u
l
d
b
e
th
e
n
ec
ess
ar
y
b
u
ild
in
g
b
l
o
ck
to
tr
ai
n
s
u
p
er
v
is
ed
lear
n
in
g
s
y
s
tem
s
.
RE
F
E
R
E
NC
E
S
[
1
]
T.
M
.
C
o
v
e
r
,
E
l
e
m
e
n
t
s
o
f
i
n
f
o
rm
a
t
i
o
n
t
h
e
o
ry
.
Jo
h
n
W
i
l
e
y
& S
o
n
s
,
1
9
9
9
.
[
2
]
S
.
H
a
y
k
i
n
,
U
n
su
p
e
r
v
i
s
e
d
a
d
a
p
t
i
v
e
f
i
l
t
e
ri
n
g
.
W
i
l
e
y
-
I
n
t
e
r
sc
i
e
n
c
e
,
2
0
0
0
.
[
3
]
C
.
E
.
S
h
a
n
n
o
n
a
n
d
W
.
W
e
a
v
e
r
,
T
h
e
m
a
t
h
e
m
a
t
i
c
a
l
t
h
e
o
ry
o
f
c
o
m
m
u
n
i
c
a
t
i
o
n
.
U
r
b
a
n
a
,
1
9
6
4
.
[
4
]
R
.
D
u
r
b
i
n
,
S
.
R
.
E
d
d
y
,
A
.
K
r
o
g
h
,
a
n
d
G
.
M
i
t
c
h
i
s
o
n
,
Bi
o
l
o
g
i
c
a
l
se
q
u
e
n
c
e
a
n
a
l
y
s
i
s
.
C
a
mb
r
i
d
g
e
U
n
i
v
e
r
si
t
y
P
r
e
ss,
1
9
9
8
,
d
o
i
:
1
0
.
1
0
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8
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[
9
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2
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1
5
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6
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[
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7
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[
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8
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1
9
]
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.
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