Graphics Reference
In-Depth Information
he kernel function K
stat
is monotonically nonincreasing over the interval
.
he bandwidth h is increased by a constant factor with each iteration k.hetest
statistic for (
.
)
[
,
)
(
k
)
ij
(
k
)
i
θ
(
k
−)
i
, θ
(
k
−)
j
T
=
N
K(
)
(
.
)
(
k
)
ij
. his term effectively measures
the statistical difference between the current estimates in X
i
and X
j
. In (
.
) the
term
with N
i
=
j
w
ij
is used to specify the penalty
s
θ, θ
′
K(
)
denotes the Kullback-Leibler distance of the probability measures P
θ
and P
θ
′
.
Additionally, we can introduce a kind of memory into the procedure, which en-
sures that the quality of estimation will not be lost with the iterations. his basically
means that we compare a new estimate θ
θ
(
k
)
(
k
)
i
=
(
X
i
)
with the previous estimate
θ
(
k
−)
i
(
k
)
i
in ordertodefine a memoryparameter η
i
=
K
mem
(
m
)
using a kernel func-
tion K
mem
and
θ
, θ
(
k
)
i
τ
−
(
k
)
ij
(
k
)
i
(
k
−)
i
m
=
K
loc
(
l
)K(
)
(
.
)
j
his leads to an estimate
θ
(
k
)
i
η
i
θ
(
k
)
i
θ
(
k
−)
i
=
+(
−
η
i
)
(
.
)
Adaptive Weights Smoothing
8.2.1
We now formally describe the resulting algorithm.
Initialization:
Set the initial bandwidth h
()
, k
andcompute,foreveryi,the
=
statistics
(
k
)
i
(
k
)
ij
, dS
(
k
)
i
(
k
)
ij
N
=
j
w
=
j
w
Y
j
(
.
)
and the estimates
θ
(
k
)
i
(
k
)
i
(
k
)
i
S
N
(
.
)
=
()
ij
()
ij
()
h
andh
()
using w
=
K
loc
(
l
)
.Setk
=
=
c
.
Adaptation:
For every pair i, j, compute the penalties
(
k
)
ij
h
(
k
)
,
X
i
X
j
(
.
)
l
=
−
(
k
)
ij
(
k
)
ij
(
k
−)
i
θ
(
k
−)
i
, θ
(
k
−)
j
λ
−
T
λ
−
N
s
=
=
K(
)
(
.
)
(
k
)
ij
as
Now compute the weights w
(
k
)
ij
(
k
)
ij
(
k
)
ij
w
=
K
loc
l
K
stat
s
(
k
)
i
(
k
)
i
,...,w
(
k
)
in
and specify the local model by W
=
w
.