Graphics Reference
In-Depth Information
allpoints x,andonlyonebandwidth h hastobespecified.Inthelocalmodelselection
approach, the bandwidth h may vary with the point x. See Fan et al. (
) for more
details.
Weemployarelatedbutmoregeneralapproach.Weconsiderafamilyoflocalizing
models,one perdesign point X
i
,and denote them as W
i
.
Every W
i
is built inan iterative data-driven way,andits supportmayvary frompoint
to point. he method used to construct such localizing schemes is discussed in the
next section.
W
X
i
w
i
,...,w
in
=
(
)=
Structural Adaptation
8.2
Let us assume that for each design point X
i
the regression function θ can be well
approximated by a constant within a local vicinity U
(
X
i
)
containing X
i
.hisserves
as our structural assumption.
Our estimation problem can now be viewed as consisting of two parts. In order
to e
ciently estimate the function θ in a design point X
i
we need to describe a local
model,i.e.,to assign weights W
(
X
i
)=
w
i
,...,w
in
.If we knew the neighborhood
U
(
X
i
)
viaanoraclewewoulddefinethelocalweightsasw
ij
=
w
j
(
X
i
)=
I
X
j
U
(
X
i
)
and use these weights to estimate θ
are
unknown, the assignments will have to depend on the information on θ that we can
extract from the observed data. If we have good estimates θ
j
(
X
i
)
.However,sinceθ and therefore U
(
X
i
)
θ
=
(
X
j
)
of θ
(
X
j
)
,we
can use this information to infer the set U
(
X
i
)
by testing the hypothesis
H
θ
(
X
j
)=
θ
(
X
i
)
(
.
)
Aweightw
ij
can be assigned based on the value of a test statistic T
ij
, assigning zero
weights if θ
j
and θ
i
aresignificantlydifferent.hisprovidesuswithasetofweights
W
that determines a local model in X
i
.
Given the local model we can then estimate our function θ at each design point
X
i
by (
.
).
We utilize both steps in an iterative procedure.We start with a very local modelat
each point X
i
given by weights
X
i
w
i
,...,w
in
(
)=
()
ij
()
ij
()
ij
h
()
.
w
=
K
loc
(
l
)
with
l
=
X
i
−
X
j
(
.
)
he initial bandwidth h
()
is chosen very small. K
loc
is a kernel function supported
on
()
i
of radius h
()
centered on X
i
.We
then iterate two steps: estimation and local model refinement. In the kth iteration
new weights are generated as
[−
,
]
; i.e., weights vanish outside a ball U
(
k
)
ij
(
k
)
ij
(
k
)
ij
w
K
loc
K
stat
with
(
.
)
=
(
l
)
(
s
)
(
k
)
ij
(
k
)
ij
(
k
)
ij
h
(
k
)
l
=
X
i
−
X
j
and
s
=
T
λ.
(
.
)
