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a grid (e.g., for images), the initial bandwidth h () is chosen as the distance between
neighboring pixels.hebandwidth isincreased atereachiteration byadefaultfactor
c h
. d .
=
An Illustrative Univariate Example
8.3
We use a simple example to illustrate the behavior of the algorithm. he data in the
upper let of Fig. . follow a univariate regression model
Y i
=
θ
(
X i
)+
ε i .
( . )
he unknown parameter (i.e., the regression function θ) is piecewise constant, the
errors ε i are i.i.d. N
,
,andtheobservedX i
i form a univariate grid. In this
(
)
=
situation the statistical penalty takes the form
( k −)
i
σ λ
N
θ
θ
( k )
ij
( k −)
i
( k −)
j
s
=
(
)
( . )
where σ
denotes the variance of the errors. A robust estimate of the variance is
obtained from the data using the interquartile range (IQR) as
=
σ
=(
IQR
(
Y i +
Y i
)
.
)
( . )
i = ,...,n
andthisisusedasaplug-inforσ .hepropagationcondition( . )suggestsavalue
of λ
, disabling the adaptive control step.
We have four regions, differing in size and contrast, where the function θ is con-
stant. heregression function isdisplayedasablacklineinthe upperright ofFig. . .
helower part of Fig. . illustrates the evolution of the weights w ij as the number
of iterations increases. he horizontal and vertical axes correspond to indices i and
j, respectively. he upper row provides K loc
=
. . We employ a value of τ
=
( k )
ij
(
l
)
for iterations k
=
(h
=
), k
=
(h
=
), k
=
(h
=
)and k
=
(h
=
).hecentralrowshowsthecorresponding
( k )
ij
values K stat
(
s
)
. he grayscale ranges from black for to white for . he weights
( k )
ij (lower row) used in the algorithm are the products of both terms.
heletcolumncorrespondstotheinitializationstep.Herethelocationpenaltyef-
fectivelyrestrictsthelocalmodelin X i tothepoint X i itself.Ifcomputed,thestochas-
tic penalty would contain some weak information about the structure of the regres-
sion function. When we reach step k
w
, the location penalty allows for positive
weights for up to observations, and therefore less variable estimates. At this stage
the test ( . ) shows a significant difference between estimates at points within the
third homogeneous interval and estimates at locations outside this interval. his is
reflected in the statistical penalty andtherefore the weights. In step k
=
the second
interval is also clearly identified. he last column, referring to the rd iteration and
a final bandwidth of h
=
=
, shows the final situation, where the statistical penalty
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