Graphics Reference
In-Depth Information
Smoothing in One Dimension
9.2
here are many ways in which a nonparametric regression curve can be constructed.
heseincludeorthogonalbasisfunctions,awidevarietyofapproachesbasedinsplines
and,morerecently,methodsbasedonwavelets.Whilethereareimportantdifferences
between theseapproachesfromatechnical perspective,theparticular choiceof tech-
nique for the construction of a nonparametric regression curve is less important in
a graphical setting. he principal issue is how an estimate can be used to best effect,
rather than the details of its construction.
For convenience, this chapter will make use of local linear methods of nonpara-
metric regression. hese have the advantages of being simple to explain and easy
to implement, as well as having theoretical properties which are amenable to rela-
tively straightforward analysis. A further advantage lies in the link with the chap-
teronsmoothingbyLoader(
)inanearlierComputational Statistics Handbook,
where many of the technical details can be found. he basic ideas of the method are
describedbelow,buttheemphasisthereaterisontheuseofthetechniquetoenhance
graphical displays.
With regression data of the form
,wherey denotes a re-
sponse variable and x a covariate, a general prescription of a model is provided by
(
x
i
, y
i
)
i
=
,...,n
y
i
=
m
(
x
i
)+
ε
i
,
where m denotesaregressionfunctionandthe ε
i
denoteindependenterrors.Apara-
metricformfor m caneasilybefittedbythemethodofleastsquares.Anonparametric
estimate of m can beconstructed simplybyfitting aparametric modellocally. Forex-
ample, an estimate of m at the covariate value x arises fromminimising the weighted
least squares
n
i
=
w
y
i
−
α
−
β
(
x
i
−
x
)
(
x
i
−
x; h
)
(
.
)
over α and β.heestimate m
isthefittedvalueoftheregressionatx,namelyα.
By choosing the weight function w
(
x
)
,thelinear
regression is fitted locally as substantial weight is placed only on those observations
near x. Unless otherwise noted, this chapter will adopt a weight function w,whichis
a normal density centred on
with standard deviation h. he parameter h controls
the width of the weight function and therefore the extent of its local influence. his
in turn dictates the degree ofsmoothness ofthe estimate. Forthis reason, h is usually
referred to as the smoothing parameter or bandwidth.
Computationally, the solution of the weighted least squares (
.
) is straightfor-
ward, leading to an estimate of the form m
(
x
i
−
x; h
)
to be decreasing in
x
i
−
x
v
y, where the vector v is a simple
function of x,thecovariatevaluesx
i
and the weights w
(
x
)=
(
x
i
−
x; h
)
.Specifically,the
ith element of v is
s
x; h
s
x; h
x
i
x
w
x
i
x; h
n
(
)−
(
)(
−
)
(
−
)
v
i
=
,
s
(
x; h
)
s
(
x; h
)−
s
(
x; h
)
