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means of selecting an appropriate degree of smoothness. However, any method of
automatic selection must be used carefully.
Intheinitial plotsoftheDOdatainFig. . itwasnotedthatthedayeffectisaperi-
odic one. his can easily be accommodated in the construction of a smooth estimate
by employing a periodic weight function. Since a linear model is not appropriate for
periodic data, a locally weighted mean offers a simple solution. A smooth estimate is
then available as the value of α which minimises the weighted least squares
n
i =
h cos
exp
y i
α
(
π
(
x i
x
)
)
.
his uses an unscaled von Mises density as a weight function, with a period of
days to allow for leap years. In order to allow the estimate to express shapes beyond
astandard trigonometric pattern, theapproximatedegreesoffreedomweresettothe
slightly highervalue of inconstructing the estimate of the seasonal effect inFig. . .
Nonparametric curve estimates are very useful as a means of highlighting the
potential shapes of underlying regression relationships. However, like any estimate
based on limitedinformation, they aresubject tothe effects of variability. Indeed, the
flexibility which is the very motivation for a nonparametric approach will also in-
crease the sensitivity of the estimate to sampling variation in the data. It is therefore
important not only to examine curve estimates but also to examine their associated
variability.
he linear representation of the estimate as m
v y,wherev is a known vec-
tor as discussed above, means that its variance is readily available as var
(
x
)=
m
(
x
) =
n
σ ,whereσ is the common variance of the errors ε i . he calculation of
standard errors then requires an estimate of σ . Pursuing the analogy with the linear
modelsmentioned above leads to proposals suchas σ
i = v i
(
)
df,wheredf is an ap-
propriate value for the degrees of freedom for error. Other approaches are based on
local differencing. A particularly effective proposal of Gasser et al. ( ) is based on
the deviations of each observation from the linear interpolation between its neigh-
bours, as ε i
RSS
=
,under
the assumption that the data have been ordered by increasing x value. his leads to
the estimate
=
y i
a i y i
−(
a i
)
y i + ,wherea i
=(
x i +
x i
)(
x i +
x i
)
n
i =
ε i
σ
=
.
a i
n
+
+(
a i
)
n
he standard error of m
σ.
he let-hand plot of Fig. . shows the estimate of the seasonal effect in the Clyde
data, with curves indicating a distance of two standard errors from the estimated re-
gression function. Some care has to be exercised in the interpretation of this band.
he smoothing inherent in the construction of a nonparametric regression curve in-
evitably leads to bias, as discussed by Loader ( ). he band cannot therefore be
given a strict interpretation in terms of confidence. However, it does give a good in-
dication of the degree of variability in the estimate and so it is usually referred to as
a variability band.
i = v i
(
x
)
is then available as
(
)
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