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the values
n to estimate α, and the remaining matrices construct the component
estimates as m
j
P
j
y.hestandarderrorsof m
j
at each observed covariate value
are then available as the square roots of the diagonal entries of P
j
P
=
j
σ
,wherethe
estimate of error variance σ
can be constructed as RSS
df and df denotes the de-
grees of freedom for error. Hastie and Tibshirani give the details on how that can be
constructed, again by analogy with its characterisation in linear models.
helet-handpanelsofFig.
.
showthe resultsoffitting anadditive modelinlati-
tudeandlongitude tothe Reef data. hetoptwopanels showthe estimated functions
for each covariate, together with the partial residuals, while the bottom panel shows
the resulting surface. he additive nature of the surface is apparent as slices across
latitude always show the same shape of longitude effect and vice versa. (Notice that
colour has been used here simply to emphasise the heights of the surface at different
locations.)
he level of smoothing was determined by setting the number of approximate de-
grees of freedom to four for each covariate. An alternative approach, advocated by
Wood(
),applies cross-validation asan automatic methodof smoothing param-
eter selection at each iteration of the estimation process defined by (
.
). he effects
ofthis strategy onthe Reef data are displayedin theright-hand panels ofFig.
.
.he
estimate of the longitude effect is very similar but a very large smoothing parameter
has been selected for latitude, leading to a linear estimate. Based on the earlier es-
timate for the latitude effect, using four degrees of freedom, a linear model for this
term is a reasonable strategy to adopt. his leads to a semiparametric model, where
one component is linear and the other is nonparametric. his hybrid approach takes
advantage of the strength of parametric estimation where model components of this
type are justified.
For a further example of additive models, the Clyde data are revisited. When wa-
tersamples arecollected, avariety of measurements are madeon these. hisincludes
temperature and salinity and it is interesting to explore the extent to which the DO
level in the samples can be explained by these physical parameters. Clearly, tempera-
turehasastrongrelationship withthedayoftheyear.Infact,salinity alsohasastrong
relationship withthisvariable, asitmeasurestheextenttowhichfreshwaterfromthe
river and salt water from the sea mix together, and this has a strong seasonal compo-
nent related to the volume of river flow.It is therefore inappropriate touse all three of
these variables inamodelforDO.AsHastie andTibshirani (
)observe,the effect
of concurvity, where explanatory variables have strong curved relationships, creates
di
cultiesanalogoustothoseassociatedwithcollinearityinalinearmodel.hethree
explanatory variables to be considered are therefore year, temperature and salinity,
with the latter variable on a log(salinity +
) scale to reduce substantial skewness.
he top two panels of Fig.
.
show nonparametric curve estimates based on re-
gressions of DO on temperature and salinity separately. helowerthree panels of the
figure show the effects of fitting an additive model which expresses the DO values as
asumofyear,temperatureandsalinitycomponentssimultaneously.Onestrikingfea-
ture expressed in the partial residuals is the substantial reduction in the variability of
the data compared to that displayed in the marginal scatterplots, as each component
