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he actual choice of the smoothing parameters is oten driven by the context. In
our application, we pick the location and number of knots to reflect the bid-arrival
distribution, which is densest for the last day, and in particular for the last few mo-
ments of the auction. he choice of p depends, among other things, on whether
higher order derivatives of the curve are also desired. he value of the penalty term
λ is chosen by inspecting the resulting functional objects in order to ensure satisfac-
tory results (Ramsay and Silverman,
).An alternative approach is to pick λ soas
to balance the smoothness and the data fit (Wang et al.,
). In particular, one can
measure the degree of smoothness of the spline via its distance tothe smoothest pos-
sible fit, a straight line through the data. he data fit, on the other hand, can be mea-
suredasthedistancebetween thesplineandtheactual data points.Onethenchooses
avalueofλ that balances the two. We investigate and compare different smoothing
parameters for our dataset in what follows.
heprocessofmovingfromobserveddatatofunctionaldataisthenasfollows.For
asetofn functional objects, let t
ij
denotethe time ofthe jth observation
(
j
n
i
)
of the ith object
(
i
n
)
,andlety
ij
=
y
(
t
ij
)
denote the corresponding mea-
surements. Let f
i
denote the penalized smoothing spline fitted to y
i
,...,y
in
i
.
hen, functional data analysis is performed on the continuous curves f
i
(
t
)
rather
than on the noisy observations y
i
,...,y
in
i
. hat is, ater creating the functional ob-
jects f
i
(
t
)
,theobserveddatay
i
,...,y
in
i
are discarded and subsequent modeling,
estimation and inference are based on the functional objects only.
One important implication of this practice is that any error or inaccuracy in the
smoothingstepwillpropagateintotheinferencesandconclusionsmadebasedonthe
functional model. To make matters worse, the observed data are discarded ater the
functional data are created and are therefore oten hard to retrieve, and any violation
of the functional model is confounded with the error at the smoothing step. hat
is, it is hard to know whether a model violation is due to model misspecification or
due to anomalies at the smoothing step. For this reason, it is important to carefully
monitor the functional object recovery process and to detect inaccuracies early in
the process using appropriate tools. Although measures for evaluating the goodness
of fit of the functional object to the observed data are available (such as those based
on the residual sums of squares, or criteria that include the roughness penalty), it is
unwise to rely on these measures alone, and visualization becomes an indispensable
tool in the process.
Consider Figs.
.
-
.
for illustration. he figures compare the functional objects
recovered forthree different smoothing scenarios. Specifically, for bidding data from
different eBay online auctions, Fig.
.
shows the functional objects obtained from
penalized smoothing splines using a spline order p
(
t
)
=
andasmallsmoothingpa-
rameter λ
=
.
.Figure
.
on the other hand corresponds to the same spline order
(p
=
) but a larger smoothing parameter (λ
=
).InFig.
.
weuseasplineorder
p
,andadata preprocessingstepvia interpolation.
heexactdetails ofthesmoothing arenotofinteresthereand canbefoundelsewhere
Jank and Shmueli (
). What is of interest here though is the fact that Figs.
.
-
.
correspond to three different approaches to recovering functional objects from the
same data. he researcher could have taken either one of these three approaches and
=
,asmoothing parameter λ
=
