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In-Depth Information
points that covertheentire range oftheseven-dayauction. heresultsobtained from
doing this can be seen in Fig.
.
. Now the functional objects appear to be very rep-
resentative of the price evolution, much better than in the previous two approaches.
Equally importantly, therearenownomissingorpartial functional objects. Inference
based on the objects in Fig.
.
is likely to yield the most reliable insights about the
price evolution in online auctions.
he previous examples illustrate the importance of visualization at the object re-
covery stage. Although the causes of problems at this stage may oten be quite trivial
(e.g., unfortunate sotware default settings or poor parameter choices), they are typ-
ically hard to diagnose without the use of proper visualizations.
Visualizing Functional Observations
5.4
Visualizing Individual Objects and Their Dynamics
5.4.1
hefirststepinstatisticalanalysisistypicallytoscrutinizedatasummariesandgraphs.
Data summariesincludemeasuresofcentral tendency,variability, skewness,etc.Tra-
ditionally, summarystatistics arepresentedinnumerical form.However,inthefunc-
tional setting, eachsummarystatistic isactually afunctional object,suchasthe mean
function or the standard deviation function. Since there are usually no analytical,
closed-form representations of these functions, one resorts to graphical representa-
tions of the summary measures. helet panel inFig.
.
showsthe (pointwise) mean
pricecurve(solidthickline)together withthe
%pointwiseupperandlowerconfi-
dencecurves(brokenthicklines)forthe
auctionsfromSect.
.
.Wecomputethese
pointwise measures in the following way. For an equally spaced grid t
i
,we
compute the mean and standard deviation for the
auction prices at each grid point
t
i
. Weuse these two measures toconstruct
% upperand lower confidence bounds
at each grid point. By interpolating the results, we obtain the mean and confidence
curves in Fig.
.
. Notice that since we only consider
auctions in this example, one
can easily identify the minimum and maximum prices of all curves. In larger data
sets, one may also want to add a curve for the (pointwise) minimum and maximum,
respectively.
One of the main advantages of functional data analysis is that it allows for an esti-
mation of derivatives. he nonparametric approach to the recovery of the functional
object guarantees that local changes inthe data arewell-reflected, andyet the object's
smoothness properties also allow for a reliable estimation of partial derivatives. For
instance, setting m
[
,
]
in the penalty term in (
.
) guarantees smooth first and sec-
ond derivatives. Knowledge of the derivatives can result in an important advantage,
especially for applications that experience change. Take the online auction setting as
an example. While the price curve f
=
describes the exact position of the price at
any time point t, it does not reveal how fast the price is moving.Attributesthatwe
typically associate with a moving object are its velocity (or its speed)anditsaccelera-
tion. Velocity and acceleration can be computed via the first and second derivatives
(
t
)
