Graphics Reference
In-Depth Information
Visualizing Relationships Among Functional Data
5.4.2
Ater examining each variable individually, the next step in exploratory data analy-
sis is typically to investigate relationships across several variables. For two numerical
variables, thisisotenaccomplishedwiththehelpofscatterplots. Onewayofgeneral-
izing the traditional scatterplot to the functional setting is, again, to drawa sequence
of pointwise scatterplots. Figure . shows scatterplots at days , and for the auc-
tion price versus the opening bid (on a log scale). We can see that the relationship
between the two variables changes over the course of time. While there is a strong
positive effect at the beginning of the auction (let panel), the magnitude of the ef-
fect decreases at day (middle panel), and there is barely any effect at all (possibly
even a slightly negative effect) at the end of the auction (right panel). his suggests
that the relationship between the opening bid and the auction price can be mod-
eled well using a time-varying coe cient model. Of course, one aspect that remains
unexplored in this pointwise approach is a possible three-way interaction between
the opening bid, the price and the time. Such an interaction could be detected using
athree-dimensionalscatterplot.However,astheletpanelinFig. . illustrates,three-
dimensional graphs have the disadvantage that they are oten cluttered and di cult
to read. We can improve the interpretability by using smoothing. he right panel in
Figure . . Relationships among functional objects: scatterplots of (log) price vs. (log) opening bid at
days , and for a sample of eBay online auctions. he solid gray line corresponds to a cubic
smoothing spline with three degrees of freedom
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