Graphics Reference
In-Depth Information
Introduction
5.1
Technological advances in the measurement, collection, and storage of data have led
tomoreandmorecomplexdatastructures.Examples ofsuchstructuresincludemea-
surements of the behavior of individuals over time, digitized two- or three-dimen-
sional images of the brain, and recordings of three- or even four-dimensional move-
ments of objects traveling through space and time. Such data, although recorded in
a discrete fashion, are usually thought of as continuous objects that are represented
by functional relationships. his gave rise to functional data analysis (FDA), which
was made popular by the monographs of Ramsay and Silverman (
,
), where
thecenterofinterestisasetofcurves,shapes,objects,or,moregenerally,asetof func-
tional observations, in contrast to classical statistics where interest centers on a set of
data vectors. In that sense, functional data is not only different from the data struc-
ture studied in classical statistics, but it actually generalizes it. Many of these new
data structures require new statistical methods to unveil the information that they
carry.
here are many examples of functional data. he year-round temperature at
a weather station can be thought of as a continuous curve, starting in January and
endinginDecember,wheretheamplitude ofthecurvesignifies thetemperaturelevel
ateachdayorateachhour.Acollection oftemperaturecurvesfromdifferentweather
stations is then a set of functional data. Similarly, the price during an online auction
of a certain product can be represented by a curve, and a sample of multiple auction
price curves for the same product is then a set of functional objects. Alternatively,
the digitized image ofacar passing througha highwaytoll booth can bedescribed by
a two-dimensional curve measuring the pixel color or intensity of that image. A col-
lection of image curves from all of the cars passing through the toll booth during
asingledaycanthenbeconsideredtobeasetoffunctionaldata.Lastly,themove-
mentofapersonthroughtimeandspacecan bedescribedbyafour-(orevenhigher)
dimensional hyperplane in x-, y-, z- and time coordinates. he collection of all such
hyperplanes from people passing through the same space is again a set of functional
data.
Datavisualizationisanimportantpartofanystatisticalanalysisanditservesmany
different purposes. Visualization is useful for understanding the general structure
and nature of the data, such as the types of variables contained in the data (categori-
cal, numerical, text, etc.), their value ranges, and the balance between them. Visual-
ization isusefulfordetecting missing data, anditcan also aidinpinpointing extreme
observations andoutliers.Moreover,unknown trendsandpatterns inthedata areof-
tenuncoveredwiththehelpofvisualization. Ateridentifying suchpatterns, theycan
then beinvestigated moreformally using statistical models.heexactnature ofthese
models (e.g., linear vs. log-linear) is again oten based on insight learned from visu-
alization. Finally, model assumptions are typically verified through the visualization
of residuals and other model-related variables.
While visualization is an important step in comprehending any set of data, differ-
enttypesofdatarequiredifferenttypesofvisualization.Takeforinstancetheexample
