Graphics Reference
In-Depth Information
Akeyelementinthesuccessofdataanalysisisthestrongcontributionofvisualiza-
tion:itexploitsthehumancapabilityto perceive the
-D space.Ontheotherhand,the
role of the geometric approach in mathematics has a centuries-old story. Let us take
into account that many theorems were first enunciated in geometric notation and
mathematically formalized many, perhaps hundreds of years later. he Pythagorean
theorem is a well-known example.
Ourperceptionofthereal worldistheresultofageometric spacecharacterizedby
orthogonal axes, the concept of distance, and the effects of light. he combination of
thefirsttwoelements defines ametric space.Cartesian spacespermitonetovisualize
positions of a set of dimensionless points. Exploiting capabilities of current graphics
cards acting on brightness, points are enriched by point markers, characterized by
different sizes, shapes, and colors, that add information, helping the user interpret
results more easily and quickly.
Our mathematics based on the decimal system is clearly the result of having ten
fingers; similarly, it is obvious that our geometry, Euclidean geometry, is based on
a system of orthogonal axes due to our perception of the horizon line. As the binary
and hexadecimal numerical systems represent possible alternatives to the decimal
system, similarly there exist different geometries based on nonorthogonal systems
whereparallellinesconvergeinafinitespace.However,evenifalternative geometries
exist, Euclidean geometry remains the only geometry that we apply in the solution
of real-world problems.
heconcepts of far and close are native concepts. Itis not necessary tobe amathe-
matician to understand them. Distance represents the measure of closeness in space.
his contribution will introduce the concept of factorial space and of dendro-
grams;itintendstofurnishguidelinesforgiving acorrectrepresentation ofdisplayed
data. It will also show how it is possible to obtain enhanced representation where,
thanks to modern graphics cards, it is possible to obtain millions of colors, trans-
parencies, and man-machine interactions.
The Geometric Approach
to the Statistical Analysis
4.2
here are several reasons to revisit principal coordinates and dendrograms. When
these methods appeared
years ago, the capabilities of computer devices were very
poor. his lack of capability in obtaining satisfactory graphical representations led
to ingenious and original solutions. Text ASCII printers were used to draw factorial
plans using the (base) ASCII character set, and
-column printers were preferred
to
-column printers to obtain wider and clearer representations. In some papers
from that time we also find hand-drawn factorial plans and dendrograms. Never-
theless, the data analysis approach prospered despite these technical di
culties and
even more data analysis methods appeared in the specialized literature. he reasons
for this success, presumably, lie behind the very easy interpretation keys. In fact, in-
terpretation is based on the notion of distance, which is a human concept.
