comparison of graphical data analysis methods - Hochschule ...

esulting structure of the data table together with the domain structures for each attribute
represents the formalized meaning of the data. This structure is called here the data structure.
Each data analysis method represents these data structures in further structures and the
graphical data analysis methods finally lead to a representation which shows some image of
the original data structure in a graphical structure, e.g. the plane or the 3-space. But what is
„the plane“ or „the 3-space“? There are many structures on the set R 2 (or on R 3 ) which may
be used to represent data, e.g. the Euclidean distance, the structure of a real vector space or of
an affine or projective geometry, the closure system of the convex sets, orders of different
types or many other possibilities. This graphical structure should be described also in detail to
study the data representations from the original data structure into the final graphical structure.
What does it mean to „visualize structures of the data“? Looking at some graphical
representation of the data we hope to understand in terms of the graphical concepts some
structures in the original data. Hence the data representations from the original data into the
final graphical structure should preserve at least some parts of the information given in the
original data.
Therefore the main problem in the graphical representation of multidimensional data and
more generally in all knowledge representation methods is the Information Representation
Problem IRP: Given a class R of representations from a knowledge system K 1 into a
knowledge system K 2 : Which „informations about K 1 “ are preserved by the representations of
R, i.e. which parts of the knowledge system K 1 can be reconstructed from an r-image of K 1 in
K 2 and the knowledge that r is a representation of R?
Example: In many graphical methods this problem arises for a class R of projections in the
form: „What does the distance between two points in the plane mean for the two objects
represented by the two points?“.
The Information Representation Problem has been studied e.g. in Measurement Theory
(D.H.Krantz, R.D. Luce, P. Suppes, A.Tversky, Foundations of Measurement, Vol. I-III),
where the knowledge systems are described by relational structures. Representations
preserving the „whole“ structure of the first relational structure are defined as „fundamental
measurements“ or „total homomorphisms“. The IRP has been studied also in Formal Concept
Analysis (cf. GANTER, WILLE 1989), where the knowledge systems are formal contexts (i.e.
triples (G,M,I) of sets, where I is a binary relation between the set of objects G and the set of
attributes M, i.e. I ⊆ G×M) and the representations are scale measures, i.e. functions from the
set of objects of the first context into the set of objects of the second context such that the
preimage of every extent of the second context is an extent of the first context. Full scale
measures, i.e. scale measures with the property that every extent of the first context is a
preimage of an extent of the second context, are representations which preserve the whole
conceptual structure of the first context.
2 Interpretation principles of graphical data analysis methods
Each graphical data analysis method interprets the given data in a way which usually depends
heavily on some principles of knowledge representation. These principles influence the design
of the experiments and the choice of measurements. From the large number of these
principles we select some which are central for the description of graphical data analysis
methods. A very important one is the interpretation principle of distance: The objects under
consideration are viewed as members of a usually not explicitly defined metric space. To
construct such a metric space from the given many-valued context the values of each attribute
are interpreted as elements of a metric space, hence the n-tuple of all measurements for a
2