100
R. Bruggemann and A. Kerber
As can be seen, the truth values of these implications are remarkably low,
showing that the transformation into binary values must be considered with care.
More details and the role of the selection of another t-norm can be inspected in
Bruggemann and Kerber 2018. Furthermore it is of interest to state that based on
binary transformed data Zn implies Cd, whereas here the implication Zn implies Cd
(however with a small truth value) is found.
5 Discussion
The example, discussed in Sect. 4.2 shows remarkably that accepting a continuous
range of data and implications based no more on a pure binary point of view will
also lead in general to truth values less 1. The main problematic point is that in the
binary case a “1” stands not for the maximum of a possible range [0, 1] but for the
presence of an indicator, so to say, in a close interpretation of the has-relation. In the
case of a continuous scale also the has-relation has truth values and the truth values
in consequence can also take values between 0 and 1. At which truth value can we
speak of a has-relation? Therefore, one of the main tasks in the future is, to provide
tools, how a coarsening of truth values can be performed. Is for example 0.476 for
the implication Cd ⇒ Zn big enough to establish contextually that Cd implies Zn?
The very simple and still pretty arbitrary example for a correlation analysis shows
that the generation of hypotheses can better be based on correlation coefficients than
on the truth values of implications, if the nature of the data allows a correlation
analysis (either Spearman or Pearson, just to denote two famous methods).
Nevertheless, the theoretical concept behind implications based on continuous
data opens another tool, which is worth to be examined further. The tasks for the
future are:
How can the theoretical framework, presented in BK and partially in this chapter
be embedded into the general Formal Concept Analysis, where the concepts,
i.e. pairs, for which (A’)’ = A is valid, play an important role in deriving
implications. Here up to now, there was no mentioning of concepts, although
the theoretical framework is general enough, to establish concepts even for data
continuous in concept. Nevertheless, first approaches indicate that even with
data continuous in concept, the requirement that the “second derivative” of A
(=A” = (A’)’), has to be equal to A itself works well as a method to find
concepts. The fact that the second derivative is to be formed, makes a posteriori
understandable, that in BK the formulation of A as a set [0,1] |Q| was selected: The
first derivative may deliver a tuple of length |Q| whose components are indeed
values taken from [0,1].
The number of implications in the binary case can be large. If -as shown in the
former section- truth values are to be accepted which are not equal 1, then still
the number of implications increases dramatically. Therefore the construction of a
basis, the Duquenne, Guigues basis (Duquenne 1987), is urgently needed. So, even
R. Bruggemann and A. Kerber
As can be seen, the truth values of these implications are remarkably low,
showing that the transformation into binary values must be considered with care.
More details and the role of the selection of another t-norm can be inspected in
Bruggemann and Kerber 2018. Furthermore it is of interest to state that based on
binary transformed data Zn implies Cd, whereas here the implication Zn implies Cd
(however with a small truth value) is found.
5 Discussion
The example, discussed in Sect. 4.2 shows remarkably that accepting a continuous
range of data and implications based no more on a pure binary point of view will
also lead in general to truth values less 1. The main problematic point is that in the
binary case a “1” stands not for the maximum of a possible range [0, 1] but for the
presence of an indicator, so to say, in a close interpretation of the has-relation. In the
case of a continuous scale also the has-relation has truth values and the truth values
in consequence can also take values between 0 and 1. At which truth value can we
speak of a has-relation? Therefore, one of the main tasks in the future is, to provide
tools, how a coarsening of truth values can be performed. Is for example 0.476 for
the implication Cd ⇒ Zn big enough to establish contextually that Cd implies Zn?
The very simple and still pretty arbitrary example for a correlation analysis shows
that the generation of hypotheses can better be based on correlation coefficients than
on the truth values of implications, if the nature of the data allows a correlation
analysis (either Spearman or Pearson, just to denote two famous methods).
Nevertheless, the theoretical concept behind implications based on continuous
data opens another tool, which is worth to be examined further. The tasks for the
future are:
How can the theoretical framework, presented in BK and partially in this chapter
be embedded into the general Formal Concept Analysis, where the concepts,
i.e. pairs, for which (A’)’ = A is valid, play an important role in deriving
implications. Here up to now, there was no mentioning of concepts, although
the theoretical framework is general enough, to establish concepts even for data
continuous in concept. Nevertheless, first approaches indicate that even with
data continuous in concept, the requirement that the “second derivative” of A
(=A” = (A’)’), has to be equal to A itself works well as a method to find
concepts. The fact that the second derivative is to be formed, makes a posteriori
understandable, that in BK the formulation of A as a set [0,1] |Q| was selected: The
first derivative may deliver a tuple of length |Q| whose components are indeed
values taken from [0,1].
The number of implications in the binary case can be large. If -as shown in the
former section- truth values are to be accepted which are not equal 1, then still
the number of implications increases dramatically. Therefore the construction of a
basis, the Duquenne, Guigues basis (Duquenne 1987), is urgently needed. So, even