Evaluations as Sets over
Lattices – Application Point of View
Rainer Bruggemann and Adalbert Kerber
1 Introduction
This contribution is based on Bruggemann and Kerber 2018 and on the theory
developed there (in the following referred as BK). The essential point is that an
evaluation, as well as a subsequent exploration starts not always with an ordinal
data matrix or even with a data matrix built of binary data, but on data continuous
in concept, for example: data taken from [0,1] m , as described for m=3 in BK.
Therefore, it is of interest how far the powerful and elegant method of Formal
Concept Analysis (Ganter and Wille 1996) can be applied. Especially, without the
crucial and often arbitrary scaling method which was developed in the school of
Wille. The mathematical concept of an alternative method, avoiding the scaling
procedure (Ganter and Wille 1989) is explained in BK. Here the methods are
examined in more detail with respect to typical tasks in decision support systems and
taking the point of view of a statistician. I.e. a special focus was set on the point,
how the lattice theoretical method can be compared with conventional statistical
methods, as for example correlation analysis.
A technical problem arises: The lattice theoretical method in BK is selected as
one of the modules of the PyHasse software, see BK, because the method is basically
simple however manually performed, very tedious. (For the PyHasse software itself,
see e.g. Bruggemann et al. 2014). Therefore, the mathematical notation is to be
replaced by a notation, which also can be used in the programming language Python.
R. Bruggemann ()
Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany
e-mail: brg_home@web.de
A. Kerber
Department of Mathematics, University Bayreuth, Bayreuth, Germany
e-mail: kerber@uni-bayreuth.de
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. Bruggemann et al. (eds.), Measuring and Understanding Complex Phenomena,
https://doi.org/10.1007/978-3-030-59683-5_7
91
Lattices – Application Point of View
Rainer Bruggemann and Adalbert Kerber
1 Introduction
This contribution is based on Bruggemann and Kerber 2018 and on the theory
developed there (in the following referred as BK). The essential point is that an
evaluation, as well as a subsequent exploration starts not always with an ordinal
data matrix or even with a data matrix built of binary data, but on data continuous
in concept, for example: data taken from [0,1] m , as described for m=3 in BK.
Therefore, it is of interest how far the powerful and elegant method of Formal
Concept Analysis (Ganter and Wille 1996) can be applied. Especially, without the
crucial and often arbitrary scaling method which was developed in the school of
Wille. The mathematical concept of an alternative method, avoiding the scaling
procedure (Ganter and Wille 1989) is explained in BK. Here the methods are
examined in more detail with respect to typical tasks in decision support systems and
taking the point of view of a statistician. I.e. a special focus was set on the point,
how the lattice theoretical method can be compared with conventional statistical
methods, as for example correlation analysis.
A technical problem arises: The lattice theoretical method in BK is selected as
one of the modules of the PyHasse software, see BK, because the method is basically
simple however manually performed, very tedious. (For the PyHasse software itself,
see e.g. Bruggemann et al. 2014). Therefore, the mathematical notation is to be
replaced by a notation, which also can be used in the programming language Python.
R. Bruggemann ()
Leibniz-Institute of Freshwater Ecology and Inland Fisheries, Berlin, Germany
e-mail: brg_home@web.de
A. Kerber
Department of Mathematics, University Bayreuth, Bayreuth, Germany
e-mail: kerber@uni-bayreuth.de
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
R. Bruggemann et al. (eds.), Measuring and Understanding Complex Phenomena,
https://doi.org/10.1007/978-3-030-59683-5_7
91