xii
Indicators and Partial Orders – An Introduction
handled within the framework of partial ordering, however at the price of its
elegance. Furthermore, if the weights in linear sums of uncertain indicator values
are not sharp, application of partial ordering, as we know it today, comes to its
limits. Studies in this direction are for now just landmarks on a long way.
For readers interested in the mathematical aspect of partial order, we recommend
(Maggino 2017; Trotter 1992; Neggers and Kim 1998; Schröder 2003; Davey and
Priestley 2002).
This book, Indicators and their Analysis in different Scientific Fields reports
recent developments in the field of partial order applications. Some chapters are
based on presentations at the International Conference on Partial Orders in Applied
Sciences in Neuchatel, October 2018. This conference series was initialized 1998
(cf. Table 2) and is a forum for the scientific community with special interests in the
theory and application of indicators.
In the 18 chapters, a variety of new developments within the area of partial
ordering can be found.
Indicators and Theoretical Developments
As mentioned above, indicators play an increasing role in characterizing complex
systems and in decision problems. Indicators are necessary to understand system
behaviour. Hence, several chapters focus on the various aspects of indicators,
addressing subjects like scaling level, relevance and the role of the inherent
characteristic of partial orders, i.e., the incomparability (See J. Wittmann, p. 3,
F. Maggino et al., p. 17). Further chapters discuss the functionality of indicators,
the workflow for building indicators, the structure of complex indicators and the
sensitivity of indicator values, as well as assessment of inhomogeneous indicatorbased typologies through the reverse clustering approach (See J. Owsinski et al.,
p. 31), using a typology of spatial units of Polish municipalities as an illustrative
example.
Indicator values often are considered as continuous in concept, thus the evaluation and exploration is of some fuzzy character. This aspect is considered in two
chapters (see pp. 83–101) where a strict generalization is given central importance.
Evaluations using parameters can usually be considered as sets over lattices. These
two chapters (See A. Kerber and R. Bruggemann, p. 83, and R. Bruggemann and
Kerber, p. 91) are devoted to this approach, whereby the theoretical concept is
exemplified in a study of heavy metals and sulphur pollution along the southern
part of river Rhine.
Very often a strict linear order is wanted, which in the case of multiple indicators
typically is obtained as a result of aggregation of indicators, e.g., leading to a
weighted sum. Although attractive due to its simplicity, the disadvantages are, e.g.,
that potential conflicts expressed by the values of single indicators are suppressed.
A chapter is devoted to the idea of combining the advantages of linearly weighted
sums and partial order theory in order to relax the requirements for a strict linear
Indicators and Partial Orders – An Introduction
handled within the framework of partial ordering, however at the price of its
elegance. Furthermore, if the weights in linear sums of uncertain indicator values
are not sharp, application of partial ordering, as we know it today, comes to its
limits. Studies in this direction are for now just landmarks on a long way.
For readers interested in the mathematical aspect of partial order, we recommend
(Maggino 2017; Trotter 1992; Neggers and Kim 1998; Schröder 2003; Davey and
Priestley 2002).
This book, Indicators and their Analysis in different Scientific Fields reports
recent developments in the field of partial order applications. Some chapters are
based on presentations at the International Conference on Partial Orders in Applied
Sciences in Neuchatel, October 2018. This conference series was initialized 1998
(cf. Table 2) and is a forum for the scientific community with special interests in the
theory and application of indicators.
In the 18 chapters, a variety of new developments within the area of partial
ordering can be found.
Indicators and Theoretical Developments
As mentioned above, indicators play an increasing role in characterizing complex
systems and in decision problems. Indicators are necessary to understand system
behaviour. Hence, several chapters focus on the various aspects of indicators,
addressing subjects like scaling level, relevance and the role of the inherent
characteristic of partial orders, i.e., the incomparability (See J. Wittmann, p. 3,
F. Maggino et al., p. 17). Further chapters discuss the functionality of indicators,
the workflow for building indicators, the structure of complex indicators and the
sensitivity of indicator values, as well as assessment of inhomogeneous indicatorbased typologies through the reverse clustering approach (See J. Owsinski et al.,
p. 31), using a typology of spatial units of Polish municipalities as an illustrative
example.
Indicator values often are considered as continuous in concept, thus the evaluation and exploration is of some fuzzy character. This aspect is considered in two
chapters (see pp. 83–101) where a strict generalization is given central importance.
Evaluations using parameters can usually be considered as sets over lattices. These
two chapters (See A. Kerber and R. Bruggemann, p. 83, and R. Bruggemann and
Kerber, p. 91) are devoted to this approach, whereby the theoretical concept is
exemplified in a study of heavy metals and sulphur pollution along the southern
part of river Rhine.
Very often a strict linear order is wanted, which in the case of multiple indicators
typically is obtained as a result of aggregation of indicators, e.g., leading to a
weighted sum. Although attractive due to its simplicity, the disadvantages are, e.g.,
that potential conflicts expressed by the values of single indicators are suppressed.
A chapter is devoted to the idea of combining the advantages of linearly weighted
sums and partial order theory in order to relax the requirements for a strict linear