Indicators in the Framework of Partial Order
25
by Fattore 2016, Fattore et al. 2011 allows powerful conclusions and should be
called the value-based approach, because the realization of a profile by objects
does not play a leading role.
2. Metric data: The set Q(k) is not finite and
Q(k), k=,1, . . . ,m, is an infinite
set. Although a powerful approach exists to analyse this infinite set by methods
described by Kerber 2017, Kerber and Bruggemann (this book, and 2015),
Bruggemann and Kerber 2018, in practical applications the set of objects is
considered, together with their profiles. This approach is focusing on objects,
which have a profile, corresponding to the set of indicators and should therefore
be called the object-based approach. As in biology, chemistry and physics
measured data are the basis for research, the object-based approach is usually
applied. See for instance Carlsen and Bruggemann 2017, Carlsen 2018.
2.2 Interplay: Indicators and Objects
The interplay of indicators and the objects and thus the role of graph theory, is more
easily seen in the object-based-approach, which will be described in more details in
the following.
In the field of indicators applied to objects the graph -theoretical relations can
be
• Between indicators
• Between objects (being described and quantified by indicators) and
• Between objects and indicators.
In many cases indicators are applied in the framework of ranking studies, hence
the relations studied by graph theory are directed relations; and the graphs directed
graphs. Ranking has to do with ordering, hence the directed relations should obey
axioms, namely those of partial order theory: They have to be reflexive (an object
must be comparable with itself), antisymmetric, if an object x is better than object
y then the reverse can only be true if the two objects are identical (often in partial
order theory this axiom is relaxed by replacing ‘identical’ with ‘equivalent’) and
finally transitive, if object x is better than object y, and object y is better then object
z, then this implies that object x is better than object z.
The study of directed graph based on order relations leads to a certain type of
directed graphs, namely to acyclic triangle free directed graphs, often called Hasse
diagrams (or simply line diagrams). Hasse diagrams are an appropriate visualization
of partially ordered sets (but not the only ones). Although the axioms of order theory
sound plausible, they impose a very important additional requirement on indicators
(if applied in ranking studies): Any single indicator implies an order among objects.
A consequence is that even if indicators induce an order among objects they may
be counter current. A drastic example is provided by environmental chemistry: A
chemical may be considered as hazardous, if its tendency to accumulate in biota is
high. This accumulation tendency can be measured or estimated Log KOW. Another
25
by Fattore 2016, Fattore et al. 2011 allows powerful conclusions and should be
called the value-based approach, because the realization of a profile by objects
does not play a leading role.
2. Metric data: The set Q(k) is not finite and
Q(k), k=,1, . . . ,m, is an infinite
set. Although a powerful approach exists to analyse this infinite set by methods
described by Kerber 2017, Kerber and Bruggemann (this book, and 2015),
Bruggemann and Kerber 2018, in practical applications the set of objects is
considered, together with their profiles. This approach is focusing on objects,
which have a profile, corresponding to the set of indicators and should therefore
be called the object-based approach. As in biology, chemistry and physics
measured data are the basis for research, the object-based approach is usually
applied. See for instance Carlsen and Bruggemann 2017, Carlsen 2018.
2.2 Interplay: Indicators and Objects
The interplay of indicators and the objects and thus the role of graph theory, is more
easily seen in the object-based-approach, which will be described in more details in
the following.
In the field of indicators applied to objects the graph -theoretical relations can
be
• Between indicators
• Between objects (being described and quantified by indicators) and
• Between objects and indicators.
In many cases indicators are applied in the framework of ranking studies, hence
the relations studied by graph theory are directed relations; and the graphs directed
graphs. Ranking has to do with ordering, hence the directed relations should obey
axioms, namely those of partial order theory: They have to be reflexive (an object
must be comparable with itself), antisymmetric, if an object x is better than object
y then the reverse can only be true if the two objects are identical (often in partial
order theory this axiom is relaxed by replacing ‘identical’ with ‘equivalent’) and
finally transitive, if object x is better than object y, and object y is better then object
z, then this implies that object x is better than object z.
The study of directed graph based on order relations leads to a certain type of
directed graphs, namely to acyclic triangle free directed graphs, often called Hasse
diagrams (or simply line diagrams). Hasse diagrams are an appropriate visualization
of partially ordered sets (but not the only ones). Although the axioms of order theory
sound plausible, they impose a very important additional requirement on indicators
(if applied in ranking studies): Any single indicator implies an order among objects.
A consequence is that even if indicators induce an order among objects they may
be counter current. A drastic example is provided by environmental chemistry: A
chemical may be considered as hazardous, if its tendency to accumulate in biota is
high. This accumulation tendency can be measured or estimated Log KOW. Another