Indicators and Partial Orders – An Introduction
xi
Fig. 1 Hasse diagram of
Germany, years 2008–2015.
Details in a publication,
submitted
2015
2011
2012
2010
2013
2014
2009
2008
two elements are downwards and upwards, respectively, related to only one other
graph theoretical neighbored element. Such posets are called lattices and a special
realization is the formal concept analysis, deeply studied by the school of Wille
(Ganter and Wille 1986; Ganter 1987; Ganter and Wille 1996) and Kerber (2017).
The resulting lattices, formal concept lattices, are powerful tools in the analysis
of multi-indicator systems, especially when the indicators can only take discrete
numbers. The extension of the so-called formal concept analysis to indicators,
having continuous data in concept, bears additional theoretical difficulties; see, e.g.,
(Kerber 2017).
When data are metric data, the obvious question is, how to deal which such data
and what is the role of order relations compared to powerful statistical methods, such
as correlation or regression analyses, principal component and cluster analyses, just
to mention a few tools most often used in (multivariate) statistics.
When data are measured, then automatically data uncertainty comes into play.
By comparing partial order and (conventional) statistical tools it should initially
be made clear that partial order as a method to analyse data clearly belongs to
statistics. So why is a discussion needed? The reason is that multivariate statistics is
commonly associated with tools, which are already exemplified above. The aspect
of evaluation, especially evaluation in multi-indicator systems makes partial order
an important tool in this respect. Whereas applying conventional tools, a ‘good’ or
‘bad’ within a data set is not known, and partial ordering is specifically adapted
to that. Applied partial order methodology, together with the analysis of the graph
theoretical structure could be a relevant tool in decision making, operation research
and, to some degree optimization.
The role of uncertainty in data analysis by partial ordering goes back to papers
of Sørensen et al. (1998, 2000). Data, continuous in concept, cannot be considered
as ideally suitable items for partial ordering. There are two main reasons: (i) The
aforementioned role of uncertainty which often arises when data are measured. (ii)
The information due to distances is lost. Both aspects can be methodologically
xi
Fig. 1 Hasse diagram of
Germany, years 2008–2015.
Details in a publication,
submitted
2015
2011
2012
2010
2013
2014
2009
2008
two elements are downwards and upwards, respectively, related to only one other
graph theoretical neighbored element. Such posets are called lattices and a special
realization is the formal concept analysis, deeply studied by the school of Wille
(Ganter and Wille 1986; Ganter 1987; Ganter and Wille 1996) and Kerber (2017).
The resulting lattices, formal concept lattices, are powerful tools in the analysis
of multi-indicator systems, especially when the indicators can only take discrete
numbers. The extension of the so-called formal concept analysis to indicators,
having continuous data in concept, bears additional theoretical difficulties; see, e.g.,
(Kerber 2017).
When data are metric data, the obvious question is, how to deal which such data
and what is the role of order relations compared to powerful statistical methods, such
as correlation or regression analyses, principal component and cluster analyses, just
to mention a few tools most often used in (multivariate) statistics.
When data are measured, then automatically data uncertainty comes into play.
By comparing partial order and (conventional) statistical tools it should initially
be made clear that partial order as a method to analyse data clearly belongs to
statistics. So why is a discussion needed? The reason is that multivariate statistics is
commonly associated with tools, which are already exemplified above. The aspect
of evaluation, especially evaluation in multi-indicator systems makes partial order
an important tool in this respect. Whereas applying conventional tools, a ‘good’ or
‘bad’ within a data set is not known, and partial ordering is specifically adapted
to that. Applied partial order methodology, together with the analysis of the graph
theoretical structure could be a relevant tool in decision making, operation research
and, to some degree optimization.
The role of uncertainty in data analysis by partial ordering goes back to papers
of Sørensen et al. (1998, 2000). Data, continuous in concept, cannot be considered
as ideally suitable items for partial ordering. There are two main reasons: (i) The
aforementioned role of uncertainty which often arises when data are measured. (ii)
The information due to distances is lost. Both aspects can be methodologically