Special Issue: Partially ordered sets
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Applied Statistics is certainly devoted to extract information from any kind of data.
Data which characterize objects of interest. Ranking, i.e. finding a complete order
among objects, belongs to the tasks of Applied Statistics. Not anticipating a historical
research, it seems, however, as if the task of ranking in terms of the theory of
partially ordered sets does not play that role in statistics which it should do, taking
into account the enormous multitude, diversity and popularity of ranking studies.
The idea is simple, not to say trivial. Let objects x1, …, xn be characterized by m data,
attributes, say qj(j = l, …, m), then xi1 < xi2 if and only if qj (xi1) = qj (xi2) for all j, and with at least one qj* with a strict inequality. It can be easily seen that this definition (being the basis of the ‘‘Hasse diagram technique’’ (HDT)) does not guarantee that every object is in a <-relation with every other object; objects can in fact be
incomparable with others. Hence in general we arrive at a partially ordered set starting from a data matrix qj (xi). Partially ordered sets can be visualized by directed
acyclic graphs. Considered as ordinary graphs, they are triangle free due to the
axiom of transitivity of order relations. Drawn in a special manner, this kind of directed
graph is known as a Hasse diagram, which is an extremely useful tool to analyze
partially ordered sets. Examples and some lines of interpretation will be found
everywhere in this Special Issue.
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