Special Issue: Partially ordered sets - Rainer Brüggemann - Vita e Pensiero - Articolo Vita e Pensiero

Special Issue: Partially ordered sets

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digital
Special Issue: Partially ordered sets
Articolo
rivista STATISTICA & APPLICAZIONI
fascicolo STATISTICA & APPLICAZIONI - 2011 - Special issue. Partial orders in applied sciences
titolo Special Issue: Partially ordered sets
Autore
Editore Vita e Pensiero
formato Articolo | Pdf
online da 02-2011
issn 1824-6672 (stampa) | 2283-6659 (digitale)
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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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