Estimation of averaged ranks by a local partial order model.

This paper continues the series of publications about applications of partial ordering. The focus of this publication is the derivation of approximate analytical expressions for the averaged rank and the ranking probabilities. To derive such combinatorial formulas a local partial order is suggested as an approximation. The performance of the approximation is rather high; we therefore conclude that three very simple descriptors of the local partial order seem to be sufficient to get a rough impression of the linear order, induced by the averaged ranks and the ranking probabilities of empirical partially ordered sets. Linear order derived from the partial order, ranking probabilities, and other characteristics are considered as parts of a so-called "General Ranking Model" (GRM). Following the local partial order, the averaged rank of an object x can be estimated applying the following simple formula: Rk(av) = (S+1)*(N+1)/(N+1-U). S is the number of successors of the object x, N is the total number of objects (of the quotient set), and U is the number of objects incomparable with x. More complex formulas for the ranking probabilities are given in the text. A list of abbreviations and symbols can be found in Tables 3 and 4.