Order and metric geometry compatible stochastic processing

A traditional random variable X is a function that maps from a stochastic process to the real line. Here, “real line” refers to the structure (R, |⋅|, ≤) , where R is the set of real numbers, ≤ is the standard linear order relation on R , and d (x, y) ≜ |x − y| is the usual metric on R . The traditional expectation value E(X) of X is thenoften apoor choice of a statisticwhen the stochastic process that X maps from is a structure other than the real line or some substructure of the real line. If the stochastic process is a structure that is not linearly ordered (including structures totally unordered) and/or has a metric space geometry very different from that induced by the usual metric, then statistics such as E(X) are often of poor qualitywith regards to qualitative intuition and quantitative variance (expected error) measurements. For example, the traditional expected value of a fair die is E(X) = 1 6 (1 + 2 +⋯+ 6) = 3.5. But this result has no relationship with reality or with intuition because the result implies that we expect the value of ⚂ or ⚃ more than we expect the outcome of say ⚀ or ⚁. The fact is, that for a fair die, we would expect any pair of values equally. The reason for this is that the values of the face of a fair die are merely symbols with no order, and with no metric geometry other than the discrete metric geometry. On a fair die, ⚁ is not greater or less than ⚀ ; rather ⚁ and ⚀ are simply symbols without order. Moreover, ⚀ is not “closer” to ⚁ than it is to ⚂ ; rather, ⚀ , ⚁ , and ⚂ are simply symbols without any inherit order or metric geometry. This paper proposes an alternative statistical system, based somewhat on graph theory, that takes into account the order structure and metric geometry of the underlying stochastic process. 2010Mathematics SubjectClassification 60B99,60G05 (primary); 60E15,60G20,62C20,62F03,62P10,92D20 (secondary)