Order and metric geometry compatible stochastic processing

Abstract

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,<=,|x-y|), 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 then often a poor choice of a statistic when 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 quality with 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 [ooo] (die face value "3") or [oooo] (dice face value "4") more than we expect the outcome of say [o] or [oo]. 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, [oo] is not greater or less than [o]; rather [oo] and [o] are simply symbols without order. Moreover, [o] is not "closer" to [oo] than it is to [ooo]; rather, [o], [oo], and [ooo] 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.

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“…Here is a review of some key definitions Greenhoe (2015b) which purposed a stochastic processing based on graph theory. Definition 2.1 A triple ≜ ( , ≤, ) is an ordered distance space if ( , ) is a distance space (Definition A.1 page 45) and ( , ≤) is an ordered set (Definition 1.16 page 5).…”

“…24 Greenhoe (2015b) 25 page 130 26 Papoulis (1991), page 63 27 Greenhoe (2015b) Definition 2.5 29 The structure ≜ ( {⚀, ⚁, ⚂, ⚃, ⚄, ⚅},,≤,̇) is the fair die outcome subspace if is a weighted die outcome subspace (Definition 2.4), and(⚀) =(⚁) =(⚂) =(⚃) =(⚄) =(⚅) = 1 /6. Definition 2.6 30 The structure ≜ ( {⚀, ⚁, ⚂, ⚃, ⚄, ⚅},, ∅,̇) is the weighted real die outcome subspace if is an outcome subspace, and metriċis defined as in the table to the right.̇( , ) ⚀ ⚁ ⚂ ⚃ ⚄ ⚅ ⚀ 0 1 1 1 1 2 ⚁ 1 0 1 1 2 1 ⚂ 1 1 0 2 1 1 ⚃ 1 1 2 0 1 1 ⚄ 1 2 1 1 0 1 ⚅ 2 1 1 1 1 0 Definition 2.7 31 The structure ≜ ( {⚀, ⚁, ⚂, ⚃, ⚄, ⚅},, ∅,̇) is the real die outcome subspace if is a weighted real die outcome subspace (Definition 2.6 page 12) witḣ (⚀) =(⚁) =(⚂) =(⚃) =(⚄) =(⚅) = 1 /6.…”

“…Definition 2.6 30 The structure ≜ ( {⚀, ⚁, ⚂, ⚃, ⚄, ⚅},, ∅,̇) is the weighted real die outcome subspace if is an outcome subspace, and metriċis defined as in the table to the right.̇( , ) ⚀ ⚁ ⚂ ⚃ ⚄ ⚅ ⚀ 0 1 1 1 1 2 ⚁ 1 0 1 1 2 1 ⚂ 1 1 0 2 1 1 ⚃ 1 1 2 0 1 1 ⚄ 1 2 1 1 0 1 ⚅ 2 1 1 1 1 0 Definition 2.7 31 The structure ≜ ( {⚀, ⚁, ⚂, ⚃, ⚄, ⚅},, ∅,̇) is the real die outcome subspace if is a weighted real die outcome subspace (Definition 2.6 page 12) witḣ (⚀) =(⚁) =(⚂) =(⚃) =(⚄) =(⚅) = 1 /6. 28 Greenhoe (2015b) 29 Greenhoe (2015b) 30 Greenhoe (2015b) 31 Greenhoe (2015b) version 0.5…”

“…A traditional random variable (Definition 1.55 page 11) is a function that maps from a set of outcomes to the real line (ℝ, |⋅|, ≤), where ℝ is the set of real numbers,≤ is the standard linear order relation on ℝ, and ( , ) ≜ | − | is the usual metric (Euclidean metric) on ℝ. Greenhoe (2015b) demonstrated that this definition of random variable is often a poor choice of a statistic when the stochastic process that maps from is a structure other than the real line or some substructure of the real line, such as the integer line (ℤ, |⋅|, ≤). Greenhoe (2015b) further proposed an alternative statistical system, that rather than mapping stochastic processes to the real line, instead maps to a weighted graph that has order and metric geometry structure compatible to that of the underlying stochastic process.…”

“…Greenhoe (2015b) demonstrated that this definition of random variable is often a poor choice of a statistic when the stochastic process that maps from is a structure other than the real line or some substructure of the real line, such as the integer line (ℤ, |⋅|, ≤). Greenhoe (2015b) further proposed an alternative statistical system, that rather than mapping stochastic processes to the real line, instead maps to a weighted graph that has order and metric geometry structure compatible to that of the underlying stochastic process. In particular, ideally the weighted graph and stochastic process are isomorphic and isometric with respect to each other.…”