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.
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle inequality does not hold but in which we may still like to perform analysis. This paper investigates what happens if the triangle inequality is removed all together, leaving what is called a distance space, and also what happens if the triangle inequality is replaced with a much more general two parameter relation, which is herein called the "power triangle inequality". The power triangle inequality represents an uncountably large class of inequalities, and includes the triangle inequality, relaxed triangle inequality, and inframetric inequality as special cases. The power triangle inequality is defined in terms of a function that is herein called the power triangle function. The power triangle function is itself a power mean, and as such is continuous and monotone with respect to its exponential parameter, and also includes the operations of maximum, minimum, mean square, arithmetic mean, geometric mean, and harmonic mean as special cases.
A traditional random variable X is a function that maps from a stochastic process to the real line (X,<=,d,+,.), where R is the set of real numbers, <= is the standard linear order relation on R, d(x,y)=|x-y| is the usual metric on R, and (R, +, .) is the standard field on R. has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that X maps from has structure that is dissimilar to that of the real line. Greenhoe(2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure X maps from and the structure X maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable.for example the expectation EX of X can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis), in performing sequence processing (using for example FIR filtering or IIR filtering), in making diagnostic measurements (using a post-transform metric space), or in making goptimal h decisions (based on gdistance h measurements in a metric space or more generally a distance space). Rather than mapping to a weighted graph, this paper proposes instead mapping to an ordered distance linear space Y=(R^n,<=,d,+,.,R,+,x), where (R,+,x) is a field, + is the vector addition operator on R^n Order and Metric Compatible Symbolic Sequence Processing DANIEL J. GREENHOEAbstract: A traditional random variable is a function that maps from a stochastic process to the "real line" (ℝ, ≤, , +, ⋅), where ℝ is the set of real numbers, ≤ is the standard linear order relation on ℝ, ( , ) ≜ | − | is the usual metric on ℝ, and (ℝ, +, ⋅) is the standard field on ℝ.Greenhoe (2015b) has demonstrated that this definition of random variable is often a poor choice for computing statistics when the stochastic process that maps from has structure that is dissimilar to that of the real line. Greenhoe (2015b) has further proposed an alternative statistical system, that rather than mapping a stochastic process to the real line, instead maps to a weighted graph that has order and metric geometry structures similar to that of the underlying stochastic process. In particular, ideally the structure maps from and the structure maps to are, with respect to each other, both isomorphic and isometric.Mapping to a weighted graph is useful for analysis of a single random variable-for example the expectation of can be defined simply as the center of its weighted graph. However, the mapping has limitations with regards to a sequence of random variables in performing sequence analysis (using for example Fourier analysis or wavelet analysis),...
The spherical metric d_r operates on the surface of a sphere with radius r centered at the origin in a linear space R^n. Thus, for any pair of points (p,q) on the surface of this sphere, (p,q) is in the domain of d_r and d_r(p,q) is the "distance" between those points. However, if x and y are both in R^n but are not on the surface of a common sphere centered at the origin, then (p,q) is not in the domain of d_r and d_r(p,q) is simply undefined. In certain applications, however, it would be useful to have an extension d of d_r to the entire space R^n (rather than just on a surface in R^n). Real world applications for such an extended metric include calculations involving near earth objects, and for certain distance spaces useful in symbolic sequence processing. This paper introduces an extension to the spherical metric using a polar form of linear interpolation. The extension is herein called the "Lagrange arc distance". It has as its domain the entire space R^n, is homogeneous, and is continuous everywhere in R^n except at the origin. However the extension does come at a cost: The Lagrange arc distance d(p,q), as its name suggests, is a distance function rather than a metric. In particular, the triangle inequality does not in general hold. Moreover, it is not translation invariant, does not induce a norm, and balls in the distance space (R^n,d) are not convex. On the other hand, empirical evidence suggests that the Lagrange arc distance results in structure similar to that of the Euclidean metric in that balls in R^2 and R^3 generated by the two functions are in some regions of R^n very similar in form.
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