scholarly journals Order and metric geometry compatible stochastic processing

2015 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Real Line ◽  
Order Relation ◽  
Random Variable ◽  
Poor Quality ◽  
Order Structure ◽  
Metric Geometry ◽  
The Real ◽  
R And D ◽  
The Face

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.

scholarly journals Order and metric geometry compatible stochastic processing

2015 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Real Line ◽  
Order Relation ◽  
Random Variable ◽  
Poor Quality ◽  
Order Structure ◽  
Metric Geometry ◽  
The Real ◽  
R And D ◽  
The Face

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.

scholarly journals Order and Metric Compatible Symbolic Sequence Processing

2016 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Sequence Analysis ◽  
Linear Space ◽  
Metric Space ◽  
Real Line ◽  
Distance Geometry ◽  
Weighted Graph ◽  
Random Variable ◽  
The Real ◽  
Sequence Processing

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. Greenhoe(2015b) 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 x R^n, and . is the scalar-vector multiplication operator on R x R^n. The linear space component of Y provides a much more convenient (as compared to the weighted graph) framework for sequence analysis and processing. The ordered set and distance space components of Y allow one to preserve the order structure and distance geometry inherent in the underlying stochastic process, which in turn likely provides a less distorted (as compared to the real line) framework for analysis, diagnostics, and optimal decision making.

scholarly journals Order and Metric Compatible Symbolic Sequence Processing

2016 ◽  
Author(s):  
Daniel J Greenhoe
Keyword(s):  
Stochastic Process ◽  
Sequence Analysis ◽  
Linear Space ◽  
Metric Space ◽  
Real Line ◽  
Distance Geometry ◽  
Weighted Graph ◽  
Random Variable ◽  
The Real ◽  
Sequence Processing

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. Greenhoe(2015b) 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 x R^n, and . is the scalar-vector multiplication operator on R x R^n. The linear space component of Y provides a much more convenient (as compared to the weighted graph) framework for sequence analysis and processing. The ordered set and distance space components of Y allow one to preserve the order structure and distance geometry inherent in the underlying stochastic process, which in turn likely provides a less distorted (as compared to the real line) framework for analysis, diagnostics, and optimal decision making.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

On random motions with velocities alternating at Erlang-distributed random times

2001 ◽  
Vol 33 (03) ◽  
pp. 690-701 ◽  
Author(s):  
Antonio Di Crescenzo
Keyword(s):  
Stochastic Process ◽  
Random Process ◽  
Bessel Function ◽  
Real Line ◽  
Renewal Process ◽  
The Real ◽  
Alternating Renewal Process ◽  
Transition Densities ◽  
Random Motions ◽  
Random Times

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

Sojourns and extremes of a diffusion process on a fixed interval

1982 ◽  
Vol 14 (4) ◽  
pp. 811-832 ◽  
Author(s):  
Simeon M. Berman
Keyword(s):  
Diffusion Process ◽  
Lebesgue Measure ◽  
Stationary Distribution ◽  
Real Line ◽  
Diffusion Coefficients ◽  
Limit Theorems ◽  
Random Variable ◽  
Fixed Interval ◽  
The Real ◽  
And Diffusion

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt(u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt(u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t]X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (2) ◽  
pp. 473-475 ◽  
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

On a three-state sojourn time problem

1971 ◽  
Vol 8 (04) ◽  
pp. 716-723 ◽  
Author(s):  
A. E. Gibson ◽  
B. W. Conolly
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Sojourn Time ◽  
Random Variable ◽  
The Real ◽  
Time Problem ◽  
Arbitrary Space ◽  
Image Position

Consider the real-valued stochastic process {S(t), 0 ≦ t &lt; ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t 0 at which the visit to state T begins.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (02) ◽  
pp. 473-475
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

On a three-state sojourn time problem

1971 ◽  
Vol 8 (4) ◽  
pp. 716-723 ◽  
Author(s):  
A. E. Gibson ◽  
B. W. Conolly
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Sojourn Time ◽  
Random Variable ◽  
The Real ◽  
Time Problem ◽  
Arbitrary Space ◽  
Image Position

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t0 at which the visit to state T begins.

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