Abstract
Various problems in physics and engineering lead to the problem of run- length probability distribution function (pdf) in a finite time series. In this study to find the pdf of run-lengths in such a series, first infinite sequence properties are reviewed and then finite series run length pdf is derived on the basis of simple set theory. This paper presents the derivation of exact run length pdf in finite length dependent series. In the derivation, two different definitions of runs are considered as the integration method for infinite series and combinatorial analysis for finite time series. The analytical derivations are solved numerically and the results are presented in forms of cumulative pdf, expectation, variance and higher order moment changes with the sample lengths. On the other hand, homogenous run properties based on Bernoulli trials are used in many physical and engineering applications for many decades. Heterogeneous regional Bernoulli trial probability distribution model is not available so far in applications and numerical calculations. Herein, a plausible, rational and logical mathematical derivation of the heterogeneous case is derived, which reduces to the classical homogeneous Bernoulli trial case. This paper provides regional probabilistic success and failure period areal coverage modeling, which is useful for temporal and spatial pattern recognition of spatial risk predictions and parameter assessments. The basis of the methodology is mutually exclusive and independent sub-areal (site) success and failure occurrences’ heterogeneous probabilities.
Highlights
From the statistical point of view, a run is defined by Mood (1940) as a succession of similar kind of elements proceeded and succeeded by different kinds
Let Si = s = i be an event denoting the occurrence of s successive runs of length i in an infinite sequence of Bernoulli trials
The necessary analytical derivations are achieved through the combinatorial methodology
Summary
From the statistical point of view, a run is defined by Mood (1940) as a succession of similar kind of elements proceeded and succeeded by different kinds. It is possible to truncate the given sequence at x0 level. Such a truncation procedure leads to a new sequence of two basic elements, namely, a success when xi - x0 0 and a failure if x1 - x0 0. These two basic elements are, the Bernoulli trials. In the classical statistics literature, a success run of lengths s is defined as an uninterrupted sequence of either s or at least s successes. In water resources engineering x0 might represent the demand necessary for water supply
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