Temporal and Spatially Heterogeneous Finite Length Runs Analysis

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.