Modeling Nonlinear Beta Probability Fields

Abstract

Experiments show how proportions correspond to nonlinear conditional Beta probability distributions. The encountered Beta probability (Beta p-) fields are not classic p-fields. Instead, the shape of the local probability density function (pdf) for the proportion p(x) random variable at each location x changes with the Beta distribution parameters, α and β which are in turn functions of the local proportions themselves. In addition, Beta random variables with different shapes of pdf’s have nonlinear pairwise relations; therefore, Beta p-fields are nonlinear. A novel approach was devised to transform these complex Beta random variables to the Gaussian domain which can be modeled with linear geostatistics. A property of the proposed numerical approach is that kriging estimates, when back transformed to the Beta domain using Riemann’s integral produce unbiased Beta parameters. This contribution completely redefines the p-field concept with a theoretical framework enabling the simulation of Beta p-fields of proportions. This work represents new findings in geostatistics and motivates further studies of probability or proportion as a random variable.