Abstract: |
This study compares direct estimation to kriging under normalizing transform approaches to obtain unbiased parameters for the modeling of local likelihood distributions of proportions or conditional probabilities of a P-field. The motivation comes from a simple exercise that shows how proportions and their probabilities respond to nonlinear and nonstationary Beta models. Perturbed real data and modeled conditional random variables *y(x)* of an attribute, when projected on the axis of global cumulative probability, *F(y)* as *p(x)=F(y(x))* ? the inverse operation to stochastic simulation ? result in an array of conditional histogram shapes of the [0,1] valued probability random variate *p(x)*. The array of exhaustive histograms resembles the Beta family, *f(p(x);,)*, comprised of a succession of highly flexible shapes on [0,1] ranging from highly skewed, to symmetric, and bimodal for contrasting extreme proportions. The exercise provides empirical evidence that the probability density function (pdf) of conditional probability is Beta-distributed, motivating the investigation of the characteristics and modeling approaches for complex non-linear and non-stationary Beta random fields.
P-field simulation introduced the idea of simulating correlated fields of probabilities. The local distributions of probability in P-field simulation were assumed to be uniform. This paper shows that these distributions of conditional probabilities are not uniformly distributed; instead, they follow a wide variety of shapes that can be modeled by the Beta distribution function.
Two pathways: direct kriging and simulation, and kriging under a normalizing transformation of the proportions are explored and compared to provide unbiased Beta distribution parameters for simulation. The Beta random field modeling approach is applied to real data, providing a cutting edge analysis and revision of the P-field paradigm for practical and theoretical Geostatistics. Proposals for further development lead to new probability constraint schemes and tools for modeling Beta distributed probability fields of spatially heterogeneous attributes. |