Stochastic models have become a powerful necessary statistical tool to estimate parameters where data are represented by regions of interests under uncertain areas. Due to the high dimensionality of the spatial patterns, investigation of the stochastic modeling simulations must be applied based on spatial variability. Models for spreading diseases are given based on whether or not the disease succeeds or fails to appear in the region. Based on this assumption, an Ising/Potts random fields model has to be introduced to analyze the spatial pattern of spreading. In this work, the spatial pattern models for spreading diseases have been analyzed considering Markov random fields auto-models. The Gibbs sampler would be used to simulate example images for various parameter combinations. In this work, a spatial analysis methodology based on Bayesian analysis was introduced, and procedures to solve the problem with spatial variability are described based on spatial model estimation techniques.
TopBayesian Uncertainty
Bayesian statistical methods under uncertainty conditions could be implemented by calculating probabilistic predictions. The procedure has three stages: (1) determination of the prior probability distribution for model parameters, (2) construction of a likelihood function for the statistical model, and (3) derivation of the posterior probability distribution for the parameters by using the Bayes rule to adjust the prior distribution based on the observed data (Katz, 2002).
Statistical estimation techniques can be used to determine the parameters of the distribution. These techniques are helpful to estimate probability distributions from available data or by collecting a large amount of them (Figure 1) (Jansen et al., 1998)
Figure 1. Bayesian statistic uncertainty analysis (Jansen et al., 1998)
Under Bayesian analysis, modeling of the uncertain regions could be illustrated by introducing hidden models due to the high dimensionality of the process (Morgan and Hemion, 1990, Klepper, 1997, Katz, 2002). In confidential information, hidden models could be denoted, and an estimation procedure could be applied based on hidden Markov Models. These models can be used to estimate confidential information based on Markov random fields models where spatial homogeneity between regions is taking place (Zimeras, 1997; Zimeras and Matsinos, 2011; 2012; 2019). The advantage of these models is that they need only the local characteristics between neighboring regions. Models that can explain the spatial patterns considering conditional p.d.f. functions are defined as auto-models (Zimeras, 1997).