Analyzing Spatial Data Using Skew-Gaussian Processes
Spatial statistics has become a tool-box of methods useful for attacking a range of problems in scientiﬁc ﬁelds such as petroleum engineering, civil engineering, geography, geology, hydrology. The most useful technique for statistical spatial prediction is kriging (Cressie 1993). Most theories related to spatial prediction assume that the data are generated from a Gaussian random ﬁeld. However non-Gaussian characteristics, such as (non-negative) continuous variables with skewed distribution, appear in many datasets from scientiﬁc ﬁelds. For example potential, porosity and permeability measurements from petroleum engineering applications usually follow skewed distribution. A common way to model this type of data is to assume that the random ﬁeld of interest is the result of an unknown nonlinear transformation of a Gaussian random ﬁeld. Trans-Gaussian kriging is the kriging variant used for prediction in transformed Gaussian random ﬁelds, where the normalizing transformation is assumed known. This approach has some potential weaknesses (De Oliveira et al. 1997, Azzalini and Capitanio 1999) such as: (i) the transformations are usually on each component separately, and achievement of joint normality is only hoped for;
(ii) the transformed variables are more diﬃcult to interpret, especially when each variable is transformed by using a diﬀerent function;
(iii) even though the normalizing transformation can be estimated by maximum likelihood, it may be unwise to select a particular transformation;
(iv) sometimes the back-transformed ﬁtted model is severely biased (Miller 1984, Cressie 1993). Alternatively, we can use more general, ﬂexible parametric classes of
multivariate distributions to represent features of the dataset aiming to reduce the unrealistic assumptions. The pioneering work in this ﬁeld started with Zellner (1976) who dealt with the regression model with multivariate Student-t error terms.