This paper proposes a new flexible non-parametric data transformation to Gaussian distribution. This option is often required because kriging is the best predictor under squared-error minimization criterion only if the data follow multivariate Gaussian distribution, while environmental data are often best described by skewed distributions with nonnegative values and a heavy right tail.
We assume that the modeling random field is the result of some nonlinear transformation of a Gaussian random field. In this case, the researchers commonly use a certain parametric monotone (for example, power or logarithmic) or normal score transformations. We discuss drawbacks of these methods and propose a new flexible non-parametric transformation.
We show how to find a nonlinear function which transforms the observed data to multivariate Gaussian distribution. This function has the following features: it is monotonously increasing, it has at least two derivatives, it is scalable and shift resistant. We also discuss unbiased back-transformation.
Our method is flexible and it can be used for automatic data transformation, for example, in the black-box kriging models in emergency situations. We show how our transformation can be used as an option for the non-stationary empirical Bayesian kriging. In this case, the transformation function becomes a random variable and a large number of transformations are used to account for the uncertainty of the estimated sampling distribution.