Title: A unified theory for exact stochastic modelling of univariate and multivariate processes with continuous, mixed type, or discrete marginal distributions and any correlation structure,
Abstract: Hydroclimatic processes are characterized by heterogeneous spatiotemporal
correlation structures and marginal distributions that can be continuous,
mixed-type, discrete or even binary. Simulating exactly such processes can
greatly improve hydrological analysis and design. Yet this challenging task is
accomplished often by ad hoc and approximate methodologies that are devised for
specific variables and purposes. In this study, a single framework is proposed
allowing the exact simulation of processes with any marginal and any
correlation structure. We unify, extent, and improve of a general-purpose
modelling strategy based on the assumption that any process can emerge by
transforming a parent Gaussian process with a specific correlation structure. A
novel mathematical representation of the parent-Gaussian scheme provides a
consistent and fully general description that supersedes previous specific
parameterizations, resulting in a simple, fast and efficient simulation
procedure for every spatiotemporal process. In particular, introducing a simple
but flexible procedure we obtain a parametric expression of the correlation
transformation function, allowing to assess the correlation structure of the
parent-Gaussian process that yields the prescribed correlation of the target
process after marginal back transformation. The same framework is also
applicable for cyclostationary and multivariate modelling. The simulation of a
variety of hydroclimatic variables with very different correlation structures
and marginals, such as precipitation, stream flow, wind speed, humidity,
extreme events per year, etc., as well as a multivariate application,
highlights the flexibility, advantages, and complete generality of the proposed
methodology.