A unified theory for exact stochastic modelling of univariate and multivariate processes with continuous, mixed type, or discrete marginal distributions and any correlation structure

TL;DR

This paper introduces a comprehensive framework for the exact simulation of univariate and multivariate processes with diverse marginal distributions and correlation structures, significantly improving hydrological modeling accuracy.

Contribution

A novel, unified mathematical approach that enables exact simulation of processes with any marginal and correlation structure, surpassing previous ad hoc methods.

Findings

Successfully simulated diverse hydroclimatic variables

Demonstrated flexibility across different marginal types

Enabled multivariate process modeling

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.