Analysis of rainfall data and subsequent modelling of the many variables concerning rainfall is fundamental to many areas such as agricultural, ecological and engineering disciplines and, due to the complexity of the underlying hydrological system, it relies heavily on historical records. Daily rainfall series are arguably the most used. In this context, we initially investigate the modelling of daily rainfall interarrival times through a family of discrete probability distributions known as the Hurwitz-Lerch-Zeta family, along with two other distributions which are deeply related to the latter and have never been considered with this aim. Building up on the relationships between the interarrival times and certain other temporal variables, fundamental in describing the alternation between periods of continuous rainfall and periods of drought, we delineate a methodology particularly well-suited for statistical applications. The latter procedure and the fitting performance of the aforementioned distributions is shown on a dataset composed of a variety of rainfall regimes. Additionally, the multivariate modelling of rainfall variables has never been more important, as a perceivable shift in the inter-relationships between these variables could reflect climate changes in a region. In this context, copulas are well known and valued for their flexibility. However, they lose their charm when dealing with discrete random vectors. In this case, the uniqueness of the copula is compromised, leading to inconsistencies which basically break down the theoretical underpinnings of the inferential procedures commonly used in the continuous case. Recently, Gery Geenens made a compelling case for a new approach, grounding its beliefs in historical ideas regarding contingency tables. The theoretical insights he gives, coupled with a computational tool known as iterative proportional fitting procedure, open up the path to our development of novel (semi-parametric or parametric) models for finitely supported bivariate discrete random vectors. With this aim, we shall prove a sklar-like decomposition of a bivariate discrete probability mass function 3 between its margins and a copula probability mass function. Statistical models hinging upon this representation are built and related inferential procedures are studied both theoretically and empirically. Of the same significance as modelling the behaviour of rainfall is its impact on water bodies and land surfaces. For instance, understanding the time it takes for rainfall to cause river levels to exceed a flood stage is of paramount importance for flood prediction and management. More in general, it is often crucial to determine the time at which certain hydrological thresholds are crossed by some hydrological quantity. When the latter’s value in time is modelled by a stochastic process, this problem can be restated in terms of the first passage time. In this context, a practical computation of the first passage time probability density and distribution function is a delicate issue. Within this framework, we propose an approximation method based on a series expansion. Theoretical results are accompanied by discussions on the computational aspects. Extensive numerical experiments are carried out for the geometric Brownian motion and the Cox-Ingersoll-Ross process, showing the usefulness of the proposed method.
Statistical and Probabilistic Approaches to Hydrological Data Analysis: Rainfall Patterns, Copula-like Models and First Passage Time Approximations(2024 Dec 19).
Statistical and Probabilistic Approaches to Hydrological Data Analysis: Rainfall Patterns, Copula-like Models and First Passage Time Approximations
MARTINI, TOMMASO
2024-12-19
Abstract
Analysis of rainfall data and subsequent modelling of the many variables concerning rainfall is fundamental to many areas such as agricultural, ecological and engineering disciplines and, due to the complexity of the underlying hydrological system, it relies heavily on historical records. Daily rainfall series are arguably the most used. In this context, we initially investigate the modelling of daily rainfall interarrival times through a family of discrete probability distributions known as the Hurwitz-Lerch-Zeta family, along with two other distributions which are deeply related to the latter and have never been considered with this aim. Building up on the relationships between the interarrival times and certain other temporal variables, fundamental in describing the alternation between periods of continuous rainfall and periods of drought, we delineate a methodology particularly well-suited for statistical applications. The latter procedure and the fitting performance of the aforementioned distributions is shown on a dataset composed of a variety of rainfall regimes. Additionally, the multivariate modelling of rainfall variables has never been more important, as a perceivable shift in the inter-relationships between these variables could reflect climate changes in a region. In this context, copulas are well known and valued for their flexibility. However, they lose their charm when dealing with discrete random vectors. In this case, the uniqueness of the copula is compromised, leading to inconsistencies which basically break down the theoretical underpinnings of the inferential procedures commonly used in the continuous case. Recently, Gery Geenens made a compelling case for a new approach, grounding its beliefs in historical ideas regarding contingency tables. The theoretical insights he gives, coupled with a computational tool known as iterative proportional fitting procedure, open up the path to our development of novel (semi-parametric or parametric) models for finitely supported bivariate discrete random vectors. With this aim, we shall prove a sklar-like decomposition of a bivariate discrete probability mass function 3 between its margins and a copula probability mass function. Statistical models hinging upon this representation are built and related inferential procedures are studied both theoretically and empirically. Of the same significance as modelling the behaviour of rainfall is its impact on water bodies and land surfaces. For instance, understanding the time it takes for rainfall to cause river levels to exceed a flood stage is of paramount importance for flood prediction and management. More in general, it is often crucial to determine the time at which certain hydrological thresholds are crossed by some hydrological quantity. When the latter’s value in time is modelled by a stochastic process, this problem can be restated in terms of the first passage time. In this context, a practical computation of the first passage time probability density and distribution function is a delicate issue. Within this framework, we propose an approximation method based on a series expansion. Theoretical results are accompanied by discussions on the computational aspects. Extensive numerical experiments are carried out for the geometric Brownian motion and the Cox-Ingersoll-Ross process, showing the usefulness of the proposed method.File | Dimensione | Formato | |
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