Flood frequency analysis by a bivariate model based on copulas

La adecuada estimación de avenidas de diseño asociadas a altos periodos de retorno es necesaria para el diseño y gestión de estructuras hidráulicas como presas. En la práctica, la estimación de estos cuantiles se realiza normalmente a través de análisis de frecuencia univariados, basados en su mayoría en el estudio de caudales punta. Sin embargo, la naturaleza de las avenidas es multivariada, siendo esencial tener en cuenta características representativas de las avenidas, tales como caudal punta, volumen y duración del hidrograma, con el fin de llevar a cabo un análisis apropiado; especialmente cuando el caudal de entrada se transforma en un caudal de salida diferente durante el proceso de laminación en un embalse o llanura de inundación. Los análisis de frecuencia de avenidas multivariados han sido tradicionalmente llevados a cabo mediante el uso de distribuciones bivariadas estándar con el fin de modelar variables correlacionadas. Sin embargo, su uso conlleva limitaciones como la necesidad de usar el mismo tipo de distribuciones marginales para todas las variables y la existencia de una relación de dependencia lineal entre ellas. Recientemente, el uso de cópulas se ha extendido en hidrología debido a sus beneficios en relación al contexto multivariado, permitiendo superar los inconvenientes de las técnicas tradicionales. Una copula es una función que representa la estructura de dependencia de las variables de estudio, y permite obtener la distribución de frecuencia multivariada de dichas variables mediante sus distribuciones marginales, sin importar el tipo de distribución marginal utilizada. La estimación de periodos de retorno multivariados, y por lo tanto, de cuantiles multivariados, también se facilita debido a la manera en la que las cópulas están formuladas. La presente tesis doctoral busca proporcionar metodologías que mejoren las técnicas tradicionales usadas por profesionales para estimar cuantiles de avenida más adecuados para el diseño y la gestión de presas, así como para la evaluación del riesgo de avenida, mediante análisis de frecuencia de avenidas bivariados basados en cópulas. Las variables consideradas para ello son el caudal punta y el volumen del hidrograma. Con el objetivo de llevar a cabo un estudio completo, la presente investigación abarca: (i) el análisis de frecuencia de avenidas local bivariado centrado en examinar y comparar los periodos de retorno teóricos basados en la probabilidad natural de ocurrencia de una avenida, con el periodo de retorno asociado al riesgo de sobrevertido de la presa bajo análisis, con el fin de proporcionar cuantiles en una estación de aforo determinada; (ii) la extensión del enfoque local al regional, proporcionando un procedimiento completo para llevar a cabo un análisis de frecuencia de avenidas regional bivariado para proporcionar cuantiles en estaciones sin aforar o para mejorar la estimación de dichos cuantiles en estaciones aforadas; (iii) el uso de cópulas para investigar tendencias bivariadas en avenidas debido al aumento de los niveles de urbanización en una cuenca; y (iv) la extensión de series de avenida observadas mediante la combinación de los beneficios de un modelo basado en cópulas y de un modelo hidrometeorológico. Accurate design flood estimates associated with high return periods are necessary to design and manage hydraulic structures such as dams. In practice, the estimate of such quantiles is usually done via univariate flood frequency analyses, mostly based on the study of peak flows. Nevertheless, the nature of floods is multivariate, being essential to consider representative flood characteristics, such as flood peak, hydrograph volume and hydrograph duration to carry out an appropriate analysis; especially when the inflow peak is transformed into a different outflow peak during the routing process in a reservoir or floodplain. Multivariate flood frequency analyses have been traditionally performed by using standard bivariate distributions to model correlated variables, yet they entail some shortcomings such as the need of using the same kind of marginal distribution for all variables and the assumption of a linear dependence relation between them. Recently, the use of copulas has been extended in hydrology because of their benefits regarding dealing with the multivariate context, as they overcome the drawbacks of the traditional approach. A copula is a function that represents the dependence structure of the studied variables, and allows obtaining the multivariate frequency distribution of them by using their marginal distributions, regardless of the kind of marginal distributions considered. The estimate of multivariate return periods, and therefore multivariate quantiles, is also facilitated by the way in which copulas are formulated. The present doctoral thesis seeks to provide methodologies that improve traditional techniques used by practitioners, in order to estimate more appropriate flood quantiles for dam design, dam management and flood risk assessment, through bivariate flood frequency analyses based on the copula approach. The flood variables considered for that goal are peak flow and hydrograph volume. In order to accomplish a complete study, the present research addresses: (i) a bivariate local flood frequency analysis focused on examining and comparing theoretical return periods based on the natural probability of occurrence of a flood, with the return period associated with the risk of dam overtopping, to estimate quantiles at a given gauged site; (ii) the extension of the local to the regional approach, supplying a complete procedure for performing a bivariate regional flood frequency analysis to either estimate quantiles at ungauged sites or improve at-site estimates at gauged sites; (iii) the use of copulas to investigate bivariate flood trends due to increasing urbanisation levels in a catchment; and (iv) the extension of observed flood series by combining the benefits of a copula-based model and a hydro-meteorological model.

Learning Bayesian networks from data by the incremental compilation of new network polynomials

Probabilistic graphical models are a huge research field in artificial intelligence nowadays. The scope of this work is the study of directed graphical models for the representation of discrete distributions. Two of the main research topics related to this area focus on performing inference over graphical models and on learning graphical models from data. Traditionally, the inference process and the learning process have been treated separately, but given that the learned models structure marks the inference complexity, this kind of strategies will sometimes produce very inefficient models. With the purpose of learning thinner models, in this master thesis we propose a new model for the representation of network polynomials, which we call polynomial trees. Polynomial trees are a complementary representation for Bayesian networks that allows an efficient evaluation of the inference complexity and provides a framework for exact inference. We also propose a set of methods for the incremental compilation of polynomial trees and an algorithm for learning polynomial trees from data using a greedy score+search method that includes the inference complexity as a penalization in the scoring function.

Conditional density approximations with mixtures of polynomials

Mixtures of polynomials (MoPs) are a non-parametric density estimation technique especially designed for hybrid Bayesian networks with continuous and discrete variables. Algorithms to learn one- and multi-dimensional (marginal) MoPs from data have recently been proposed. In this paper we introduce two methods for learning MoP approximations of conditional densities from data. Both approaches are based on learning MoP approximations of the joint density and the marginal density of the conditioning variables, but they differ as to how the MoP approximation of the quotient of the two densities is found. We illustrate and study the methods using data sampled from known parametric distributions, and we demonstrate their applicability by learning models based on real neuroscience data. Finally, we compare the performance of the proposed methods with an approach for learning mixtures of truncated basis functions (MoTBFs). The empirical results show that the proposed methods generally yield models that are comparable to or significantly better than those found using the MoTBF-based method.

Orthogonal MCMC algorithms

Monte Carlo (MC) methods are widely used in signal processing, machine learning and stochastic optimization. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In this work, we introduce a novel parallel interacting MCMC scheme, where the parallel chains share information using another MCMC technique working on the entire population of current states. These parallel ?vertical? chains are led by random-walk proposals, whereas the ?horizontal? MCMC uses a independent proposal, which can be easily adapted by making use of all the generated samples. Numerical results show the advantages of the proposed sampling scheme in terms of mean absolute error, as well as robustness w.r.t. to initial values and parameter choice.