A Generalized Regression Methodology for Bivariate Heteroscedastic Data

We present a methodology for reducing a straight line fitting regression problem to a Least Squares minimization one. This is accomplished through the definition of a measure on the data space that takes into account directional dependences of errors, and the use of polar descriptors for straight lines. This strategy improves the robustness by avoiding singularities and non-describable lines. The methodology is powerful enough to deal with non-normal bivariate heteroscedastic data error models, but can also supersede classical regression methods by making some particular assumptions. An implementation of the methodology for the normal bivariate case is developed and evaluated.

Efficient sampling from truncated bivariate Gaussians via Box-Muller transformation

Many practical simulation tasks demand procedures to draw samples efficiently from multivariate truncated Gaussian distributions. In this work, we introduce a novel rejection approach, based on the Box-Muller transformation, to generate samples from a truncated bivariate Gaussian density with an arbitrary support. Furthermore, for an important class of support regions the new method allows us to achieve exact sampling, thus becoming the most efficient approach possible. RESUMEN. Método específico para generar muestras de manera eficiente de Gaussianas bidimensionales truncadas con cualquier zona de truncamiento basado en la transformación de Box-Muller.

A bivariate return period based on copulas for hydrologic dam design: accounting for reservoir routing in risk estimation

A multivariate analysis on flood variables is needed to design some hydraulic structures like dams, as the complexity of the routing process in a reservoir requires a representation of the full hydrograph. In this work, a bivariate copula model was used to obtain the bivariate joint distribution of flood peak and volume, in order to know the probability of occurrence of a given inflow hydrograph. However, the risk of dam overtopping is given by the maximum water elevation reached during the routing process, which depends on the hydrograph variables, the reservoir volume and the spillway crest length. Consequently, an additional bivariate return period, the so-called routed return period, was defined in terms of risk of dam overtopping based on this maximum water elevation obtained after routing the inflow hydrographs. The theoretical return periods, which give the probability of occurrence of a hydrograph prior to accounting for the reservoir routing, were compared with the routed return period, as in both cases hydrographs with the same probability will draw a curve in the peak-volume space. The procedure was applied to the case study of the Santillana reservoir in Spain. Different reservoir volumes and spillway lengths were considered to investigate the influence of the dam and reservoir characteristics on the results. The methodology improves the estimation of the Design Flood Hydrograph and can be applied to assess the risk of dam overtopping

Univariate and bivariate truncated von Mises distributions

In this article we study the univariate and bivariate truncated von Mises distribution, as a generalization of the von Mises distribution (\cite{jupp1989}), (\cite{mardia2000directional}). This implies the addition of two or four new truncation parameters in the univariate and, bivariate cases, respectively. The results include the definition, properties of the distribution and maximum likelihood estimators for the univariate and bivariate cases. Additionally, the analysis of the bivariate case shows how the conditional distribution is a truncated von Mises distribution, whereas the marginal distribution that generalizes the distribution introduced in \cite{repe}. From the viewpoint of applications, we test the distribution with simulated data, as well as with data regarding leaf inclination angles (\cite{safari}) and dihedral angles in protein chains (\cite{prote}). This research aims to assert this probability distribution as a potential option for modelling or simulating any kind of phenomena where circular distributions are applicable.\par

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.

A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.

Bivariate Krawtchouk polynomials: Inversion and connection problems with the NAVIMA algorithm

In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic case and also for Krawtchouk families, several expansion formulas are given, but for polynomials written in scalar form.