Desarrollo de metodologías para mejorar la estimación de los hidrogramas de diseño para el cálculo de los órganos de desagüe de las presas

En la presente Tesis se ha llevado a cabo el contraste y desarrollo de metodologías que permitan mejorar el cálculo de las avenidas de proyecto y extrema empleadas en el cálculo de la seguridad hidrológica de las presas. En primer lugar se ha abordado el tema del cálculo de las leyes de frecuencia de caudales máximos y su extrapolación a altos periodos de retorno. Esta cuestión es de gran relevancia, ya que la adopción de estándares de seguridad hidrológica para las presas cada vez más exigentes, implica la utilización de periodos de retorno de diseño muy elevados cuya estimación conlleva una gran incertidumbre. Es importante, en consecuencia incorporar al cálculo de los caudales de diseño todas la técnicas disponibles para reducir dicha incertidumbre. Asimismo, es importante hacer una buena selección del modelo estadístico (función de distribución y procedimiento de ajuste) de tal forma que se garantice tanto su capacidad para describir el comportamiento de la muestra, como para predecir de manera robusta los cuantiles de alto periodo de retorno. De esta forma, se han realizado estudios a escala nacional con el objetivo de determinar el esquema de regionalización que ofrece mejores resultados para las características hidrológicas de las cuencas españolas, respecto a los caudales máximos anuales, teniendo en cuenta el numero de datos disponibles. La metodología utilizada parte de la identificación de regiones homogéneas, cuyos límites se han determinado teniendo en cuenta las características fisiográficas y climáticas de las cuencas, y la variabilidad de sus estadísticos, comprobando posteriormente su homogeneidad. A continuación, se ha seleccionado el modelo estadístico de caudales máximos anuales con un mejor comportamiento en las distintas zonas de la España peninsular, tanto para describir los datos de la muestra como para extrapolar a los periodos de retorno más altos. El proceso de selección se ha basado, entre otras cosas, en la generación sintética de series de datos mediante simulaciones de Monte Carlo, y el análisis estadístico del conjunto de resultados obtenido a partir del ajuste de funciones de distribución a estas series bajo distintas hipótesis. Posteriormente, se ha abordado el tema de la relación caudal-volumen y la definición de los hidrogramas de diseño en base a la misma, cuestión que puede ser de gran importancia en el caso de presas con grandes volúmenes de embalse. Sin embargo, los procedimientos de cálculo hidrológico aplicados habitualmente no tienen en cuenta la dependencia estadística entre ambas variables. En esta Tesis se ha desarrollado un procedimiento para caracterizar dicha dependencia estadística de una manera sencilla y robusta, representando la función de distribución conjunta del caudal punta y el volumen en base a la función de distribución marginal del caudal punta y la función de distribución condicionada del volumen respecto al caudal. Esta última se determina mediante una función de distribución log-normal, aplicando un procedimiento de ajuste regional. Se propone su aplicación práctica a través de un procedimiento de cálculo probabilístico basado en la generación estocástica de un número elevado de hidrogramas. La aplicación a la seguridad hidrológica de las presas de este procedimiento requiere interpretar correctamente el concepto de periodo de retorno aplicado a variables hidrológicas bivariadas. Para ello, se realiza una propuesta de interpretación de dicho concepto. El periodo de retorno se entiende como el inverso de la probabilidad de superar un determinado nivel de embalse. Al relacionar este periodo de retorno con las variables hidrológicas, el hidrograma de diseño de la presa deja de ser un único hidrograma para convertirse en una familia de hidrogramas que generan un mismo nivel máximo en el embalse, representados mediante una curva en el plano caudal volumen. Esta familia de hidrogramas de diseño depende de la propia presa a diseñar, variando las curvas caudal-volumen en función, por ejemplo, del volumen de embalse o la longitud del aliviadero. El procedimiento propuesto se ilustra mediante su aplicación a dos casos de estudio. Finalmente, se ha abordado el tema del cálculo de las avenidas estacionales, cuestión fundamental a la hora de establecer la explotación de la presa, y que puede serlo también para estudiar la seguridad hidrológica de presas existentes. Sin embargo, el cálculo de estas avenidas es complejo y no está del todo claro hoy en día, y los procedimientos de cálculo habitualmente utilizados pueden presentar ciertos problemas. El cálculo en base al método estadístico de series parciales, o de máximos sobre un umbral, puede ser una alternativa válida que permite resolver esos problemas en aquellos casos en que la generación de las avenidas en las distintas estaciones se deba a un mismo tipo de evento. Se ha realizado un estudio con objeto de verificar si es adecuada en España la hipótesis de homogeneidad estadística de los datos de caudal de avenida correspondientes a distintas estaciones del año. Asimismo, se han analizado los periodos estacionales para los que es más apropiado realizar el estudio, cuestión de gran relevancia para garantizar que los resultados sean correctos, y se ha desarrollado un procedimiento sencillo para determinar el umbral de selección de los datos de tal manera que se garantice su independencia, una de las principales dificultades en la aplicación práctica de la técnica de las series parciales. Por otra parte, la aplicación practica de las leyes de frecuencia estacionales requiere interpretar correctamente el concepto de periodo de retorno para el caso estacional. Se propone un criterio para determinar los periodos de retorno estacionales de forma coherente con el periodo de retorno anual y con una distribución adecuada de la probabilidad entre las distintas estaciones. Por último, se expone un procedimiento para el cálculo de los caudales estacionales, ilustrándolo mediante su aplicación a un caso de estudio. The compare and develop of a methodology in order to improve the extreme flow estimation for dam hydrologic security has been developed. First, the work has been focused on the adjustment of maximum peak flows distribution functions from which to extrapolate values for high return periods. This has become a major issue as the adoption of stricter standards on dam hydrologic security involves estimation of high design return periods which entails great uncertainty. Accordingly, it is important to incorporate all available techniques for the estimation of design peak flows in order to reduce this uncertainty. Selection of the statistical model (distribution function and adjustment method) is also important since its ability to describe the sample and to make solid predictions for high return periods quantiles must be guaranteed. In order to provide practical application of previous methodologies, studies have been developed on a national scale with the aim of determining a regionalization scheme which features best results in terms of annual maximum peak flows for hydrologic characteristics of Spanish basins taking into account the length of available data. Applied methodology starts with the delimitation of regions taking into account basin’s physiographic and climatic characteristics and the variability of their statistical properties, and continues with their homogeneity testing. Then, a statistical model for maximum annual peak flows is selected with the best behaviour for the different regions in peninsular Spain in terms of describing sample data and making solid predictions for high return periods. This selection has been based, among others, on synthetic data series generation using Monte Carlo simulations and statistical analysis of results from distribution functions adjustment following different hypothesis. Secondly, the work has been focused on the analysis of the relationship between peak flow and volume and how to define design flood hydrographs based on this relationship which can be highly important for large volume reservoirs. However, commonly used hydrologic procedures do not take statistical dependence between these variables into account. A simple and sound method for statistical dependence characterization has been developed by the representation of a joint distribution function of maximum peak flow and volume which is based on marginal distribution function of peak flow and conditional distribution function of volume for a given peak flow. The last one is determined by a regional adjustment procedure of a log-normal distribution function. Practical application is proposed by a probabilistic estimation procedure based on stochastic generation of a large number of hydrographs. The use of this procedure for dam hydrologic security requires a proper interpretation of the return period concept applied to bivariate hydrologic data. A standard is proposed in which it is understood as the inverse of the probability of exceeding a determined reservoir level. When relating return period and hydrological variables the only design flood hydrograph changes into a family of hydrographs which generate the same maximum reservoir level and that are represented by a curve in the peak flow-volume two-dimensional space. This family of design flood hydrographs depends on the dam characteristics as for example reservoir volume or spillway length. Two study cases illustrate the application of the developed methodology. Finally, the work has been focused on the calculation of seasonal floods which are essential when determining the reservoir operation and which can be also fundamental in terms of analysing the hydrologic security of existing reservoirs. However, seasonal flood calculation is complex and nowadays it is not totally clear. Calculation procedures commonly used may present certain problems. Statistical partial duration series, or peaks over threshold method, can be an alternative approach for their calculation that allow to solve problems encountered when the same type of event is responsible of floods in different seasons. A study has been developed to verify the hypothesis of statistical homogeneity of peak flows for different seasons in Spain. Appropriate seasonal periods have been analyzed which is highly relevant to guarantee correct results. In addition, a simple procedure has been defined to determine data selection threshold on a way that ensures its independency which is one of the main difficulties in practical application of partial series. Moreover, practical application of seasonal frequency laws requires a correct interpretation of the concept of seasonal return period. A standard is proposed in order to determine seasonal return periods coherently with the annual return period and with an adequate seasonal probability distribution. Finally a methodology is proposed to calculate seasonal peak flows. A study case illustrates the application of the proposed methodology.

Heavy tails, importance sampling and cross-entropy

We consider the problem of estimating P(Yi + (...) + Y-n > x) by importance sampling when the Yi are i.i.d. and heavy-tailed. The idea is to exploit the cross-entropy method as a toot for choosing good parameters in the importance sampling distribution; in doing so, we use the asymptotic description that given P(Y-1 + (...) + Y-n > x), n - 1 of the Yi have distribution F and one the conditional distribution of Y given Y > x. We show in some specific parametric examples (Pareto and Weibull) how this leads to precise answers which, as demonstrated numerically, are close to being variance minimal within the parametric class under consideration. Related problems for M/G/l and GI/G/l queues are also discussed.

Regularisation of mixture density networks

Mixture Density Networks are a principled method to model conditional probability density functions which are non-Gaussian. This is achieved by modelling the conditional distribution for each pattern with a Gaussian Mixture Model for which the parameters are generated by a neural network. This thesis presents a novel method to introduce regularisation in this context for the special case where the mean and variance of the spherical Gaussian Kernels in the mixtures are fixed to predetermined values. Guidelines for how these parameters can be initialised are given, and it is shown how to apply the evidence framework to mixture density networks to achieve regularisation. This also provides an objective stopping criteria that can replace the `early stopping' methods that have previously been used. If the neural network used is an RBF network with fixed centres this opens up new opportunities for improved initialisation of the network weights, which are exploited to start training relatively close to the optimum. The new method is demonstrated on two data sets. The first is a simple synthetic data set while the second is a real life data set, namely satellite scatterometer data used to infer the wind speed and wind direction near the ocean surface. For both data sets the regularisation method performs well in comparison with earlier published results. Ideas on how the constraint on the kernels may be relaxed to allow fully adaptable kernels are presented.

Regularisation of mixture density networks

Mixture Density Networks are a principled method to model conditional probability density functions which are non-Gaussian. This is achieved by modelling the conditional distribution for each pattern with a Gaussian Mixture Model for which the parameters are generated by a neural network. This thesis presents a novel method to introduce regularisation in this context for the special case where the mean and variance of the spherical Gaussian Kernels in the mixtures are fixed to predetermined values. Guidelines for how these parameters can be initialised are given, and it is shown how to apply the evidence framework to mixture density networks to achieve regularisation. This also provides an objective stopping criteria that can replace the `early stopping' methods that have previously been used. If the neural network used is an RBF network with fixed centres this opens up new opportunities for improved initialisation of the network weights, which are exploited to start training relatively close to the optimum. The new method is demonstrated on two data sets. The first is a simple synthetic data set while the second is a real life data set, namely satellite scatterometer data used to infer the wind speed and wind direction near the ocean surface. For both data sets the regularisation method performs well in comparison with earlier published results. Ideas on how the constraint on the kernels may be relaxed to allow fully adaptable kernels are presented.

On the comparative performance of socially responsible and Islamic mutual funds

This is the first study to provide comprehensive analyses of the relative performance of both socially responsible investment (SRI) and Islamic mutual funds. The analysis proceeds in two stages. In the first, the performance of the two categories of funds is measured using partial frontier methods. In the second stage, we use quantile regression techniques.By combining two variants of the Free Disposal Hull (FDH) methods (order-m and order-?) in the first stage of analysis and quantile regression in the second stage, we provide detailed analyses of the impact of different covariates across methods and across different quantiles. In spite of the differences in the screening criteria and portfolio management of both types of funds, variation in the performance is only found for some of the quantiles of the conditional distribution of mutual fund performance. We established that for the most inefficient funds the superior performance of SRI funds is significant. In contrast, for the best mutual funds this evidence vanished and even Islamic funds perform better than SRI.These results show the benefits of performing the analysis using quantile regression.

On the comparative performance of socially responsible and Islamic mutual funds

This is the first study to provide comprehensive analyses of the relative performance of both socially responsible investment (SRI) and Islamic mutual funds. The analysis proceeds in two stages. In the first, the performance of the two categories of funds is measured using partial frontier methods. In the second stage, we use quantile regression techniques. By combining two variants of the Free Disposal Hull (FDH) methods (order- m and order- α) in the first stage of analysis and quantile regression in the second stage, we provide detailed analyses of the impact of different covariates across methods and across different quantiles. In spite of the differences in the screening criteria and portfolio management of both types of funds, variation in the performance is only found for some of the quantiles of the conditional distribution of mutual fund performance. We established that for the most inefficient funds the superior performance of SRI funds is significant. In contrast, for the best mutual funds this evidence vanished and even Islamic funds perform better than SRI. These results show the benefits of performing the analysis using quantile regression. © 2013 Elsevier B.V.

Fuzzy sets: Abstraction Axiom, Statistical Interpretation, Observations of Fuzzy Sets

The issues relating fuzzy sets definition are under consideration including the analogue for separation axiom, statistical interpretation and membership function representation by the conditional Probabilities.

Extreme Situations Prediction by MultidimenSional Heterogeneous Time Series Using Logical Decision Functions

* The work is supported by RFBR, grant 04-01-00858-a

Measuring, Modeling, and Forecasting Volatility and Correlations from High-Frequency Data

This dissertation contains four essays that all share a common purpose: developing new methodologies to exploit the potential of high-frequency data for the measurement, modeling and forecasting of financial assets volatility and correlations. The first two chapters provide useful tools for univariate applications while the last two chapters develop multivariate methodologies. In chapter 1, we introduce a new class of univariate volatility models named FloGARCH models. FloGARCH models provide a parsimonious joint model for low frequency returns and realized measures, and are sufficiently flexible to capture long memory as well as asymmetries related to leverage effects. We analyze the performances of the models in a realistic numerical study and on the basis of a data set composed of 65 equities. Using more than 10 years of high-frequency transactions, we document significant statistical gains related to the FloGARCH models in terms of in-sample fit, out-of-sample fit and forecasting accuracy compared to classical and Realized GARCH models. In chapter 2, using 12 years of high-frequency transactions for 55 U.S. stocks, we argue that combining low-frequency exogenous economic indicators with high-frequency financial data improves the ability of conditionally heteroskedastic models to forecast the volatility of returns, their full multi-step ahead conditional distribution and the multi-period Value-at-Risk. Using a refined version of the Realized LGARCH model allowing for time-varying intercept and implemented with realized kernels, we document that nominal corporate profits and term spreads have strong long-run predictive ability and generate accurate risk measures forecasts over long-horizon. The results are based on several loss functions and tests, including the Model Confidence Set. Chapter 3 is a joint work with David Veredas. We study the class of disentangled realized estimators for the integrated covariance matrix of Brownian semimartingales with finite activity jumps. These estimators separate correlations and volatilities. We analyze different combinations of quantile- and median-based realized volatilities, and four estimators of realized correlations with three synchronization schemes. Their finite sample properties are studied under four data generating processes, in presence, or not, of microstructure noise, and under synchronous and asynchronous trading. The main finding is that the pre-averaged version of disentangled estimators based on Gaussian ranks (for the correlations) and median deviations (for the volatilities) provide a precise, computationally efficient, and easy alternative to measure integrated covariances on the basis of noisy and asynchronous prices. Along these lines, a minimum variance portfolio application shows the superiority of this disentangled realized estimator in terms of numerous performance metrics. Chapter 4 is co-authored with Niels S. Hansen, Asger Lunde and Kasper V. Olesen, all affiliated with CREATES at Aarhus University. We propose to use the Realized Beta GARCH model to exploit the potential of high-frequency data in commodity markets. The model produces high quality forecasts of pairwise correlations between commodities which can be used to construct a composite covariance matrix. We evaluate the quality of this matrix in a portfolio context and compare it to models used in the industry. We demonstrate significant economic gains in a realistic setting including short selling constraints and transaction costs.

An efficient DSE using conditional multivariate complex Gaussian distribution

This paper presents an efficient noniterative method for distribution state estimation using conditional multivariate complex Gaussian distribution (CMCGD). In the proposed method, the mean and standard deviation (SD) of the state variables is obtained in one step considering load uncertainties, measurement errors, and load correlations. In this method, first the bus voltages, branch currents, and injection currents are represented by MCGD using direct load flow and a linear transformation. Then, the mean and SD of bus voltages, or other states, are calculated using CMCGD and estimation of variance method. The mean and SD of pseudo measurements, as well as spatial correlations between pseudo measurements, are modeled based on the historical data for different levels of load duration curve. The proposed method can handle load uncertainties without using time-consuming approaches such as Monte Carlo. Simulation results of two case studies, six-bus, and a realistic 747-bus distribution network show the effectiveness of the proposed method in terms of speed, accuracy, and quality against the conventional approach.

Computational learning of the conditional phase-type (C-Ph) distribution: Learning C-Ph distributions

This paper presents a new algorithm for learning the structure of a special type of Bayesian network. The conditional phase-type (C-Ph) distribution is a Bayesian network that models the probabilistic causal relationships between a skewed continuous variable, modelled by the Coxian phase-type distribution, a special type of Markov model, and a set of interacting discrete variables. The algorithm takes a dataset as input and produces the structure, parameters and graphical representations of the fit of the C-Ph distribution as output.The algorithm, which uses a greedy-search technique and has been implemented in MATLAB, is evaluated using a simulated data set consisting of 20,000 cases. The results show that the original C-Ph distribution is recaptured and the fit of the network to the data is discussed.

The Conditional Phase-type distribution: Graphical Models for Continuous Non-Normal Data:The Conditional Phase-type (C-Ph) distribution:

Conditional Gaussian (CG) distributions allow the inclusion of both discrete and continuous variables in a model assuming that the continuous variable is normally distributed. However, the CG distributions have proved to be unsuitable for survival data which tends to be highly skewed. A new method of analysis is required to take into account continuous variables which are not normally distributed. The aim of this paper is to introduce the more appropriate conditional phase-type (C-Ph) distribution for representing a continuous non-normal variable while also incorporating the causal information in the form of a Bayesian network.