Maximization of the Likelihood Function of Probability Distributions using Genetic Algorithms

[ES] Tradicionalmente, para obtener los parámetros de una función de distribución con el método de máxima verosimilitud se acostumbra igualar a cero la derivada del logaritmo de la función de verosimilitud y resolver el sistema de ecuaciones no lineales que resulta. La popularidad del procedimiento se debe a su sencillez; sin embargo, cuando la función de verosimilitud no es suficientemente regular, puede llevar a obtener un valor muy alejado del máximo Por ese motivo, en este documento se presenta el uso de un algoritmo genético que permite encontrar los parámetros de la función de distribución (con los que se maximiza directamente la función de verosimilitud, o su logaritmo), sin recurrir a la derivada de los logaritmos de dicha función. Se halló buena concordancia de los resultados respecto a los obtenidos usando un software de uso frecuente en México, para el caso las funciones Gumbel y Gumbel de dos poblaciones. [EN] Traditionally, to get the parameters of a distribution function with the maximum likelihood method is usually equaled to zero the derivative of the logarithm of the likelihood function and then the resulting non-linear system of equations is solved. The popularity of the procedure is due to its simplicity; however, when the likelihood function is not regular enough, can lead to obtain a value very far away from the maximum sought. This document presents the use of a genetic algorithm that allows to find the parameters of the distribution function by directly maximizing the likelihood function, or its logarithm, without need to resort to the derivative of the logarithms of the function. The results are compared with those obtained using a software frequently used in Mexico, for the case functions Gumbel and Gumbel of two populations.Fuentes Mariles, OA.; Arganis Juárez, ML.; Domínguez Mora, R.; Fuentes Mariles, GE.; Rodríguez Vázquez, K. (2015). Maximización de la función de Verosimilitud de Distribuciones de Probabilidad usando Algoritmos Genéticos. Ingeniería del agua. 19(1):17-29. https://doi.org/10.4995/ia.2015.3225OJS1729191Arganis-Juárez, M.L., Domínguez-Mora, R., González-Villarreal, F., Carrizosa-Elizondo, E., Esquivel-Garduño, G., Hollands, A.J., Ramírez-Salazar, L.E. (2009). Estudio Integral de la Cuenca Alta del Río Grjialva. Actualización de Avenidas de Diseño. Para CFE. Informe Final.Baker, J.E. (1985). Adaptive Search Selection Methods for Genetic Algorithms, in Proceedings of the First International Conference on Genetic Algorithms (Grefenstette, ed), Lawrence Erlbaum, 101-111.Clark, C., Whu, Y.Z. (2006). Integrated hydraulic model and genetic algorithm optimization for informed analysis of a real water system. Asce 8th Annual International Symposium On Water Distribution System Analysis, Cincinnati, August 27-30, Ohio.Domínguez-Mora, R., Carrizosa-Elizondo, E., Fuentes-Mariles, G.E., Arganis-Juárez, M.L. (2000). Estudio de diferentes aspectos sobre el funcionamiento de la obra de excedencias del Proyecto Hidroeléctrico, la Angostura, Chiapas y actualización de la hidrología para el sistema de presas del Río Grijalva. "Estudio Hidrológico de la Cuenca alta del Río Grijalva". Para CFE. Informe final.Domínguez-Mora, R., Fuentes-Mariles, G.E., Arganis-Juárez, M.L. (2004). Optimación de los parámetros de la función de distribución doble gumbel usando algoritmos genéticos en una serie de gastos máximos anuales. XXI Congreso Latinoamericano de Hidráulica, Sao Paulo, Brasil.Domínguez-Mora, R., Arganis-Juárez, M.L., Carrizosa-Elizondo, E., Fuentes-Mariles, G.E., Echeverri, C.A. (2006). Determinación de Avenidas de Diseño y Ajuste de los Parámetros del Modelo de Optimización de las Políticas de Operación del Sistema de Presas del Río Grijalva. Para CFE. Informe Final.Escalante-Sandoval, C., Reyes-Chávez, L. (2002). Técnicas Estadísticas en Hidrología. Facultad de Ingeniería. Universidad Nacional Autónoma de México.Fuentes-Mariles, O.A., Fuentes-Mariles, G.E., Domínguez-Mora, R. (2005). Optimación de los parámetros de algunas funciones de distribución de probabilidad de gastos máximos anuales usando un algoritmo genético simple. 4a. Conferencia Iberoamericana en Sistemas Cibernética e Informática, Cicsi, Orlando, Flo., Usa, Vol. 2, 156-159.Fuentes-Mariles, O.A. Domínguez-Mora, R., Fuentes-Mariles, G.E., Arganis-Juárez, M.L., Rodríguez-Vázquez, K. (2006). Estimación de los parámetros de funciones de distribución empleadas en hidrología usando ecuaciones de máxima verosimilitud y algoritmos genéticos. XXII Congreso Latinoamericano De Hidráulica, Ciudad Guayana, Venezuela.Goldberg, D.E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. Addison-Wesley, USA.González-Villarreal, F. (1970). Contribución al análisis de frecuencias de valores extremos de los gastos máximos en un río. Serie Azul, Instituto de Ingeniería, UNAM.Gumbel, E.J. (1958). Statistics of Extremes, Columbia University Press, New York. (citado por Koutsoyiannis, D., 2003)Holland, J.H. (1975). Adaptation in Natural and Artificial Systems. The University of Michigan Press.Horbelt, W., Timmer, J., Voss, H.U. (2002). Parameter estimation in nonlinear delayed feedback systems from noisy data. Physics Letters A. 299(5-6): 513-521. doi:10.1016/S0375-9601(02)00748-XJenkinson, A.F. (1955). The frequency distribution of the annual maximum (or minimum) value of meteorological elements, Quarterly Journal of the Royal Meteorological Society 81, 158-171. (citado por Koutsoyiannis, D.,2003)Jenkinson, A.F. (1969). Estimation of maximum floods, World Meteorological Organization, Technical Note No. 98, ch. 5, 183-257. (citado por Koutsoyiannis, D.,2003)Jha, M.K., Nanda G., Samuel, M.P. (2004). Determining hydraulic characteristics of production wells using genetic algorithm Water Resources Management, 18(4): 353-377. doi:10.1023/B:WARM.0000048485.62254.1cJiménez-Espinoza. M. (1996). Programa Ax. Área De Riesgos Hidrometeorológicos. Centro Nacional de Prevención de Desastres. México.Kite, G.W. (1988). Frequency And Risk Analyses In Hidrology. Littletown, Colorado.USA.Koutsoyiannis, D. (2003). On the appropriateness of the gumbel distribution in modelling extreme rainfall. Hydrological Risk: recent advances in peak river flow modelling, prediction and real-time forecasting. Assessment of the impacts of land-use and climate changes. Proceedings of the ESF LESC Exploratory Workshop held at Bologna, Italy, October 24-25, 303-319.Liu, Y., Khu, S.T., Savic D. (2004). A Hybrid Optimization Method Of Multi-Objective Genetic Algorithm (Moga) And K-Nearest Neighbor (Knn) Classifier for Hydrological Model Calibration. Lecture Notes In Computer Sciences, Volume 3177, 546-551. doi:10.1007/978-3-540-28651-6_80Mazariegos, B.R., Raynal-V., J.A. (2002). Paquete Interactivo Para La Estimación De Parámetros De La Distribución Weibull, B14. Memorias Del XX Congreso Latinoamericano De Hidráulica, La Habana, Cuba.Myung, I.J. (2003). Tutorial on maximum likelihood estimation. Journal of Mathematical Psychology, 47(1): 90-100, doi:10.1016/S0022-2496(02)00028-7.Nicklow, J.W., Ozkurt O., Bringer Jr, J.A. (2003). Control of Channel Bed Morphology in Large-Scale River Networks using a Genetic Algorithm, Water Resources Management, 17(2): 113-132. doi:10.1023/A:1023609806431O-Matrix Statistical Time Series Analysis. Stsa Toolbox Version 2. (2005). The Time Series Analysis Toolbox For O-Matrix, http://www.omatrix.com/Stsav2.htmlRao, A.R., Hamed, K.H. (2000). Flood Frequency Analysis. Crc Press, USA, Web Site: Google.Books.ComRossi, F., Florentino, M., Versace, P. (1984). Two-Component Extreme Value Distribution for Flood Frequency Analysis, Water Resources Research 20(7), 847-856. doi:10.1029/WR020i007p00847Smith, R.L. (1988). Forecasting Records By Maximum Likelihood. Journal Of The American Statistical Association, 83(402): 331-338. doi:10.2307/2288847.The Mathworks, Inc. (1992). The Mathworks Matlab Reference Guide

Distributed Indexing Schemes for k-Dominant Skyline Analytics on Uncertain Edge-IoT Data

Skyline queries typically search a Pareto-optimal set from a given data set to solve the corresponding multiobjective optimization problem. As the number of criteria increases, the skyline presumes excessive data items, which yield a meaningless result. To address this curse of dimensionality, we proposed a k-dominant skyline in which the number of skyline members was reduced by relaxing the restriction on the number of dimensions, considering the uncertainty of data. Specifically, each data item was associated with a probability of appearance, which represented the probability of becoming a member of the k-dominant skyline. As data items appear continuously in data streams, the corresponding k-dominant skyline may vary with time. Therefore, an effective and rapid mechanism of updating the k-dominant skyline becomes crucial. Herein, we proposed two time-efficient schemes, Middle Indexing (MI) and All Indexing (AI), for k-dominant skyline in distributed edge-computing environments, where irrelevant data items can be effectively excluded from the compute to reduce the processing duration. Furthermore, the proposed schemes were validated with extensive experimental simulations. The experimental results demonstrated that the proposed MI and AI schemes reduced the computation time by approximately 13% and 56%, respectively, compared with the existing method.Comment: 13 pages, 8 figures, 12 tables, to appear in IEEE Transactions on Emerging Topics in Computin

Application of Fractional Calculus to Rainfall-Streamflow Modelling

There is evidence that hydrologic systems exhibit memory processes that may be represented by fractional order systems. A new theory is developed in this work that generalises the classical unit hydrograph technique for the rainfall-runoff transformation. The theory is based upon a fractional order linear deterministic systems approach subject to an initial condition and is taken to apply to the entire rainfallstreamflow transformation (i.e. including baseflow). The general equation for a cascade of time-lagged linear reservoirs of fractional order subject to a constant initialisation function is derived, and is shown to be a form of fractional relaxation model. Dooge's (1959) general theory of the instantaneous unit hydrograph is shown to fit within the new theoretical framework. Similarly the relationship to the general storage equation of Chow and Kulandaiswamy (1971) is demonstrated. It is shown that the correct initialisation of cascade models requires a substantial number of initial conditions which may limit the viability of applying them in practice. Consequently, the differential formulation of the classical Nash cascade has been corrected and reinterpreted. The unbounded nature of the solution to the convolution integral form of the single fractional relaxation model is overcome by application of the Laplace transform of the pulse rainfall hyetograph following Wang and Wu (1983). The model parameters are fitted using the genetic algorithm. The fractional order cascade equations are tested for classical rainfall-runoff modelling using a set of 22 events for the River Nenagh. The cascade of 2 unequal fractionalorder reservoirs is shown to converge to that of the integer order case, whilst the cascade of equal reservoirs shows some differences. For the modelling of the total rainfall-streamflow process the single fractional order reservoir model with a constant initialisation function is tested on a selection of events for a range of UK catchment scales (22km^ to 510km ). A rainfall loss model is incorporated to account for infiltration and evapotranspiration. The results show that the new approach is viable for modelling the rainfall-streamflow transformation at the lumped catchment scale, although the parameter values are not constant for a given catchment. Further work is recommended on determining the nature of the initialisation function using field studies to improve the identification of the model parameters on an event-by-event basis

The Analysis and Application of Artificial Neural Networks for Early Warning Systems in Hydrology and the Environment

Final PhD thesis submissionArtificial Neural Networks (ANNs) have been comprehensively researched, both from a computer scientific perspective and with regard to their use for predictive modelling in a wide variety of applications including hydrology and the environment. Yet their adoption for live, real-time systems remains on the whole sporadic and experimental. A plausible hypothesis is that this may be at least in part due to their treatment heretofore as “black boxes” that implicitly contain something that is unknown, or even unknowable. It is understandable that many of those responsible for delivering Early Warning Systems (EWS) might not wish to take the risk of implementing solutions perceived as containing unknown elements, despite the computational advantages that ANNs offer. This thesis therefore builds on existing efforts to open the box and develop tools and techniques that visualise, analyse and use ANN weights and biases especially from the viewpoint of neural pathways from inputs to outputs of feedforward networks. In so doing, it aims to demonstrate novel approaches to self-improving predictive model construction for both regression and classification problems. This includes Neural Pathway Strength Feature Selection (NPSFS), which uses ensembles of ANNs trained on differing subsets of data and analysis of the learnt weights to infer degrees of relevance of the input features and so build simplified models with reduced input feature sets. Case studies are carried out for prediction of flooding at multiple nodes in urban drainage networks located in three urban catchments in the UK, which demonstrate rapid, accurate prediction of flooding both for regression and classification. Predictive skill is shown to reduce beyond the time of concentration of each sewer node, when actual rainfall is used as input to the models. Further case studies model and predict statutory bacteria count exceedances for bathing water quality compliance at 5 beaches in Southwest England. An illustrative case study using a forest fires dataset from the UCI machine learning repository is also included. Results from these model ensembles generally exhibit improved performance, when compared with single ANN models. Also ensembles with reduced input feature sets, using NPSFS, demonstrate as good or improved performance when compared with the full feature set models. Conclusions are drawn about a new set of tools and techniques, including NPSFS and visualisation techniques for inspection of ANN weights, the adoption of which it is hoped may lead to improved confidence in the use of ANN for live real-time EWS applications.EPSRCUKWIRThe Environment Agenc

EG-ICE 2021 Workshop on Intelligent Computing in Engineering

The 28th EG-ICE International Workshop 2021 brings together international experts working at the interface between advanced computing and modern engineering challenges. Many engineering tasks require open-world resolutions to support multi-actor collaboration, coping with approximate models, providing effective engineer-computer interaction, search in multi-dimensional solution spaces, accommodating uncertainty, including specialist domain knowledge, performing sensor-data interpretation and dealing with incomplete knowledge. While results from computer science provide much initial support for resolution, adaptation is unavoidable and most importantly, feedback from addressing engineering challenges drives fundamental computer-science research. Competence and knowledge transfer goes both ways

Renewable Energy Resource Assessment and Forecasting

In recent years, several projects and studies have been launched towards the development and use of new methodologies, in order to assess, monitor, and support clean forms of energy. Accurate estimation of the available energy potential is of primary importance, but is not always easy to achieve. The present Special Issue on ‘Renewable Energy Resource Assessment and Forecasting’ aims to provide a holistic approach to the above issues, by presenting multidisciplinary methodologies and tools that are able to support research projects and meet today’s technical, socio-economic, and decision-making needs. In particular, research papers, reviews, and case studies on the following subjects are presented: wind, wave and solar energy; biofuels; resource assessment of combined renewable energy forms; numerical models for renewable energy forecasting; integrated forecasted systems; energy for buildings; sustainable development; resource analysis tools and statistical models; extreme value analysis and forecasting for renewable energy resources

EG-ICE 2021 Workshop on Intelligent Computing in Engineering

The 28th EG-ICE International Workshop 2021 brings together international experts working at the interface between advanced computing and modern engineering challenges. Many engineering tasks require open-world resolutions to support multi-actor collaboration, coping with approximate models, providing effective engineer-computer interaction, search in multi-dimensional solution spaces, accommodating uncertainty, including specialist domain knowledge, performing sensor-data interpretation and dealing with incomplete knowledge. While results from computer science provide much initial support for resolution, adaptation is unavoidable and most importantly, feedback from addressing engineering challenges drives fundamental computer-science research. Competence and knowledge transfer goes both ways