Maximización de la función de Verosimilitud de Distribuciones de Probabilidad usando Algoritmos Genéticos

Abstract

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 lo 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.

Optimization of Parameters in the Generalized Extreme-Value Distribution Type 1 for Three Populations Using Harmonic Search

“…However, in Latin American countries, human settlements on riverbanks are very common. Material damage and loss of human life happens The incorporation of technology in hydrological studies, specifically in frequency distribution analyses, has allowed the exploration of the use of genetic algorithms [2] and meta-heuristic techniques [17].…”

“…For this reason, this paper considers the evolution of maximum flows within a region highly exposed to hurricanes. This work also creates the possibility that with the proposed methodology, other climatic variables can be studied.Regarding frequency analysis, particularly for extreme events, the objective is to define the events associated with a return period that provide information to carry out the design of hydraulic works, decreasing the uncertainty of the forecast [2].The development of the theory of extreme values comes from the past century, with an intense exploration of applications to different disciplinary areas in the last 30 years. This theory allows the modeling of queues in distribution and can extrapolate information in extreme regions with little or no information [3].…”

“…A detailed study of the origin, characteristics, properties, and applications in hydrology for the function of the Generalized Extreme-Value Distribution (GEV) was presented in Mexico [5,6].The study of 31 historical records of flows, maximum annual precipitation, and maximum wind speeds obtained in rivers and regions of Canada, the United States, England, Kenya, Mexico, New Zealand, and Switzerland was carried out using the GEV distribution, comparing its performance using the expressions of standard error and absolute error of adjustment [7].The graphical, Log-Pearson III, and Gumbel methods were applied to obtain maximum rainfall intensities in pluviographic stations in eastern Venezuela, concluding that Gumbel's methodology is applicable for periods less than 25 years [8].The existence of areas in which the distribution of extreme values for extra-tropical winds presents two populations, where local winds may be the product of two different systems with different physical origins [9]. Each system has its own distribution function, so we can speak of mixed functions in the modeling of information from two populations [10].The presence of cold fronts or tropical cyclones generates atypical values in the series of maximum annual flows (Q) registered by hydrometric stations installed in the study basins [2]. This behavior has been studied using the Gumbel probability distribution function of two populations (EV1-2P) [11], which later gave rise to the development of a new type of probability function named the mixed probability distribution function [12].Considering the statistical analysis of the record of maximum annual scale readings of a hydrometric station in southeastern Mexico, it was concluded that the Gumbel double distribution function carried out the best adjustment.…”

“…Regarding frequency analysis, particularly for extreme events, the objective is to define the events associated with a return period that provide information to carry out the design of hydraulic works, decreasing the uncertainty of the forecast [2].…”

“…The presence of cold fronts or tropical cyclones generates atypical values in the series of maximum annual flows (Q) registered by hydrometric stations installed in the study basins [2]. This behavior has been studied using the Gumbel probability distribution function of two populations (EV1-2P) [11], which later gave rise to the development of a new type of probability function named the mixed probability distribution function [12].…”