On The Dirichlet Distribution

The Dirichlet distribution is a multivariate generalization of the Beta distribution. It is an important multivariate continuous distribution in probability and statistics. In this report, we review the Dirichlet distribution and study its properties, including statistical and information-theoretic quantities involving this distribution. Also, relationships between the Dirichlet distribution and other distributions are discussed. There are some different ways to think about generating random variables with a Dirichlet distribution. The stick-breaking approach and the Pólya urn method are discussed. In Bayesian statistics, the Dirichlet distribution and the generalized Dirichlet distribution can both be a conjugate prior for the Multinomial distribution. The Dirichlet distribution has many applications in different fields. We focus on the unsupervised learning of a finite mixture model based on the Dirichlet distribution. The Initialization Algorithm and Dirichlet Mixture Estimation Algorithm are both reviewed for estimating the parameters of a Dirichlet mixture. Three experimental results are shown for the estimation of artificial histograms, summarization of image databases and human skin detection.

Modelo de regressão beta para teste de estresse em risco de crédito de instituições financeiras

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Estatistica, 2015.

A mathematical model for estimating the extent of solute- and water-flux heterogeneity in multiple sample percolation experiments

Multiple sampling is widely used in vadose zone percolation experiments to investigate the extent in which soil structure heterogeneities influence the spatial and temporal distributions of water and solutes. In this note, a simple, robust, mathematical model, based on the beta-statistical distribution, is proposed as a method of quantifying the magnitude of heterogeneity in such experiments. The model relies on fitting two parameters, alpha and zeta to the cumulative elution curves generated in multiple-sample percolation experiments. The model does not require knowledge of the soil structure. A homogeneous or uniform distribution of a solute and/or soil-water is indicated by alpha = zeta = 1, Using these parameters, a heterogeneity index (HI) is defined as root 3 times the ratio of the standard deviation and mean. Uniform or homogeneous flow of water or solutes is indicated by HI = 1 and heterogeneity is indicated by HI > 1. A large value for this index may indicate preferential flow. The heterogeneity index relies only on knowledge of the elution curves generated from multiple sample percolation experiments and is, therefore, easily calculated. The index may also be used to describe and compare the differences in solute and soil-water percolation from different experiments. The use of this index is discussed for several different leaching experiments. (C) 1999 Elsevier Science B.V. All rights reserved.

Elicitation of subjective crop yield PDF

En este trabajo se investiga la coherencia y confiabilidad de estimaciones de funciones de densidad de probabilidad (FDP) subjetivas de rendimientos de cultivos realizadas por un amplio grupo de agricultores. Se utilizaron tres técnicas de elicitación diferentes: el método de estimación de FDP en dos pasos, la distribución Triangular y la distribución Beta. Los sujetos entrevistados ofrecieron estimaciones para los valores puntuales de rendimientos de cultivos (medio, máximo posible, más frecuente y mínimo posible) y para las FDP basadas en la estimación de intervalos. Para evaluar la persistencia, se utilizaron los conceptos de persistencia temporal y persistencia metodológica. Los resultados son interesantes para juzgar la adecuación de las técnicas de estimación de probabilidades subjetivas a los sistemas de ayuda en la toma de decisiones en agricultura.

Characterization of Probability Distributions Using the Residual Entropy Function

The present study on the characterization of probability distributions using the residual entropy function. The concept of entropy is extensively used in literature as a quantitative measure of uncertainty associated with a random phenomenon. The commonly used life time models in reliability Theory are exponential distribution, Pareto distribution, Beta distribution, Weibull distribution and gamma distribution. Several characterization theorems are obtained for the above models using reliability concepts such as failure rate, mean residual life function, vitality function, variance residual life function etc. Most of the works on characterization of distributions in the reliability context centers around the failure rate or the residual life function. The important aspect of interest in the study of entropy is that of locating distributions for which the shannon’s entropy is maximum subject to certain restrictions on the underlying random variable. The geometric vitality function and examine its properties. It is established that the geometric vitality function determines the distribution uniquely. The problem of averaging the residual entropy function is examined, and also the truncated form version of entropies of higher order are defined. In this study it is established that the residual entropy function determines the distribution uniquely and that the constancy of the same is characteristics to the geometric distribution

Exercise sheet 5

Exercises and solutions in LaTex

Exercise sheet 5 pdf

Exercises and solutions in PDF

Closed-loop feedback computation model of dynamical reputation based on the local trust evaluation in business-to-consumer e-commerce

Trust and reputation are important factors that influence the success of both traditional transactions in physical social networks and modern e-commerce in virtual Internet environments. It is difficult to define the concept of trust and quantify it because trust has both subjective and objective characteristics at the same time. A well-reported issue with reputation management system in business-to-consumer (BtoC) e-commerce is the “all good reputation” problem. In order to deal with the confusion, a new computational model of reputation is proposed in this paper. The ratings of each customer are set as basic trust score events. In addition, the time series of massive ratings are aggregated to formulate the sellers’ local temporal trust scores by Beta distribution. A logical model of trust and reputation is established based on the analysis of the dynamical relationship between trust and reputation. As for single goods with repeat transactions, an iterative mathematical model of trust and reputation is established with a closed-loop feedback mechanism. Numerical experiments on repeated transactions recorded over a period of 24 months are performed. The experimental results show that the proposed method plays guiding roles for both theoretical research into trust and reputation and the practical design of reputation systems in BtoC e-commerce.

Modelo iterativo para estimativa da evapotranspiração de referência provável

O presente trabalho teve por objetivo a apresentação de um modelo iterativo para utilização em microcomputadores capaz de estimar valores de evapotranspiracão de referencia em diversos níveis de probabilidade, a partir de uma série de dados. O procedimento matemático envolvido na técnica iterativa empregada relaciona-se à utilização das funções de densidade gama incompleta e beta. Para tanto, foram utilizados dados quinzenais de evapotranspiracão do mês de março, a partir de uma série de 30 anos para a região de Piracicaba-SP. Através do teste de Kolmogorov-Smirnov, verificou-se que dados estimados através do modelo apresentaram alto grau de ajuste com dados relatados em literatura, justificando assim sua utilização.

ESTUDO DA DISTRIBUIÇÃO DA EVAPOTRANSPIRAÇÃO DE REFERÊNCIA VISANDO O DIMENSIONAMENTO DE SISTEMAS DE IRRIGAÇÃO

Visando caracterizar a distribuição de freqüência da evapotranspiração de referência na região de Piracicaba, SP, e avaliar as práticas usuais de estimativa da necessidade de água para fins de dimensionamento de sistemas de irrigação, foram utilizados 30 anos de dados de evapotranspiração do mês de setembro, que foi dividido em períodos de 5, 10, 15 e 30 dias. As distribuições Beta e Normal foram aplicadas aos dados e ambas mostraram-se aptas para representá-los. Para valores de probabilidade de ocorrência iguais ou superiores a 60%, verificou-se o aumento do valor da evapotranspiração de referência com a diminuição do período. Adotando como parâmetros o período de máxima exigência hídrica de 2 a 3 semanas e a evapotranspiração de referência ao nível de 75% de probabilidade, verificou-se que o uso do valor médio mensal da evapotranspiração no dimensionamento de sistemas de irrigação conduz ao subdimensionamento, enquanto a adoção do máximo valor diário da evapotranspiração acarreta o superdimensionamento.

Evaluation Method for Strategic Investments

Real Options Analysis (ROA) has become a complimentary tool for engineering economics. It has become popular due to the limitations of conventional engineering valuation methods; specifically, the assumptions of uncertainty. Industry is seeking to quantify the value of engineering investments with uncertainty. One problem with conventional tools are that they may assume that cash flows are certain, therefore minimizing the possibility of the uncertainty of future values. Real options analysis provides a solution to this problem, but has been used sparingly by practitioners. This paper seeks to provide a new model, referred to as the Beta Distribution Real Options Pricing Model (BDROP), which addresses these limitations and can be easily used by practitioners. The positive attributes of this new model include unconstrained market assumptions, robust representation of the underlying asset‟s uncertainty, and an uncomplicated methodology. This research demonstrates the use of the model to evaluate the use of automation for inventory control.

Estimating the mean and standard deviation of concentration distributions using censored samples

Environmental data sets of pollutant concentrations in air, water, and soil frequently include unquantified sample values reported only as being below the analytical method detection limit. These values, referred to as censored values, should be considered in the estimation of distribution parameters as each represents some value of pollutant concentration between zero and the detection limit. Most of the currently accepted methods for estimating the population parameters of environmental data sets containing censored values rely upon the assumption of an underlying normal (or transformed normal) distribution. This assumption can result in unacceptable levels of error in parameter estimation due to the unbounded left tail of the normal distribution. With the beta distribution, which is bounded by the same range of a distribution of concentrations, $\rm\lbrack0\le x\le1\rbrack,$ parameter estimation errors resulting from improper distribution bounds are avoided. This work developed a method that uses the beta distribution to estimate population parameters from censored environmental data sets and evaluated its performance in comparison to currently accepted methods that rely upon an underlying normal (or transformed normal) distribution. Data sets were generated assuming typical values encountered in environmental pollutant evaluation for mean, standard deviation, and number of variates. For each set of model values, data sets were generated assuming that the data was distributed either normally, lognormally, or according to a beta distribution. For varying levels of censoring, two established methods of parameter estimation, regression on normal ordered statistics, and regression on lognormal ordered statistics, were used to estimate the known mean and standard deviation of each data set. The method developed for this study, employing a beta distribution assumption, was also used to estimate parameters and the relative accuracy of all three methods were compared. For data sets of all three distribution types, and for censoring levels up to 50%, the performance of the new method equaled, if not exceeded, the performance of the two established methods. Because of its robustness in parameter estimation regardless of distribution type or censoring level, the method employing the beta distribution should be considered for full development in estimating parameters for censored environmental data sets. ^

Quantifying effects of soil heterogeneity on groundwater pollution in four sites in the USA

Four sites located in the north-eastern region of the United States of America have been chosen to investigate the impacts of soil heterogeneity in the transport of solutes (bromide and chloride) through the vadose zone (the zone in the soil that lies below the root zone and above the permanent saturated groundwater). A recently proposed mathematical model based on the cumulative beta distribution has been deployed to compare and contrast the regions' heterogeneity from multiple sample percolation experiments. Significant differences in patterns of solute leaching were observed even over a small spatial scale, indicating that traditional sampling methods for solute transport, for example the gravity pan or suction lysimeters, or more recent inventions such as the multiple sample percolation systems may not be effective in estimating solute fluxes in soils when a significant degree of soil heterogeneity is present. Consequently, ignoring soil heterogeneity in solute transport studies will likely result in under- or overprediction of leached fluxes and potentially lead to serious pollution of soils and/or groundwater. The cumulative beta distribution technique is found to be a versatile and simple technique of gaining valuable information regarding soil heterogeneity effects on solute transport. It is also an excellent tool for guiding future decisions of experimental designs particularly in regard to the number of samples within one site and the number of sampling locations between sites required to obtain a representative estimate of field solute or drainage flux.

An Estimate of the Probability Pr(X

2000 Mathematics Subject Classification: 33C90, 62E99

The Market for Apples: A Theory of Identity and Consumption

This paper presents an economic model of the effects of identity and social norms on consumption patterns. By incorporating qualitative studies in psychology and sociology, I propose a utility function that features two components – economic (functional) and identity elements. This setup is extended to analyze a market comprising a continuum of consumers, whose identity distribution along a spectrum of binary identities is described by a Beta distribution. I also introduce the notion of salience in the context of identity and consumption decisions. The key result of the model suggests that fundamental economic parameters, such as price elasticity and market demand, can be altered by identity elements. In addition, it predicts that firms in perfectly competitive markets may associate their products with certain types of identities, in order to reduce product substitutability and attain price-setting power.