Pareto Parameters Research Articles

Pareto parameter estimation by merging locally weighted median of multiple neural networks and weighted least squares

Pareto parameter estimation by merging locally weighted median of multiple neural networks and weighted least squares

Uncertainty Quantification for the Noah‐MP Land Surface Model: A Case Study in a Grassland and Sandy Soil Region

AbstractLand surface models rely on a multitude of parameters to simulate land‐atmosphere interactions, but the parameter uncertainty can limit the reliability of model predictions. This study utilizes a parameter uncertainty quantification (UQ) framework to quantify and reduce the parameter uncertainty of the Noah‐MP land surface model in a grassland and sandy soil region in the Midwest of the USA. First, the sparse polynomial chaos expansion method which can quantify the interaction effect of parameters, is employed. A relatively small parameter sample size (i.e., 20 times of the number of parameters) was sufficient to identify the sensitive parameters; an additional sensitive parameter, the saturated soil hydraulic conductivity, was screened out compared to previous study. Then, based on the selected sensitive parameters, the weighted multi‐objective adaptive surrogate modeling‐based optimization algorithm is used as the parameter optimization method. The optimization results showed that the root mean square error of flux of latent heat (FLH) on about 82% of the total grids was reduced, and the number was about 57% for gross primary production (GPP) compared to the results using the original parameter settings, indicating that the Pareto parameter set by the UQ framework improved the Noah‐MP model in simulating FLH and GPP in a grassland and sandy soil region in the Midwest of the USA.

Optimization of vehicle structural parameters considering stability improvement

The main structural parameters that affect vehicle bifurcation characteristics and the safe basin are obtained, with the goal of improving the stability. First, the distribution of the system equilibrium points in three-dimensional space is studied, and the bifurcation characteristics of the front steering angles are obtained. Second, the front steering angle range corresponding to the bifurcation point (FSAR-BP) and the safe basin of the system (SBS) are determined and used as optimization objective functions. Considering the influence of the structural parameters on the FSAR-BP and SBS, a sensitivity analysis of the parameters is performed to establish their importance ranking in terms of their influence on the FSAR-BP and SBS. Third, a non-dominated sorting genetic algorithm II (NSGA-II) is utilized to optimize the parameters, and the competitive Pareto parameter optimization solution sets are obtained. Finally, the three solution sets with better comprehensive performance are selected for comparison with the pre-optimized set in the FSAR-BP and SBS. The results indicate that the optimized FSAR-BP increased by 218.81%, 237.83%, and 267.88%, and the optimized SBS increased by 207.52%, 197.06%, and 173.93%, respectively.

General Entropy with Bayes Techniques under Lindley and MCMC for Estimating the New Weibull–Pareto Parameters: Theory and Application

Censored data play a pivotal role in life testing experiments since they significantly reduce cost and testing time. Hence, this paper investigates the problem of statistical inference for a system of progressive first-failure censoring data for a new Weibull–Pareto distribution. Maximum likelihood estimates for the parameters as well as some lifetime indices such as reliability, hazard rate functions, and coefficient of variation are derived. Lindley approximation and the Markov chain Monte Carlo technique are applied to obtain the Bayes estimates relative to two different loss functions: balanced linear exponential and general entropy loss functions. The results of the Bayes estimate are computed under the consideration of informative prior function. A real-life example "the survival times in years of a group of patients given chemotherapy treatment" is presented to illustrate the proposed methods. Finally, a simulation study is carried out to determine the performance of the maximum likelihood and Bayes estimates and compare the performance of different corresponding confidence intervals.

Probabilistic Predictions of Ecologically Relevant Hydrologic Indices Using a Hydrological Model

AbstractThis study evaluated the effects of prior knowledge obtained from multi‐objective optimization on the uncertainty quantification of simulated ecologically relevant hydrologic indices (ERHIs) using a hydrological model. Two experiments were formulated considering contrasting conditions in prior knowledge when implementing Bayesian parameter estimation. In the first experiment, we reproduced a common modeling practice in the absence of previous information where uniform priors are considered when calibrating hydrological and error model parameters. In the second experiment, near‐optimal Pareto solutions from multi‐objective calibration were used to build a multivariate prior distribution for estimating hydrological and error model parameters using Bayesian calibration. In both experiments, a likelihood function considering heteroscedasticity and autocorrelation effects was employed. The Unified Non‐dominated Sorting Algorithm III was used for multi‐objective calibration, whereas the multiple‐try differential evolution adaptive metropolis algorithm was used for Markov Chain Monte Carlo sampling. The experiments were tested using a hydrologic model. We selected a subset of ERHIs comprised of 32 central tendency Indicators of Hydrologic Alteration and seven additional indices describing basic streamflow statistical properties. In general, using near‐optimal Pareto parameter distributions as prior knowledge in Bayesian calibration reduced both the bias and variability ranges in ERHIs prediction. Furthermore, there was no significant loss in the reliability of streamflow predictions (from 2% to 6%, p = 0.07) when targeting the selected ERHIs, while improving precision (from 66% to 58%, p < 0.05) and reducing the bias (from 6% to 4%, p < 0.05). Moreover, parametric uncertainty substantially shrank by 72% when linking multi‐objective calibration and Bayesian parameter estimation.

Groundwater Single- and Multiobjective Optimization Using Harris Hawks and Multiobjective Billiards-Inspired Algorithm

This is the first attempt to combine the Multiobjective Billiards-Inspired Optimization Algorithm (MOBOA) with groundwater modelling to determine pumping rates within a well-distributed range of Pareto options. In this study, in order to determine an optimum solution for groundwater drawdown, pumping rates were selected accompanied by three minimization objectives: minimizing shortage influenced by inability to supply, adjusted shortage index, and minimizing the degree of drawdown within predefined areas. To optimize hydraulic conductivity and specific yield parameters of a modular three-dimensional finite-difference (MODFLOW) groundwater model, the Harris Hawks optimization algorithm was used to minimize the sum of absolute deviation between observed and simulated water-table levels. MOBOA was then utilized to optimize pumping rate variables for an Iranian arid to semiarid groundwater environment using these parameters. As the study results, when the maximum and minimum aquifer drawdown was specified in the range of −40 to +40 cm/year, the Pareto parameter sets produced satisfactory results. Overall, the “Simulation-Optimization-Modelling” protocol was able to generate a series of optimal solutions that were shown on a Pareto front. The study concluded to an optimum approach that provides policy makers in the Iranian water stressed zones with safe groundwater management alternatives.

Experimental analysis of a game-theoretic formulation of target tracking

Optimal trajectories for two platforms with similar dynamics are calculated using a game theoretic formulation. Each platform makes noisy observations of the kinematic state of the other. The objective of each is to maximise observable information about the other while minimising the information the other is able to acquire about it. That is to say, each platform maximises the mutual information between the expected future measurement of the opposing platform and the current likelihood of the state whilst minimising the estimated mutual information between potential measurements of itself by the other and its actual state. The multi-objective optimisation problem for each platform is converted to a single optimisation using the Pareto parameter to weigh the relative importance of the two information measures. The relationship between the two Pareto parameters, and different initial track initialisations is investigated. Remarkably this complex coupled system of two platforms exhibits, for suitably chosen values of the Pareto parameters, interesting cyclical behaviours that are worthy of further exploration.

Automation and Top Income Inequality

Top income inequality has been increasing in the US. Hence, the Pareto parameter as-sociated with the top income distribution is decreasing. In this paper, we provide a theory that links automation technology to the Pareto parameter of the top income distribution. We construct a model in which the span of control is defined by the measure of labor used in production. We model this as a convex cost of labor. This convex cost generates a decreasing returns to scale production function. An improvement in automation enables entrepreneurs to substitute labor with capital and decreases the severity of diseconomies of scale. This leads to higher returns to entrepreneurial skills, a decrease in the Pareto parameter, and an increase in top income inequality. We rationalize the convex cost of labor using a theory of efficiency wages. Using cross-industry and cross-country data, we show that there is a significant correlation between automation and top income inequality.

Confidence regions for Pareto parameters from a single and independent samples

[] Based on a single and on two independent samples, joint confidence regions for parameters of Pareto distributions are proposed with minimum volume properties and without assigning the confidence level to dimensions. In the one-sample case, comparisons are made to former simultaneous confidence sets for Pareto parameters by means of simulation and a real data set. The two-sample case is studied in various set-ups and comprises simultaneous confidence regions for the shape parameters, the scale parameters, and higher-dimensional vectors of these parameters, where common shape and common scale models are also considered.

Non‐stationary peaks‐over‐threshold analysis of extreme precipitation events in Finland, 1961–2016

There is an urgent need to understand and predict how extreme precipitation events (EPEs) will change at high latitudes, both for local climate change adaptation plans and risk mitigation and as a potential proxy “anticipating” the impact of climate change elsewhere in the world. This paper illustrates that a combination of non‐stationary modelling approaches can be adopted to evaluate trends in EPEs under uncertainty. A large database of daily rainfall events from 281 sparsely distributed weather stations in Finland between 1961 and 2016 was analysed. Among the tested methods, Poisson distributions provided a powerful method to evaluate the impacts of multiple physical covariates, including temperature and atmospheric circulation patterns (ACPs), on the resulting trends. The analysis demonstrates that non‐stationarity is statistically valid for the majority of observations, independently of their location in the country and the season of the year. However, subsampling can severely hinder the statistical validity of the trends, which can be easily confused with random noise and therefore complicate the decision‐making processes regarding long‐term planning. Scaling effects have a strong impact on the estimates of non‐stationary parameters, as homogenizing the data in space and time reduces the statistical validity of the trends. Trends in EPE statistics (mean, 90 and 99% percentiles) and best‐fitted Generalized Pareto parameters in the tails of the distributions appear to be stronger when approaching the Polar region (Lapland) than away from it, consistent with the Arctic amplification of climate change. ACPs are key covariates in physically explaining these trends. In particular, the Arctic Oscillation (AO) and North Atlantic Oscillation (NAO) can explain statistically significant increases in extreme precipitation in Lapland, Bothnian and South regions of Finland, particularly during summer and fall seasons.

ESTIMATING PARETO DISTRIBUTION PARAMETERS OF TYPE I

Some of methods like (Moment Method, Maximum Likelihood Method, Term Omission Method) are selected in this paper to estimate the parameters of Pareto (Type I) using simulation of different sample sizes (n=50, 100, 200). A Comparison made between these methods to determine the best method using mean square error criterion.

Identifying the dispersion of the Pareto productivity distribution in international trade

Most of the heterogeneous firm studies use the Pareto distribution of productivity. This paper identifies the Pareto parameter by relying on guidance from monopolistic competition models. The application to Korean export industries reports that the dispersion of productivity varies across industries.

Estimation of the exponential Pareto II distribution parameters

ABSTRACTIn this article, we estimate the parameters of exponential Pareto II distribution by two new methods. The first one is based on the principle of maximum entropy (POME) and the second is by Kullback–Leibler divergence of survival function (KLS). Monte Carlo simulated data are used to evaluate these methods and compare them with the maximum likelihood method. Finally, we fit this distribution to a set of real data by estimation procedures.

Inferences on functions of Pareto parameters with application to income inequality measures

ABSTRACTThe Theil, Pietra, Éltetö and Frigyes measures of income inequality associated with the Pareto distribution function are expressed in terms of parameters defining the Pareto distribution. Inference procedures based on the generalized variable method, the large sample method, and the Bayesian method for testing of, and constructing confidence interval for, these measures are discussed. The results of Monte Carlo study are used to compare the performance of the suggested inference procedures from a population characterized by a Pareto distribution.

Sustainable groundwater modeling using single- and multi-objective optimization algorithms

This study presents the first attempt to link the multi-algorithm genetically adaptive search method (AMALGAM) with a groundwater model to define pumping rates within a well distributed set of Pareto solutions. The pumping rates along with three minimization objectives, i.e. minimizing shortage affected by the failure to supply, modified shortage index and minimization of extent of drawdown within prespecified regions, were chosen to define an optimal solution for groundwater drawdown and subsidence. Hydraulic conductivity, specific yield parameters of a modular three-dimensional finite-difference (MODFLOW) groundwater model were first optimized using Cuckoo optimization algorithm (COA) by minimizing the sum of absolute deviation between the observed and simulated water table depths. These parameters were then applied in AMALGAM to optimize the pumping rate variables for an arid groundwater system in Iran. The Pareto parameter sets yielded satisfactory results when maximum and minimum drawdowns of the aquifer were defined in a range of −40 to +40 cm/year. Overall, ‘Modelling – Optimization – Simulation’ procedure was capable to compute a set of optimal solutions displayed on a Pareto front. The proposed optimal solution provides sustainable groundwater management alternatives to decision makers in arid region.

Estimation of Pareto Parameters Using a Fuzzy Least-Squares Method and Other Known Techniques with a Comparison

The purpose of this paper is to obtain the fuzzy least-squar es estimator for the two-parameter Pareto distribution and to compare the fuzzy estimator with different types of estimators. The trimmed linear moments (TL-moments), linear moments (L-moments) and linear quantile moments (LQ-moments) formulas will be obtained for the two-parameter Pareto distribution and the TL-moments estimator, L-moments estimator and LQ-moments estimator will be derived for the Pareto distribution. Numerical comparisons between the proposed method and the existing methods are implemented. According to these comparisons, it is suggested that the proposed fuzzy least-squares estimator is preferable all times.

Optimal progressive group censoring scheme under cost considerations for pareto distribution

In this article, optimal design under the restriction of pre-determined budget of experiment is developed for the Pareto distribution when the life test is progressively group censored. We use the maximum-likelihood method to obtain the point estimator of the Pareto parameter. We propose two approaches to decide the number of test units, the number of inspections, and the length of inspection interval under limited budget such that the asymptotic variance of estimator of Pareto parameter is minimum. A numerical example is given to illustrate the proposed method. Some sensitivity analysis is also studied.

AN EMPIRICAL STUDY OF THE DISTRIBUTIONS OF THE REVENUES AND PROFITS OF FORTUNE 1000 COMPANIES WITH FOCUS ON PARETO�S LAW, ZIPF�S LAW, AND GINI INDEX

ABSTRACT In this paper, we study the empirical distributions of the revenues and profits of the Fortune 1000 companies with a focus on the applicability of Pareto’s law and Zipf’s law. Our study confirms Pareto’s 80-20 rule that the top 20% of the publicly traded companies account for more than 80% of the total corporate revenues and profits. We find the Gini Indexes for both the revenue and profit distributions to be relatively high, indicating a high degree of revenue and profit inequality among the Fortune 1000 companies. We estimate the parameters of Pareto’s distribution using three different methods: (a) Least Squares, (b) Maximum Likelihood, and (c) Method of Moments (or Quantiles). The Least Squares Method is used to estimate the slope parameter of: (a) the log-log plot of the ranks and revenues, and (b) the log-log plot of the ranks and profits of the companies. The estimation from Maximum Likelihood Method did not yield useful results. But the estimates obtained from the third method and the Gini Index were not only closer to one another, but also found to be in the proximity of the values obtained by the Least Squares Method. The revenue distribution of the Fortune 1000 companies does approximately follow Zipf’s law and its upper tail does conform to a Pareto’s distribution. The distribution of profits does not follow Zipf’s law, but the upper tail again conforms to a Pareto’s distribution. The values of the slope parameter obtained in our study are within the range of the values obtained by earlier researchers, and the results of our study are consistent with the findings reported in the literature. Keywords Pareto’s Law, Zipf’s Law, Gini Index, Fortune 1000 Companies

Computing optimal confidence sets for Pareto models under progressive censoring

This paper presents two families of joint confidence sets for the Pareto shape and precision parameters based on progressively censored data, which include the existing regions in the literature as particular cases, and derives explicit expressions for the corresponding areas. Upon using fiducial arguments, an exact and quick computational method is then proposed in order to find the smallest-area Pareto region which depends on the minimal sufficient statistic with a specified confidence level. Shortest-length confidence intervals for the Pareto parameters are also provided. The reduction in area of the minimal joint confidence region with respect to the existing sets is substantial in most situations, and impressive in some cases. For illustrative and comparative purposes, two practical studies concerning device lifetimes and business failures are considered. Applications of the smallest confidence sets include uses in hypothesis testing and estimation. In particular, it is possible to construct confidence intervals for functions of the Pareto parameters, as well as pointwise and simultaneous confidence bands for the Pareto reliability function.

Estimation of Pareto clutter parameters using order statistics and linear regression

Using a transformation of the order statistic for a series of Pareto distributed random variables, a linear regression approach is applied to estimating Pareto parameters for data sets. This method is shown to perform better than a standard maximum likelihood estimation procedure.