Modified Maximum Likelihood Method Research Articles

Analysis and efficient comparison of ten numerical methods in estimating Weibull parameters for wind energy potential: Application to the city of Bafoussam, Cameroon

Abstract The aim of this paper is to analyze and compare efficiently of 10 (ten) numerical methods namely, the empirical method of Justus (EMJ), the empirical method of Lysen (EML), the method of moments (MoM), the graphical method (GM), the Mabchour’s method (MMab), the energy pattern factor method (EPFM), the maximum likelihood method (MLM), the modified maximum likelihood method (MMLM), the equivalent energy method (EEM), and the alternative maximum likelihood method (AMLM) in order to estimate Weibull parameters for wind energy potential. They were performed by using wind speed data collected in the meteorological station of Bafoussam city, in the west region of Cameroon, in the period from 2007 to 2013. The results of this study obtained from statistical analysis show that the MLM presents relatively more excellent ability throughout the simulation tests, followed by EEM, EPFM and EMJ respectively. They also demonstrated that EEM presented minimum error in estimating the monthly wind power density and that the wind potential of Bafoussam city can be interesting for some applications such as rural electrification and water pumping in agriculture.

Comparison of six different methods of Weibull distribution for wind power assessment: A case study for a site in the Northern region of Pakistan

Pakistan is one of those countries that heavily dependent on hydrocarbon fuel for energy production which results in severe climate change. Although renewable energy resources, like wind, are available in abundance, they are not completely utilized due to which energy crises of the country increases day by day. This study comprehensively elaborates the potential of wind power of a site located in the Northern part of Pakistan. The potential of wind power for the site was accessed and analyzed by using two years (2016–18) wind data at four heights (80, 60, 40 and 20 m from sea level). Six numerical methods are used for the estimation of Weibull parameters (shape ‘k’ and scale ‘c’). Using these parameters, Weibull mean velocity, power density, and energy density are computed at all mast heights for each year. The seasonal wind characteristics parameters and power density are also calculated. Among these methods, the modified maximum likelihood method (MMLM) is found to be the most effective method to represent the wind data, while the graphical method (GM) showed the weakest fit. Root mean square error (RMSE), determination of coefficient (R2), and mean bias error (MBE) are computed to validate the computed results. Wind power potential assessment of the calculated data shows that the proposed location is appropriate for small scale wind energy generation.

Estimating Morgenstern Type Bivariate Association Parameter Using a Modified Maximum Likelihood Method

This paper investigates estimating the association parameter of Morgenstern type bivariate distribution using a modified maximum likelihood method where the regular maximum likelihood methods failed to achieve estimation. The simple random sampling, concomitant of ordered statistics and bivariate ranked set sampling methods are used and compared. Efficiency and bias of the produced estimators are compared for two specific examples, Morgenstern type bivariate uniform and exponential distributions.

Assessment of wind energy potential using wind energy conversion system

Wind energy, as a renewable resource, is the most rapidly growing source that produces electrical energy using wind turbines. Such a wind energy conversion system is both economical and is environmental friendly. It requires understanding of wind conditions at the site under study. With this intent, wind characteristics of Jhampir (district Thatta Sindh, Pakistan) are investigated and wind energy potential is determined. The study is conducted using 10-min averaged wind speed data obtained from Alternate Energy Development Board of Pakistan for a period of three years (2007–2010). Monthly, seasonal, and yearly analysis is performed by fitting measured wind speed data to a Weibull distribution function. Weibull shape and scale parameters are determined numerically using Maximum Likelihood Method, Modified Maximum Likelihood Method, and Energy Pattern Factor Methods. The suitability of the fit is assessed using goodness-of-fit tests, such as, Root Mean Square Error, Coefficient of Determination (R2), and Chi-Square (χ2) tests. In all three data analysis periods, RMSE values varied between 10−2 and 10−4. Similarly, R2 values varied between 0.989 and 0.996 and χ2-test between 10−4 and 10−8. For entire data set, all the tests showed better performance of Maximum Likelihood and Modified Maximum Likelihood Methods compared to Energy Pattern Factor. In case of monthly analysis, Maximum Likelihood Method performed better compared to Modified Maximum Likelihood Method and Energy Pattern Factor according to root mean square error and χ2 tests results. Seasonal performance of all the methods is found to be similar with marginal superiority of MLM over other methods. A very good agreement is observed between standard deviation values for measured wind speed data distribution and fitted Weibull distribution using Maximum Likelihood Method estimator. Additionally, to understand the optimum directional efficiency, directional wind power densities are calculated. Finally, a wind turbine is used to the seasonal and yearly wind speed data to determine the actual wind energy potential of the site. Extracted wind energy values for four seasons are found to be 1691, 2851, 4572, and 916 kWh with an annual yield of 10054 kWh. Wind energy values obtained for different periods and directions suggest that Jhampir is a suitable site for developing the wind power plant.

Estimation of different wind characteristics parameters and accurate wind resource assessment for Kadavu, Fiji

Wind resource assessment is carried out for a site in Kadavu, Fiji Islands. This included estimating the Weibull parameters and wind power density using ten different methods and carrying out an economic analysis. The wind speeds at 34 m and 20 m above ground level, wind direction, atmospheric pressure and temperature were measured for 18 months and statistically analyzed. The overall average wind speed at a height of 34 m above ground level was found to be 3.63 m/s. The seasonal averages for the site were 3.81 m/s and 3.40 m/s for summer and winter respectively. The diurnal variation of the wind shear for the site was correlated with the temperature variation. The Moments Method was found to be the best method for the entire period of study. The Modified Maximum Likelihood Method was found to be the best for the summer season whilst Median and Quartile method was the best for the winter period. The mean wind power density at the location was found to be 45.88 W/m2. The WAsP software was used to create the wind resource map. Five potential sites were selected installing the wind turbines and for carrying out the economic analysis, which included an estimation of annual energy production. It was found that, for the five turbines average capacity factor would be 20.05%. The payback period for installing the wind turbines at 50 m above the ground level is estimated to be between 6.99 and 8.74 years.

Comparative Study of Different Methods for Estimating Weibull Parameters: A Case Study on Jeju Island, South Korea

On Jeju Island, South Korea, an investigation was conducted to determine the best method for estimating Weibull parameters. Six methods commonly used in many fields of the wind energy industry were reviewed: the empirical, moment, graphical, energy pattern factor, maximum likelihood, and modified maximum likelihood methods. In order to improve the reliability of a research result, five-year actual wind speed data taken from nine sites with various topographical conditions were used for the estimation. Furthermore, the effect of various topographical conditions on the accuracy of the methods was analyzed and 10 bin interval types were applied to determine the most appropriate bin interval based on their performances. Weibull distributions that were estimated using these methods were compared with the observed wind speed distribution. Then the accuracy of each method was evaluated using four accuracy tests. The results showed that of the six methods, the moment method had the best performance regardless of topographical conditions, while the graphical method performed the worst. Additionally, topographical conditions did not affect the accuracy ranking of the methods for estimating the Weibull parameters, while an increase of terrain complexity resulted in an increase of discrepancy between the estimated Weibull distribution and the frequency of the observed wind speed data. In addition, the choice in bin interval greatly affected the accuracy of the graphical method while it did not depend on the accuracy of the modified maximum likelihood method.

Eddington's demon: inferring galaxy mass functions and other distributions from uncertain data

We present a general modified maximum likelihood (MML) method for inferring generative distribution functions from uncertain and biased data. The MML estimator is identical to, but easier and many orders of magnitude faster to compute than the solution of the exact Bayesian hierarchical modelling of all measurement errors. As a key application, this method can accurately recover the mass function (MF) of galaxies, while simultaneously dealing with observational uncertainties (Eddington bias), complex selection functions and unknown cosmic large-scale structure. The MML method is free of binning and natively accounts for small number statistics and non-detections. Its fast implementation in the R-package "dftools" is equally applicable to other objects, such as haloes, groups and clusters, as well as observables other than mass. The formalism readily extends to multi-dimensional distribution functions, e.g. a Choloniewski function for the galaxy mass-angular momentum distribution, also handled by dftools. The code provides uncertainties and covariances for the fitted model parameters and approximate Bayesian evidences. We use numerous mock surveys to illustrate and test the MML method, as well as to emphasise the necessity of accounting for observational uncertainties in MFs of modern galaxy surveys.

Wind potential assessment for an efficient wind farm sizing

The effectiveness of autonomous wind plants depends basically on the characterization, sizing, and environmental design and analysis of its renewable energy conversion system. This article presents an assessment on wind potential characterization to be used to compute the size of a wind farm turbine. Different methods are adopted to estimate parameters of the Weibull distribution. The modified maximum likelihood method is selected as the most accurate with reference to comparison between many approaches output results and measurements provided by the National Institute of Meteorology. Also, an artificial neural network–based algorithm is developed to optimize the MMLM parameters. The monthly wind potential distribution is consequently computed for Sfax, Tunisia. Obtained results are used to optimize the size calculation of wind turbine blades and battery capacity for a standalone wind farm. The proposed approach profitability is evaluated upon the lost produced energy.

Academic salary compression across disciplines and over time

Academic salary compression occurs when professors of lower professorial rank earn salaries close to—or even higher than—salaries of more senior faculty. We present a modified maximum likelihood method for fitting flexible Dagum distributions to limited data that provide only the minimum, maximum, mean, and sample size, and we use this method to study salary compression across 15 academic disciplines over the past 22 years. After examining mean-based compression ratios, we estimate salary percentiles and explore stochastic dominance relationships between estimated salary distributions for different disciplines and professorial ranks. Although salary compression is not seen in most academic disciplines, it is prevalent in business-related disciplines, is increasing in these disciplines, and exhibits examples of stochastic dominance. In addition, salary compression increases as competing nonacademic salaries increase. Finally, we evaluate our methodology, showing that it would likely be useful in a variety of settings.

Estimation of the number of orthogonal signals with unknown parameters

Algorithms for estimation of the number of signals with unknown amplitudes and several non-energy parameters are synthesized on the basis of the modified maximum likelihood method. Asymptotic values of the probabilities of the error in estimation of the signal number are determined. The effect of a reduced volume of the a priori domain of possible values and the dimensionality of the vector of unknown non-energy parameters on the quality of operation of the algorithms for estimation of the number of signals is investigated.

Estimation of P(Y < X) in a Four-Parameter Generalized Gamma Distribution

In this paper we consider estimation of R = P(Y < X), when X and Y are distributed as two independent four-parameter generalized gamma random variables with same location and scale parameters. A modified maximum likelihood method and a Bayesian technique have been used to estimate R on the basis of independent samples. As the Bayes estimator cannot be obtained in a closed form, it has been implemented using importance sampling procedure. A simulation study has also been carried out to compare the two methods.

Assessment of Weibull Parameter by Five Numerical Methods and Estimation of Wind Speed at Rotterdam, Netherland

In this paper Weibull parameters(c and k) have been estimated on wind speed data of Rotterdam, Netherland. We have applied five numerical methods i.e. Modified Maximum Likelihood Method, Maximum Likelihood Method, Energy Pattern Method, Empirical Method and Method of Moments, to calculate the values of c and k. The parameters k and c have used to estimate the probability distribution function and average wind speed. The wind speed data on hourly basis from 2005-2014 have obtained from “Royal Netherlands Meteorological Institute”.

A novel energy pattern factor method for wind speed distribution parameter estimation

Power output of wind turbine depends on many factors. Among them, the most crucial one is wind speed. Since wind speed data is a significant factor for wind energy analyses, it should be modeled accurately. Weibull distribution has been used extensively to model variation of wind speed. Therefore, the most appropriate distribution parameter estimation method selection is critical in order to minimize data set modeling errors. In this context, a novel, robust, efficient and better method than standard methods to estimate Weibull parameters is presented for the first time in this paper. The accuracy of the proposed method is verified using different data sets. Also, developed method is compared with Graphic Method (GM), Maximum Likelihood Method (MLM), Alternative Maximum Likelihood Method (AMLH), Modified Maximum Likelihood Method (MMLH), Moment Method (MM), Justus Moment Method (JMM), WAsP Method (WM) and Power Density Method (PD). The results indicate that the proposed novel method is adequate to determine Weibull distribution parameters.

Performance Analysis of Methods for Estimating Weibull Parameters for Wind Speed Distribution in the District of Maroua

In this study, five numerical Weibull distribution methods, namely, the maximum likelihood method, the modified maximum likelihood method (MLM), the energy pattern factor method (EPF), the graphical method (GM), and the empirical method (EM) were explored using hourly synoptic data collected from 1985 to 2013 in the district of Maroua in Cameroon. The performance analysis revealed that the MLM was the most accurate model followed by the EPF and the GM. Furthermore, the comparison between the wind speed standard deviation predicted by the proposed models and the measured data showed that the MLM has a smaller relative error of -3.33% on average compared to -11.67% on average for the EPF and -8.86% on average for the GM. As a result, the MLM was precisely recommended to estimate the scale and shape parameters for an accurate and efficient wind energy potential evaluation. Keywords : Empirical method; energy pattern factor method; graphical method; maximum likelihood method; modified maximum likelihood method; wind speed.

Bias Reduction of MLEs for Weibull Distributions under Grouped Lifetime Data

ABSTRACTAccelerated life tests (ALTs) usually contain subsampling because of the cost-effective evaluation. The two-stage method can deal with grouped data with subsampling, but the bias of maximum likelihood estimates (MLEs) can be alarmingly high. In this article, we propose reducing the bias of MLEs for grouped data via an unbiasing factor method. We introduce the procedures and give the unbiasing factor values. The proposed method is studied and compared with the modified maximum likelihood method for relative bias (RB) and mean square error (MSE) in Monte Carlo simulations. The results show that the unbiasing factor method is better in most cases.

Weibull Distribution Function for Wind Energy Estimation of Gharo (Sindh)

Weibull distribution function is fitted to a measured wind speed data set at mast height of 30 m and Gharo-Sindh (Pakistan) is selected as a case site under study. Wind speed data recorded in one minute interval for the year 2004 is used to estimate Weibull parameters (k and c). Weibull parameters are calculated using Modified Maximum Likelihood Method (MMLM), Maximum likelihood Method (MLM) and Method of Moment (MoM) and the results obtained are compared. Kolomogorov-Smirnov test, Root Mean Square Error (RMSE) and R2 tests are performed to test the goodness-of-fit of the Weibull distribution function. The analysis is based on recorded monthly and yearly wind speed data. Goodness-of-fit tests indicate a better performance of MMLM and MLM as compared to MoM. Wind power density is estimated for the site under study using MMLM and Weibull function estimator. A lowest Weibull mean wind speed of 3.73 m/s in the month of October and highest value of 7.90 m/s for August are observed and correspondingly power densities of 80.95 W/m2 and 425.87 W/m2 are obtained. Descriptive statistics for the measured wind speed data is also evaluated.

Influence of the Determination Methods of K and C Parameters on the Ability of Weibull Distribution to Suitably Estimate Wind Potential and Electric Energy

The modeling of the wind speed distribution is of great importance for the assessment of wind energy potential and the performance of wind energy conversion system. In this paper, the choice of two determination methods of Weibull parameters shows theirs influences on the Weibull distribution performances. Because of important calm winds on the site of Ngaoundere airport, we characterize the wind potential using the approach of Weibull distribution with parameters which are determined by the modified maximum likelihood method. This approach is compared to the Weibull distribution with parameters which are determined by the maximum likelihood method and the hybrid distribution which is recommended for wind potential assessment of sites having nonzero probability of calm. Using data provided by the ASECNA Weather Service (Agency for the Safety of Air Navigation in Africa and Madagascar), we evaluate the goodness of fit of the various fitted distributions to the wind speed data using the Q – Q plots, the Pearson’s coefficient of correlation, the mean wind speed, the mean square error, the energy density and its relative error. It appears from the results that the accuracy of the Weibull distribution with parameters which are determined by the modified maximum likelihood method is higher than others. Then, this approach is used to estimate the monthly and annual energy productions of the site of the Ngaoundere airport. The most energy contribution is made in March with 255.7 MWh. It also appears from the results that a wind turbine generator installed on this particular site could not work for at least a half of the time because of higher frequency of calm. For this kind of sites, the modified maximum likelihood method proposed by Seguro and Lambert in 2000 is one of the best methods which can be used to determinate the Weibull parameters.

Statistical Diagnosis of the Best Weibull Methods for Wind Power Assessment for Agricultural Applications

The best Weibull distribution methods for the assessment of wind energy potential at different altitudes in desired locations are statistically diagnosed in this study. Seven different methods, namely graphical method (GM), method of moments (MOM), standard deviation method (STDM), maximum likelihood method (MLM), power density method (PDM), modified maximum likelihood method (MMLM) and equivalent energy method (EEM) were used to estimate the Weibull parameters and six statistical tools, namely relative percentage of error, root mean square error (RMSE), mean percentage of error, mean absolute percentage of error, chi-square error and analysis of variance were used to precisely rank the methods. The statistical fittings of the measured and calculated wind speed data are assessed for justifying the performance of the methods. The capacity factor and total energy generated by a small model wind turbine is calculated by numerical integration using Trapezoidal sums and Simpson’s rules. The results show that MOM and MLM are the most efficient methods for determining the value of k and c to fit Weibull distribution curves.

Robust Regression Analysis for Non-Normal Situations under Symmetric Distributions Arising In Medical Research

In medical research, while carrying out regression analysis, it is usually assumed that the independent (covariates) and dependent (response) variables follow a multivariate normal distribution. In some situations, the covariates may not have normal distribution and instead may have some symmetric distribution. In such a situation, the estimation of the regression parameters using Tiku’s Modified Maximum Likelihood (MML) method may be more appropriate. The method of estimating the parameters is discussed and the applications of the method are illustrated using real sets of data from the field of public health.

Inference in multivariate linear regression models with elliptically distributed errors

In this study we investigate the problem of estimation and testing of hypotheses in multivariate linear regression models when the errors involved are assumed to be non-normally distributed. We consider the class of heavy-tailed distributions for this purpose. Although our method is applicable for any distribution in this class, we take the multivariate t-distribution for illustration. This distribution has applications in many fields of applied research such as Economics, Business, and Finance. For estimation purpose, we use the modified maximum likelihood method in order to get the so-called modified maximum likelihood estimates that are obtained in a closed form. We show that these estimates are substantially more efficient than least-square estimates. They are also found to be robust to reasonable deviations from the assumed distribution and also many data anomalies such as the presence of outliers in the sample, etc. We further provide test statistics for testing the relevant hypothesis regarding the regression coefficients.