A mixed-copula-based integral method for reliability analysis of a novel multi-functional rescue end-effector

To evaluate the impact of the randomness and correlation of photovoltaic (PV) and load on AC/DC systems with a multiport current flow controller (M-CFC), this paper proposes a probabilistic optimal power flow calculation for AC/DC systems based on a nonparametric kernel density estimation. First, according to the M-CFC model, the DC power flow calculation method with M-CFC was deduced, and its influence on line loss was analyzed. Second, a nonparametric kernel density estimation with an adaptive bandwidth was used to accurately describe the probability distribution of the PV and load, and correlation samples of the PV and load were obtained by the mixed copula function. Then an optimization model that considers system loss and static security was established, and a fast nondominated sorting genetic algorithm based on the elite strategy (NSGA-II) was used to calculate the multi-objective probability optimal power flow of the AC/DC system. Finally, a case study was performed on the modified IEEE39 bus system using measured PV and load data. We verified the nonparametric kernel density estimation with an adaptive bandwidth can better adapt to random component uncertainty, and M-CFC can improve the static security of the system.

Load is one of the main causes of structural failure, and the correlation among loads would affect the evaluation results of structural performance. The purpose of this paper is to analyze the influence of the correlation among multiple loads on the structural reliability. In this paper, the nonparametric kernel density estimation (NKDE) method is used to estimate the probability density function (PDF) of related loads. In addition, the mixed copula (M-Copula) model is proposed, which combines Gumbel copula, Frank copula, Clayton copula, and weight coefficient, and the model parameters are fitted by MATLAB software to get the correlation of related loads. The reliability based on the related load combination is calculated according to the constructed model. After analyzing three numerical cases, the results show that the probability characteristics of NKDE estimation are very close to the actual conditions, and the reliability calculated by the M-Copula model is larger than those calculated by JCSS, Turkstra, and Gong methods. Using the M-Copula model for load correlation would avoid underestimating the reliability of the structure, which is conducive to structural economic development.

This paper proposes a data-driven approximation of the Cumulative Distribution Function using the Finite Mixtures of the Cumulative Distribution Function of Logistic distribution. Since it is not possible to solve the logistic mixture model using the Maximum likelihood method, the mixture model is modeled to approximate the empirical cumulative distribution function using the computational intelligence algorithms. The Probability Density Function is obtained by differentiating the estimate of the Cumulative Distribution Function. The proposed technique estimates the Cumulative Distribution Function of different benchmark distributions. Also, the performance of the proposed technique is compared with the state-of-the-art kernel density estimator and the Gaussian Mixture Model. Experimental results on κ−μ distribution show that the proposed technique performs equally well in estimating the probability density function. In contrast, the proposed technique outperforms in estimating the cumulative distribution function. Also, it is evident from the experimental results that the proposed technique outperforms the state-of-the-art Gaussian Mixture model and kernel density estimation techniques with less training data.

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bounds of system failure probability. The second method is SA based Nataf approximation, in which SA is used to estimate the probability density function (PDF) and cumulative distribution function (CDF) for the approximately linearized performance function of each failure mode. After the PDF of each failure mode and the correlation coefficients among approximately linearized performance functions are estimated, Nataf distribution is employed to approximate the joint PDF of multiple structural system performance functions, and then the system failure probability can be estimated directly by numerical simulation using the joint PDF. The third method is SA based line sampling (LS). The standardization transformation is needed to eliminate the dimensions of variables firstly in this case. Then LS method can express the system failure probability as an arithmetic average of a set of failure probabilities of the linear performance functions, and the probabilities of the linear performance functions can be estimated by the SA in the non-normal variables space. By comparing basic concepts, implementations and results of illustrations, the following conclusions can be drawn: (1) The first method can only obtain the bounds of system failure probability and it is only acceptable for the linear limit state function; (2) the second method can give the estimation of system failure probability, and its error mostly results from the approximation of Nataf distribution for the joint PDF of the structural system performance functions and the linearization of the performance functions; (3) the SA based LS method can obtain the estimator of system failure probability, which converges to the actual value along with the increase of sample size. The SA based LS method considers the influence of nonlinearity of limit state function on the failure probability, and it is acceptable for the structural system both with a single failure mode and with multiple failure modes, therefore it has the widest applicability.

Modeling diameter distribution is a crucial aspect of forest management, requiring the selection of an appropriate probability density function or cumulative distribution function along with a fitting method. This study compared the suitability of eight probability density functions—A Charlier, beta, generalized beta, gamma, Gumbel, Johnson’s SB, and Weibull (two- and three-parameter)—fitted using both derivative methods (Moments) fitted in SAS/STATTM and optimization methods (MLE) fitted with the ‘optim’ function in R for diameter distribution estimation in forest stands. The A Charlier and Gumbel functions were used for the first time in this type of comparison. The data were derived from 167 permanent sample plots in an Atlantic forest (Quercus robur) and 59 temporary sample plots in tropical forests (Tectona grandis). Fit quality was assessed using various indices, including Kolmogorov–Smirnov, Cramér–von Mises, mean absolute error, bias, and mean squared error. The results indicated that Johnson’s SB function was more suitable for describing the diameter distribution of the stands. Johnson’s SB, three-parameter Weibull, and generalized beta consistently performed well across different fitting methods, while the fits produced by gamma, Gumbel, and two-parameter Weibull were of poor quality.

This research paper developed a new distribution referred as transmuted Newweighted exponential distribution. This is an extension of the popular Exponential distribution by using the transmutation map, the probability density function (pdf) and the cumulative distribution function (cdf) of the developed distribution were derived. Some distributional properties of the new distribution were extensively studied. The estimation of the distribution parameters was done using the method of maximum likelihood estimation. The performance of the TNWED has been evaluated using real life datasets from two (2) previous researches and the results show that our proposed distribution fits the datasets more better compared to the fits of the other two distributions (New Weighted Exponential distribution and Exponential distribution) considered in this study.

In this paper, a new continuous probability distribution is proposed for fitting real data using Biweight kernel function and the exponential distribution. The suggested distribution is named the Biweight-exponential distribution (BiEd). Some statistical properties of this distribution are derived and illustrated mathematically. The probability density function and the cumulative distribution function are derived. Some reliability analysis functions are defined. The moments and moment generating function are derived. Re’nyi entropy is derived. The maximum likelihood method of estimation is used to derive the parameter estimates. The Bonferroni and Lorenz curves and Gini index equations are derived. The distribution of the order statistic and the quantile function are derived as well. The mean and median absolute deviations of the new distribution are derived. A numerical study was conducted to the quantile equation. An application to real data set is conducted to investigate the usefulness of the suggested distribution. In the real data application, the values of Cramer-von misses (W), Anderson Darling statistic (A), Kolmogorov-Smirnov (D) statistic, the p-value, the maximum likelihood estimates (MLE), Akaike information criterion (AIC), consistent Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan Quinn information criterion (HQIC) and minimum value of the log-likelihood function are obtained. The distribution is compared with the exponential (base) distribution based on these criteria. The results showed that the BiED fits better than the exponential distribution. This means that the suggested distribution can replace the exponential distribution for analysis of some real data sets.

In this paper, we have derived and studied new probability distributions by extending the 2‐dimensional Rayleigh distribution (RD). First, we extend the RD to 3 dimensions and then generalize it to k dimensions for any positive integer k ≥ 3. The distributions are named the 3‐dimensional Rayleigh distribution (3‐DRD) and k‐dimensional Rayleigh distribution (k‐DRD), respectively. For both 3‐DRD and k‐DRD, detailed mathematical and statistical properties including derivations of the corresponding cumulative distribution, probability density, survival, and hazard functions, moments, moment generating functions, mode, skewness, kurtosis, and differential entropy are obtained in closed forms. Parameter estimation is done for both models using the maximum likelihood estimation method and some statistical properties of the estimator are discussed for each case. Interestingly, the commonly known Normal, Rayleigh, Maxwell–Boltzmann, chi‐square, gamma, and Erlang distributions are related to the newly developed extended RDs as special cases. For the 3‐DRD, plots of cumulative distribution, probability density, survival, and hazard functions are exhibited, a simulation study is carried out, and random samples are generated using the standard accept–reject (AR) algorithm to check the efficiency of the maximum likelihood estimates of the parameter. Moreover, the new 3‐DRD model is fitted to one simulated and three real datasets, revealing good performance compared to four existing Rayleigh‐based distributions. This study will contribute new knowledge to the field of applied statistics and probability, and the findings will be used as a basis for future research in the field.

Half-Cauchy generalized exponential (HCGE) distribution is a novel distribution that we have proposed on in this paper. The quantiles, the measures of skewness based on quartiles, and the measures of kurtosis based on octiles, survival function, the probability density function, hazard function, cumulative distribution function, cumulative hazard function, are just a few of the crucial statistical properties we have derived for the proposed distribution. To estimate the parameters of the half-Cauchy generalized exponential distribution, the maximum likelihood estimation method has been applied. For the evaluation of the new distribution’s potential, we have considered a real dataset and compared the goodness-of-fit attained by proposed distribution with some competing distribution. The suggested model fits the data much better and is more adaptable than some other models.

In general, distribution theory plays a crucial role in modeling various real-life phenomena, making it a fundamental tool in statistical analysis and decision-making. Over the years, extensive research has been conducted on different statistical distributions and estimation techniques. While the literature abounds with information regarding well-known distributions, there is always room for exploring new variants that can better capture the characteristics of complex phenomena. In this paper, we contribute to the field of distribution theory by introducing novel probability distribution called the Weibull-Weibull distribution. The Weibull-Weibull distribution is derived by compounding two Weibull distributions, and it offers a flexible framework for modeling phenomena that exhibits a complex interplay of factors. By combining the strength of the Weibull distribution with itself, we are able to capture a wider range of shapes and behaviour, providing more accurate representations of real-world occurrences. To facilitate the practical application of the Weibull-Weibull distribution, we employ the maximum likelihood estimation (MLE) approach to estimate its shape and scale parameters. The MLE method is a widely used statistical technique that allows us to determine the most likely values of the parameters based on observed data. By applying this estimation method to the Weibull-Weibull distribution, we enable researchers and practitioners to effectively utilize this new distribution in their analyses and modeling efforts. Furthermore, we delved into a comprehensive study of statistical theory and properties of the Weibull-Weibull distribution. We investigate its moments, cumulative distribution function, probability density function, and other key measures. Through rigorous analysis, we establish the theoretical foundations of the Weibull-Weibull distribution and provide insights into its behaviour and characteristics. This comprehensive examination equips researchers with a solid understanding of the distribution, enabling them make informed decisions and interpretations when working with real-life data. In conclusion, our research introduces the Weibull-Weibull distribution as a valuable addition to the existing repertoire of statistical distributions. By leveraging the power of compounding two Weibull distributions, we provide a flexible and robust framework of modeling complex phenomena. With the use of maximum likelihood estimation and the given Chernoff bound, practitioners are able to estimate the distribution's parameters as well as analyze its tails accurately. Our extensive statistical analysis further enhances the understanding of the Weibull-Weibull distribution, facilitating its practical application in a wide range of fields, including reliability analysis, survival analysis, and risk assessment.

SUMMARY In an attempt to overcome the difficulties of the full waveform inversion (FWI), several alternative objective functions have been proposed over the last few years. Many of them are based on the assumption that the residuals (differences between modelled and observed seismic data) follow specific probability distributions when, in fact, the true probability distribution is unknown. This leads FWI to converge to an incorrect probability distribution if the assumed probability distribution is different from the real one and, consequently it may lead the FWI to achieve biased models of the subsurface. In this work, we propose an objective function which does not force the residuals to follow a specific probability distribution. Instead, we propose to use the non-parametric kernel density estimation technique (KDE) (which imposes the least possible assumptions about the residuals) to explore the probability distribution that may be more suitable. As evidenced by the results obtained in a synthetic model and in a typical P-wave velocity model of the Brazilian pre-salt fields, the proposed FWI reveals a greater potential to overcome more adverse situations (such as cycle-skipping) and also a lower sensitivity to noise in the observed data than conventional L2- and L1-norm objective functions and thus making it possible to obtain more accurate models of the subsurface. This greater potential is also illustrated by the smoother and less sinuous shape of the proposed objective function with fewer local minima compared with the conventional objective functions.

The normal distribution is one of the most popular probability distributions with applications to real life data. In this research paper, an extension of this distribution together with Weibull distribution called the Weimal distribution which is believed to provide greater flexibility to model scenarios involving skewed data was proposed. The probability density function and cumulative distribution function of the new distribution can be represented as a linear combination of exponential normal density functions. Analytical expressions for some mathematical quantities comprising of moments, moment generating function, characteristic function and order statistics were presented. The estimation of the proposed distribution’s parameters was undertaken using the method of maximum likelihood estimation. Two data sets were used for illustration and performance evaluation of the proposed model. The results of the comparative analysis to other baseline models show that the proposed distribution would be more appropriate when dealing with skewed data.

Постановка задачi. З розвитком технологiй радiозв’язку, особливо, мобiльного, усе бiльш важливу
\nроль вiдiграє розробка моделей радiоканалiв. У данiй
\nстаттi розглядається одна з популярних нинi моделей,
\nа саме, двопроменевий радiоканал з розсiюванням потужностi сигналу, яку в лiтературi прийнято позначати
\nабревiатурою TWDP (вiд Two-Wave with Diffuse Power).
\nДля цiєї моделi вiдомi як точнi, так i наближенi вирази
\nдля густини розподiлу iмовiрностей (ГРI) i кумулятивної функцiї розподiлу (КФР) обвiдної сигналу. Однак,
\nзагальним їх недолiком є висока складнiсть з точки зору практичного застосування. У зв’язку iз цим у данiй
\nроботi розглядається можливiсть значно бiльш простої
\nапроксимацiї ГРI та КФР обвiдної сигналу на основi
\nметоду Накагами.
\nРозробка методу. Апроксимацiя розподiлiв по методу Накагами зводиться до обчислення другого та
\nчетвертого початкових моментiв обвiдної сигналу на
\nвиходi каналу. Такi обчислення й виконанi в роботi,
\nщо дозволяє представити зазначенi моменти у виглядi
\nфункцiй вiд двох загальноприйнятих параметрiв каналу
\nTWDP: К-фактора та параметра Д, який визначається
\nспiввiдношенням амплiтуд радiо променiв.
\nАналiз похибок апроксимацiї. Похибки оцiнювалися шляхом зiставлення розрахункiв ГРI i КФР за
\nточними формулами та розрахункiв iз використанням
\nапроксимацiї Накагами. Паралельно також проводилися розрахунки вiдповiдно до формул апроксимацiї, що
\nвiдомi у лiтературi. Аналiз показав, що похибка методу
\nНакагами при обчисленнi КФР є цiлком задовiльною
\nдля практичних потреб майже у всьому дiапазонi змiн
\nпараметрiв каналу. У той же час точнiсть апроксимацiї
\nГРI у порiвняннi з iншими вiдомими методами апроксимацiї виявилася не дуже задовiльна.
\nВисновки. Проведенi дослiдження пiдтвердили можливiсть апроксимацiї КФР обвiдної сигналу в каналi TWDP розподiлом Накагами. Результати такої ж
\nапроксимацiї для ГРI виявилися менш задовiльними.
\nТакож вiдзначається кориснiсть отриманих результатiв
\nдля трактування даних натурних експериментiв.

Oil recovery forecast relies on predictions of saturation fields from reservoir simulations. These predictions, in return, are dependent on the quality of the input data available, such as the reservoir permeability and porosity fields. However, because of inherent uncertainties in these input data, due to sparse and possibly inaccurate measurements, a stochastic framework is usually adopted to describe their variability. Probability distributions are then assigned to the input fields. In this stochastic context, predicting the saturation fields consists in determining the saturation probability distributions. For nonlinear immiscible two-phase flow in highly heterogeneous porous media, we propose an efficient distribution method to estimate the saturation cumulative distribution function (CDF) and probability density function (PDF). While high heterogeneity is traditionally regarded as an obstacle to designing stochastic methods, this distribution method exploits the physics of highly heterogeneous porous media. The method identifies hidden smooth random fields (RFs) that explain the saturation distribution, together with the nonlinear mapping from these hidden RFs to the saturation. We show that the statistics and distributions of these hidden RFs can be estimated very efficiently and simply, leading to a method that is computationally inexpensive compared to full Monte Carlo (MC) simulations. We provide numerical evidence of the agreement between MC-based and distribution-based saturation CDF, PDF and moments, in high heterogeneous cases such as high variance lognormal permeability fields, anisotropic layered fields and channelized systems. Finally, we show how the saturation CDF can be used for risk assessment by relying not only on the mean and standard deviation of the saturation, but primarily on saturation quantiles, obtained from probability of exceedance.

Two parametric models for income distributions are introduced. The models fitted to log(income) are the 4-parameter normal-Laplace (NL) and the 5-parameter generalized normal-Laplace (GNL) distributions. The NL model for log(income) is equivalent to the double-Pareto lognormal (dPlN) distribution applied to income itself. Definitions and properties are presented along with methods for maximum likelihood estimation of parameters. Both models along with 4- and 5-parameter beta distributions, are fitted to nine empirical distributions of family income. In all cases the 4-parameter NL distribution fits better than the 5-parameter generalized beta distribution. The 5-parameter GNL distribution provides an even better fit. However fitting of the GNL is numerically slow, since there are no closed-form expressions for its density or cumulative distribution functions. Given that a fairly recent study involving 83 empirical income distributions (including the nine used in this paper) found the 5-parameter beta distribution to be the best fitting, the results would suggest that the NL be seriously considered as a parametric model for income distributions.