Gumbel Distribution Research Articles

Large nearest neighbour balls in hyperbolic stochastic geometry

Consider a stationary Poisson process in a d-dimensional hyperbolic space. For R>0 define the point process xi _R^{(k)} of exceedance heights over a suitable threshold of the hyperbolic volumes of kth nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius R centred at a fixed point. The point process xi _R^{(k)} is compared to an inhomogeneous Poisson process on the real line with intensity function e^{-u} and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the hyperbolic maximum kth nearest neighbour ball with a limiting Gumbel distribution is derived.

Gumbel distribution-based technique enables quantitative comparison between streak metal artifacts of multidetector row CT and cone-beam CT: a phantom study

Cone-beam computed tomography (CBCT), derived from multidetector row CT (MDCT), has a high spatial resolution and has recently been applied to various organs. One of the severe limitations common to CBCT and MDCT is metal artifacts. In particular, streak metal artifacts (SMAs) between multiple metal materials often hinder diagnosis. However, no studies have quantitatively compared the strength of SMAs in MDCT and CBCT. Nomura et al. reported an evaluation method specialized in SMAs of CBCT using the Gumbel distribution (GD), which can also be applied to SMAs of MDCT (Oral Surg Oral Med Oral Pathol Oral Radiol 131: 494-502,2021, https://doi.org/10.1016/J.OOOO.2020.08.031 ). This study aimed to quantitatively compare SMAs occurring between titanium materials on MDCT and CBCT images using the GD-based method. The SMAs were investigated as follows: A hydroxyapatite block was sandwiched between two titanium rods to generate an SMA. They were placed in an acrylic phantom, simulating a human head, and scanned using an MDCT scanner and two CBCT scanners. The obtained images were analyzed using Gumbel plots and location parameters, and the SMA strength was calculated. The results showed that the SMAs on the MDCT images were significantly weaker than those on the CBCT images. In the CBCT scans, a smaller volume CT dose indexvalue caused stronger SMAs. These results indicate that MDCT is more advantageous than CBCT in terms of SMA reduction when bone morphology between titanium materials must be evaluated. The characteristic should be considered in clinical cases.

Statistical modelling of maximum temperature in Rwanda using extreme value analysis

This study aims to model the statistical behaviour of extreme maximum temperature values in Rwanda. To achieve such an objective, the daily temperature data from January 2000 to December 2017 recorded at nine weather stations collected from the Rwanda Meteorological Agency were used. The two methods, namely the block maxima (BM) method and the Peaks Over Threshold (POT), were applied to model and analyse extreme temperatures in Rwanda. Model parameters were estimated, while the extreme temperature return periods and confidence intervals were predicted. The model fit suggests that Gumbel and Beta distributions are the most appropriate for the annual maximum daily temperature. Furthermore, the results show that the temperature will continue to increase as estimated return levels show it.

The Cubic Rank Transmuted Gumbel Distribution

A Cubic Rank Transmuted Gumbel distribution (CTGD) in this research is extend the work of cubic transmuted distribution families. CTGD improves the flexibility of transmuted distributions and allows for the modeling of more complex data. Its hazard rate function, moment-generating function, moments, quantile function, entropy and order statistics, are only a few of the key statistical characteristics that we examine. Finally, the Cubic Transmuted Gumbel Distribution is applied to three real datasets to test its applicability and evaluate how well estimate approaches function for the CTGD, Gumbel(G), and transmuted Gumbel (TG) distributions. The observed results demonstrated that, for the used data sets, CTGD provides a superior fit than G and TG distributions.

Reliability Study of Magnesium Oxychloride-Coated Reinforced Concrete Based on Gumbel Distribution

The constant current accelerated corrosion test was used to study the durability of magnesium oxychloride-coated reinforced concrete (MOCRC) in order to solve the problem of MOCRC's durability. The relative dynamic elastic modulus was utilized as the failure threshold to evaluate the concrete durability, and the collected life data of concrete under different cover thickness were acquired. On the basis of the Gumbel distribution, the probability analysis can be used to study and foretell the life data. The results show that when the durability is evaluated by the relative mass and the relative dynamic modulus of elasticity, the durability of MOCRC with a larger protection layer thickness is better; the relative dynamic modulus of elasticity can better reflect the durability change in MOCRC than the relative mass. When the Gumbel distribution is used for durability analysis, the calculated value of the model and the life data have a relatively high degree of fit, which can provide a reference basis for the durability evaluation of concrete.

Similarities and probability distributions of chloride convection zone depth in concrete exposed to cyclic drying-wetting environments

A methodology using parabolic fitting was proposed to regress three points in chloride profile and determine the depth of convection zone in concrete. Based on exposure tests, the influences of environmental exposure, exposure time and mixture proportion of concrete on probability distributions and statistical parameters of convection depth were studied. The similarity in probability distribution of convection depth was analyzed by the Kullback-Leibler divergence. It was found that the mean, coefficient of variation and distribution type of convection depth are significantly influenced by environmental condition. Convection depth in natural environment mostly follows the Gumbel (Type I) distribution, while it follows the Beta distribution in simulated environment. The probability density functions of convection depth in simulated environment at early exposure have high similarities with those in natural environment at later exposure.

Mixture modeling of hospital charge in Zimbabwe

ObjectiveZimbabwe is one of the poorest countries in the world, just emerging from a broken health care system. The objective is to figure out variables affecting hospital charge for Zimbabwe. Material and methodsThe variables used are sex, smoking status, number of children, region, age and body mass index. The first six of these are factors and the remaining are covariates. A mixture model was fitted to describe the dependence of hospital charge on these variables. ResultsA mixture model with five components each having a reversed Gumbel distribution was found to give an adequate fit. Both the covariates and all but one of the factors were found to be significant. Estimates of value at risk of hospital charge are given for all combinations of the factors. ConclusionsThe results suggest that the hospital charge could be higher for females, higher for smokers, higher if the patient had more children and higher if the patient is older. Further, estimates of value at risk given suggest, for example, that a 90 year old female not smoking and having no children and an average body mass index will have a hospital charge less than Z$49851 with probability 0.999.

Capacity Expansion Optimization of Determined Charging Stations Based on Accessibility Analysis and Improved Discrete Choice Model

Reasonable capacity expansion optimization of determined charging stations is important for the popularization of electric vehicles (EVs). In this article, Gaussian two-step floating catchment area method and temporal clustering are adopted to study the unbalanced spatial and temporal distribution of charging station accessibility, as well as to determine candidate sites for capacity expansion optimization. Then, a two-level optimization model for capacity expansion of determined charging stations is proposed to reduce the computational complexity caused by the non-linearity and non-convexity of choice probability formulas when the discrete choice model is applied to statistically describe and analyze discrete behaviors such as users’ EV purchase preferences and charging station heterogeneity. In the proposed method, the lower-level model is described as a maximal coverage location problem by expressing the error term of the utility function as a linear combination of the random vectors from IID normal distribution and IID Gumbel distribution, which effectively simplifies the computing process. Finally, the two-level model can be transformed into an integer linear programming problem to optimize the capacity of determined charging stations. Experimental results show that the purchase rate of EVs is significantly improved, and the accessibility of charging stations during rush hours is more balanced than before.

Intensity-Duration-Frequency equations for Rio Grande do Sul - Brazil, based on stationary rainfall series

Heavy rainfall information is essential for environmental studies and water engineering. This study therefore aimed to adjust Intensity-Duration-Frequency (IDF) equations for 247 locations in the Rio Grande do Sul (RS) using stationary rainfall series. Mann-Kendall’s test was applied to identify the temporal trends in the Annual Maximum Daily Rainfall (AMDR) series of 271 rain gauges in RS. The Kappa, Generalized Extreme Value (GEV), Gumbel, two-parameters Log-Normal and three-parameters Log-Normal probabilistic distributions were adjusted to the AMDR series without significant temporal trend. The best distribution fit was given by Anderson-Darling’s test, so the AMDR was discretized up to 5 minutes. IDF equations coefficients were adjusted in RStudio, using Nash-Sutcliffe’s Coefficient and the Root-Mean-Square Error to evaluate them. In conclusion: the most suitable distributions for the AMDR were the multiparametric Kappa and GEV; the IDF equations coefficients adjustment was classified as “excellent”; coefficients a and b varied across the RS and are correlated with the AMDR and geographical positions; and the c and d coefficients were practically constant.
 Keywords: goodness of fit test, Kappa probabilistic distribution, trends test.

On the Bimodal Gumbel Model with Application to Environmental Data

The Gumbel model is a very popular statistical model due to its wide applicability for instance in the course of certain survival, environmental, financial or reliability studies. In this work, we have introduced a bimodal generalization of the Gumbel distribution thatcan be an alternative to model bimodal data. We derive the analytical shapes of the corresponding probability density function and thehazard rate function and provide graphical illustrations. Furthermore, We have discussed the properties of this density such as mode, bimodality, moment generating function and moments. Our results were verified using the Markov chain Monte Carlo simulation method. The maximum likelihood method is used for parameters estimation. Finally, we also carry out an application to real data that demonstrates the usefulness of the proposed distribution.

Generalized extreme value analysis of criticality tallies in Monte Carlo calculation

Generalized extreme value (GEV) is the standard statistical model on the occurrences of extreme observations. The trinity theorem in GEV asserts that properly normalized maxima from blocks of data converge in distribution to one of the Weibull, Gumbel and Fréchet distributions as the data size increases. In this work, the methodology of GEV is applied to criticality tallies in Monte Carlo fission source cycles in order to evaluate the utility value of the distribution tail ends. Numerical results obtained under a sufficiently large number of particles per cycle show that the extreme value index (EVI) in GEV is not capable of being differentiated for the upper and lower distribution tail ends of criticality tallies. It is also shown that the EVI of criticality tallies falls within the range of Weibull distribution including the EVI of Gumbel distribution as the role of a boundary value layer. As a demonstration aimed at practicality, GEV is utilized for the population diagnosis under an insufficient number of particles per cycle. It turns out that the transition from one equilibrium to other equilibrium due to small population size makes the EVIs of upper and lower distribution tail ends depart from each other so that one of them falls in the range of Weibull distribution and the other in that of Fréchet distribution.

Multinomial distribution as part of valuation of the number of failures

Abstract. Aim. The paper aims to examine the application of a multinomial distribution as part of valuation of the number of an object’s failures. It is assumed that the valuation is “based on past experience” (a statistical sample of the number of an object’s failures accumulated over several preceding evaluation periods).Methods. The paper uses methods of system analysis, probability theory and mathematical statistics. The author analyses the primary indicators used to define the applied dependability indicators. It is noted that the valuation of such indicators based on statistical data for complex systems appears to be promising. The problem of valuation of the number of failures using a statistical sample is examined. The primary disadvantages of the used approaches are identified that are associated with errors in defining the average values of series, variation coefficients and asymmetry. It is shown that it is possible to solve the problem using the well-known combinatorics problem “on balls and boxes”, which leads to the use of a multinomial distribution. The paper examined the definition of probabilities for compositions and partitions of the number n into m parts, as well as the probabilities of a given number of balls being in a box with their maximum number. The author also considered formulas and algorithms that allow reducing the number of calculations in case of machine computation of the probabilities of a multinomial distribution. The feasibility of approximating a discrete distribution function by the Gumbel distribution is estimated. The paper demonstrates the feasibility of valuating the number of failures for a “segment” corresponding to a part (fraction) of an object’s dimension considered on a certain part (fraction) of the time interval. It also examines examples of valuating the number of failures for an object as a whole over a 1-month evaluation interval and for ½ of an object over a 12-month evaluation interval, while the total interval for which the statistical sample is accumulated is 72 months. The paper sets forth limitations on the application of the presented method and notes some of its possible advantages. In particular, it is noted that the statistical sample is only one implementation of the multinomial distribution, so it can be said that when applying the proposed method, the results of valuation are no longer affected by the presence of unlikely combinations of series values in the statistical sample. It is also noted that when applying the proposed method of valuating the number of failures, the obtained acceptable value will never be less than or equal to the average value of the statistical sample.Results. Formulas have been obtained for calculating, based on partitions, of the discrete density number and the maximum distribution function of a multinomial distribution. The paper presents the results of algorithm analysis for machine computation. The results of applying some algorithms are presented. A formula is proposed for approximating the distribution function of the maximum of a multinomial distribution using the Gumbel distribution (for the largest values) using the method of moments. The author recommends a range of values of the estimated interval, in which the proposed method provides acceptable reliability of the results. The task of further research is defined.

Probability Density Function Models for Float Glass under Mechanical Loading with Varying Parameters

Glass as a construction material has become indispensable and is still on the rise in the building industry. However, there is still a need for numerical models that can predict the strength of structural glass in different configurations. The complexity lies in the failure of glass elements largely driven by pre-existing microscopic surface flaws. These flaws are present over the entire glass surface, and the properties of each flaw vary. Therefore, the fracture strength of glass is described by a probability function and will depend on the size of the panels, the loading conditions and the flaw size distribution. This paper extends the strength prediction model of Osnes et al. with the model selection by the Akaike information criterion. This allows us to determine the most appropriate probability density function describing the glass panel strength. The analyses indicate that the most appropriate model is mainly affected by the number of flaws subjected to the maximum tensile stresses. When many flaws are loaded, the strength is better described by a normal or Weibull distribution. When few flaws are loaded, the distribution tends more towards a Gumbel distribution. A parameter study is performed to examine the most important and influencing parameters in the strength prediction model.

A New Version of Cubic Rank Transmuted Gumbel Distribution

A Cubic Rank Transmuted Gumbel distribution (CTGD) in this article is a new generalization of the Gumbel distribution based on a cubic ranking transmutation map. Examined are the cubic transmuted Gumbel model's fundamental statistical properties, such as its hazard rate function, moment-generating function, moments, characteristic function, quantile function, entropy, and order statistics. Finally, the usefulness and applicability of the CTGD using two real data sets about waiting time at a bank is described and Wheaton River flood, and the fit has been compared with Gumbel distribution (GD) and transmuted Gumbel distribution (TGD). The results show that the proposed model provides a superior fit than transmuted Gumbel distributions and Gumbel distributions.

Statistics of synchronization times in Kuramoto oscillators

The synchronization of coupled systems is a widely studied phenomenon in nonlinear science. As a completely synchronized state emerges asymptotically, it is pertinent to quantify the statistics of the timescales in which it is attained. We study the Kuramoto model, a paradigmatic model of synchronization, and record the time at which the synchronized state is reached for the first time. The First Synchronization Time (FST), on suitable rescaling, exhibits a universal distribution for a wide range of parameter values and even in the presence of noise. We obtain an approximate time evolution of the order parameter using the celebrated Ott-Antonsen ansatz and analytically demonstrate that the FST exhibits a universal Gumbel distribution.

Reliability-based fatigue life assessment using random road load condition and local damage criteria

This study presents a fatigue reliability assessment of a vehicle coil spring based on random road load conditions for fatigue life characterisation. Fatigue life data characterisation was carried out to accelerate the durability analysis by retaining the higher amplitude cycles in the original strain signals and thus indicate the fatigue damage. To collect experimental data, road testing was conducted using various road profiles, which contributed significantly to the fatigue failure performance of the vehicle coil spring. Fatigue reliability analysis was performed to assess the failure behaviour of the coil spring utilising the probabilistic Gumbel distribution model to forecast the probability distribution of the fatigue life data. Furthermore, the highest and lowest mean-cycles-to-failure (McTF) of 5.44 × 107 and 2.46 × 105 blocks were estimated for the Coffin-Manson and Smith-Watson-Topper models from the highway and rural road profiles. The findings demonstrate that the lowest hazard rate corresponded to the highest McTF, making this an acceptable model to be implemented in reliability assessments of road profiles. Hence, the fatigue reliability assessment provided a hazard rate based on the fatigue life predictions for risk assessments when characterising the fatigue life data.

Estimation of stress‐strength reliability for beta log Weibull distribution using progressive first failure censored samples

AbstractIn this paper, we investigate classical and Bayes estimates for stress‐strength reliability (SSR) based on two independent progressive first failure censored samples from beta log Weibull distributions with different shape and scale parameters. The maximum likelihood estimation of SSR and its asymptotic confidence interval are obtained. Bayes estimates of SSR are derived under two different loss functions using the Tierney–Kadane's approximation algorithm. In addition, two different bootstrap confidence intervals are provided and the highest posterior density credible interval of SSR is also constructed by applying Monte Carlo Markov chain method. The performances of different point and interval estimates are assessed using the Monte Carlo simulation. Finally, three real data sets are analyzed for illustration purposes.

Statistical learning algorithms for dendritic neuron model artificial neural network based on sine cosine algorithm

Training of dendritic neuron model artificial neural networks is generally achieved by using nonlinear least square methods. The distribution of random error terms is ignored in training algorithms although error terms are random variables. Maximum likelihood estimators can be obtained for dendritic neuron model artificial neural networks by using some indefinite symmetric probability distributions. In this study, statistical learning algorithms are proposed for dendritic neuron model artificial neural networks. Maximum likelihood estimators for dendritic neuron model artificial neural networks are obtained by using Normal, Cauchy, Logistic, Gumbel and Laplace distributions. The Sine cosine algorithm is used for maximization of the likelihood function under error terms that have Normal, Cauchy, Logistic, Gumbel and Laplace distributions. The proposed learning algorithms are applied to Istanbul Stock Exchange time series data sets. At the end of the analysis of application results, the performance of the proposed method is statistically better than well-known deep and shallow artificial neural networks.

Uncertainty analysis of extreme mooring loads associated with environmental contours and peak tension distributions

This paper aims to assess the uncertainty on the extreme mooring loads of floating system considering short-term variability. Two environmental contour approaches based on the inverse First and Second Order Reliability Methods are employed to identify critical sea states that may give rise to extreme loads. The uncertainty related to the construction of environmental contours is addressed including significant differences due to marginal distribution fitting, parameter estimation methods and joint models. Three measured datasets are analysed using a known conditional joint distribution and proposed mixed copula model. 3-h time domain numerical simulation for each sea state is conducted and the characteristic extreme responses of mooring lines subjected to design loads are assessed. The uncertainties due to various statistical models including the average conditional exceedance rate method as well as global maxima, peak-over-threshold method combined with Gumbel distribution, Generalized Extreme Value distribution, Generalized Pareto distribution and 3-parameter Weibull distribution are investigated and quantified. It is observed that marginal distributions, joint models and parameters estimation methods have apparent effect on design loads estimation, and the extreme tensions of the semi-submersible platform shows significant difference using various probabilistic models. The results indicate that those epistemic uncertainties should be account for in the reliability analysis or safety factor calibration for mooring systems.

Estimating and Mapping Extreme Ice Accretion Hazard and Load Due to Freezing Rain at Canadian Sites

The ice accretion load in Canadian structural design codes is developed based on an operational ice accretion prediction model. In the present study, three models are employed to predict the ice accretion amount on a flat surface and horizontal wire at Canadian sites. The results confirm that the model used by Canadian practice for predicting ice accretion leads to a conservative estimate as compared to the remaining two models. The results also indicate that the use of the Gumbel distribution for the annual maximum ice accretion is adequate for regions prone to ice accretion and that the lognormal distribution may be considered for regions with a moderate or negligible amount of ice accretion. Maps of the ice accretion hazard at five selected Canadian sites are developed. Statistical analysis of an equivalent wind speed that is concurrent with the iced wire is carried out, showing that the concurrent wind speed for the 50-year return period value of the annual maximum ice accretion amount is smaller than the 50-year return period value of the annual maximum wind speed. It is shown that the statistical characteristics of the annual maximum concurrent wind speed on iced wire differ from that of the annual maximum wind speed.