Gumbel Distribution Research Articles

Gumbel–Logistic Unit Distribution with Application in Telecommunications Data Modeling

The manuscript deals with a new unit distribution that depends on two positive parameters. The distribution itself was obtained from the Gumbel distribution, i.e., by its transformation, using generalized logistic mapping, into a unit interval. In this way, the so-called Gumbel-logistic unit (abbr. GLU) distribution is obtained, and its key properties, such as cumulative distribution function, modality, hazard and quantile function, moment-based characteristics, Bayesian inferences and entropy, have been investigated in detail. Among others, it is shown that the GLU distribution, unlike the Gumbel one which is always positively asymmetric, can take both asymmetric forms. An estimation of the parameters of the GLU distribution, based on its quantiles, is also performed, together with asymptotic properties of the estimates thus obtained and their numerical simulation. Finally, the GLU distribution has been applied in modeling the empirical distributions of some real-world data related to telecommunications.

Wave Energy Potential and the Role of Extreme Events on South America's Coasts. A Regional Frequency Analysis

This study examines wave energy potential and generation capacity from extreme waves along South America's Pacific and Atlantic coasts. Utilizing the Regional Frequency Analysis (RFA) method and the WAVERYS wave reanalysis model, 27 homogeneous regions with distinct wave patterns were identified. There's a notable southward increase in median and average wave heights and energy. Unlike previous studies, this research emphasizes the significant role of extreme wave events, which contribute up to 33 % of the total energy—a critical factor often overlooked in regional energy assessments. In the Pacific, three main zones were delineated, with energy peaks reaching 120 kW/m south of 50°S, while the Atlantic identified two zones with energy values ranging from 10 kW/m to 30 kW/m. Zones between 10°S to 40°S in the Pacific and 30°S to 60°S in the Atlantic stand out as particularly promising for energy harvesting. A comprehensive statistical methodology enabled predictions of wave heights and energy for 50 and 100-year return periods, predominantly utilized generalized Pareto and Gumbel distributions. The study affirms the efficacy of the RFA in improving the understanding of wave dynamics and highlights the necessity of integrating extreme events in energy assessments. This research provides a foundation for advancing wave energy initiatives and coastal risk mitigation in South America.

Load profiles of residential off-grid solar systems on the Navajo Nation

Standalone off-grid electrical systems, no matter where they are deployed or for what user class, are designed based upon the load they are expected to serve. State-of-the-art computerized off-grid system design tools require the user to specify the expected load profile, that is, how the power consumption changes throughout the day. Often, this is at an hourly resolution, and some characterization of the distribution of power around the average values may be required. Specifying realistic and reasonable load profiles is a barrier to the appropriate design of standalone systems. This research extends previous studies on daily energy consumption of residential solar-powered off-grid systems on the Navajo Nation to provide hourly load profiles, statistical characteristics, and probabilistic models. The data analyzed come from 90 homes over a two-year period. K-means clustering is used to identify prototypical normalized load profiles when the data are grouped by year, season, weekday, and weekend. Eight parametric probability density functions are fit to the grouped data at an hourly resolution. Their fit to the data is evaluated using the Cramér-von Mises (CvM) statistic. The results show that the load profiles tend to be night-peaking and that Log Normal and Gumbel distributions can reasonably model variation in the data. The load profiles and probabilistic models can be used in off-grid design software and to synthesize load profiles for design and future research.

Grid-Based Precipitation Quantile Estimation Considering Homogeneity Using ERA5-Land Data for the Korean Peninsula

In this study, a grid-based precipitation quantile was estimated using long-term reanalysis precipitation data, considering the homogeneity of the annual maximum series (AMS) for the Korean Peninsula. For regions where significant changes in homogeneity were observed, the precipitation quantile was estimated using only the AMS from after the change point, and these results were compared with those from the original AMS. The examination of homogeneity revealed a significant increasing trend in homogeneity variability in the southeastern region of Korea. This change was particularly pronounced in the location parameter of the Gumbel distribution, resulting in an improved model fit. The change in precipitation quantile was most noticeable for a 2-year return period with a 36 h duration, with an average increase of approximately 11.5%. The results obtained from this study are anticipated to offer crucial foundational data for the design of hydraulic structures in regions with insufficient long-term ground observation data.

Landslide Hazard Prediction Based on UAV Remote Sensing and Discrete Element Model Simulation—Case from the Zhuangguoyu Landslide in Northern China

Rainfall-triggered landslides generally pose a high risk due to their sudden initiation, massive impact force, and energy. It is, therefore, necessary to perform accurate and timely hazard prediction for these landslides. Most studies have focused on the hazard assessment and verification of landslides that have occurred, which were essentially back-analyses rather than predictions. To overcome this drawback, a framework aimed at forecasting landslide hazards by combining UAV remote sensing and numerical simulation was proposed in this study. A slow-moving landslide identified by SBAS-InSAR in Tianjin city of northern China was taken as a case study to clarify its application. A UAV with laser scanning techniques was utilized to obtain high-resolution topography data. Then, extreme rainfall with a given return period was determined based on the Gumbel distribution. The Particle Flow Code (PFC), a discrete element model, was also applied to simulate the runout process after slope failure under rainfall and earthquake scenarios. The results showed that the extreme rainfall for three continuous days in the study area was 151.5 mm (P = 5%), 184.6 mm (P = 2%), and 209.3 mm (P = 1%), respectively. Both extreme rainfall and earthquake scenarios could induce slope failure, and the failure probabilities revealed by a seepage–mechanic interaction simulation in Geostudio reached 82.9% (earthquake scenario) and 92.5% (extreme rainfall). The landslide hazard under a given scenario was assessed by kinetic indicators during the PFC simulation. The landslide runout analysis indicated that the landslide had a velocity of max 23.4 m/s under rainfall scenarios, whereas this reached 19.8 m/s under earthquake scenarios. In addition, a comparison regarding particle displacement also showed that the landslide hazard under rainfall scenarios was worse than that under earthquake scenarios. The modeling strategy incorporated spatial and temporal probabilities and runout hazard analyses, even though landslide hazard mapping was not actually achieved. The present framework can predict the areas threatened by landslides under specific scenarios, and holds substantial scientific reference value for effective landslide prevention and control strategies.

Synergistic effects of CO2 and H2S on stress corrosion cracking of stainless steel 254SMo in extremely aggressive oilfield environment

Synergistic effects of CO2 and H2S on stress corrosion cracking of stainless steel 254SMo in extremely aggressive oilfield environments were explored through experiments and water chemistry modelling calculations. Fracture morphology reveals transition from ductile fracture to plastic fracture characteristics. Initiation of fracture was attributed to pitting corrosion, and depth of the pits was consistent with the Gumbel distribution. Presence of CO2 and H2S exhibits a coupling effect on stress corrosion cracking of stainless steel 254SMo. Ionization of H2S promoted by CO2 produced more HS− and S2− with increasing temperature and pressure, accelerated pit generation on the surface, and increased susceptibility to stress corrosion cracking.

Revisiting design wind speed and cyclonic factor for east coast of India

It is very much challenging to estimate the accurate design wind speed for engineering structures of India, those experiencing extreme wind events such as cyclones. However, there are different categories of structures based on their importance level and a trade-off always exists between structural safety and economy. Indian code IS15498 specifies uniform cyclonic factors. However, a wide variation has been found while considering different return periods and threshold wind speeds above which cyclones have been modelled. Accordingly, a cyclonic factor (k4) has been calculated using three distinct approaches. These include the Gumbel distribution with the Generalized Least Squares Method (GLSM), the Generalized Extreme Value distribution with the Least Squares Method, and the Peaks over Threshold approach employing the De Haan method. These methods were applied to the dataset of peak values from all cyclone events to determine the design wind speed and, subsequently, the cyclonic factor. The analysis shows that Gumbel distribution with GLSM provides the best results with 1.03 value of cyclonic factor. Reverse Weibull can also be used to fit the extreme wind speed data. The outcomes from all approaches consistently underscore the necessity of a cyclonic factor for the east coast of India.

Practical and Optimal Likelihood Intervals and Regions for Weibull and Gumbel Distributions

Profile likelihood intervals with good coverage probabilities and convenient simple expressions are presented to estimate separately and efficiently each of the parameters and quantiles of Weibull and Gumbel distributions, widely used in reliability and environmental applications. For the first time, likelihood-confidence intervals for these distributions, that were previously unavailable, can now be easily and quickly calculated with the provided R code for sample sizes n ≥ 10 . For simultaneous estimation of both parameters, likelihood-confidence regions are also provided. The proposal takes advantage of the transformation relationship between the Weibull and Gumbel distributions, of the fact that this latter distribution belongs to the Location and Scale family of distributions, and of likelihood symmetrizing parametrizations, all leading to simpler inferences about the parameters. The plot of the relative profile likelihood of a quantile of interest and its corresponding profile likelihood intervals provide decision makers with useful information in many practical applications, particularly when comparing two treatments. The proposed intervals for the Weibull shape parameter facilitate testing if an exponential distribution is reasonable for the data. The proposed methods are illustrated with practical examples.

Transmuted Power Gumbel Distribution: Estimation and Applications

In recent years, generalized distributions have been widely studied in statistics as they possess flexibility in applications. This is justified because the traditional distributions often do not provide good fit in relation to the real data set studied. This paper develops a Power Gumbel distribution using the quadratic rank transmutation map (QRTM). The new generalization is called the transmuted Power-Gumbel distribution. Various mathematical properties of this distribution including moments, moment generating function, quantile function, mean deviation and order statistics were also studied. These features support the legitimacy and robustness of the proposed distribution. The maximum likelihood method is used for estimating the model parameters, and the finite sample performance of the estimators are assessed by simulation studies indicating that their precision improves with larger sample sizes. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. Finally, the usefulness of the proposed model is illustrated in an application to two real data sets and conclude that the four-parameter transmuted Power Gumbel distribution provides better fit than the other five models.

Modelling the time-dependent mechanical properties of thermoplastic and thermosetting polymers with Gumbel distribution functions

In this paper, we propose a simple modelling method that accurately describes the time-dependent viscoelastic behaviour of thermoplastic and thermosetting solid polymers. We performed short-term creep and stress relaxation tests on representative samples of three major classes of polymers: an amorphous, a semi-crystalline thermoplastic and a thermosetting polymer. Then, to investigate the relationship between the model parameters and the material composition, we also tested thermoplastic matrix composites with different plasticiser contents. From the short-term tests, master curves were constructed with the use of the time–temperature superposition principle. Then, the temperature dependence of the obtained shift factors was described using the Arrhenius model. The resulting creep compliance and relaxation modulus master curves were normalised and modelled with a two-parameter cumulative Gumbel distribution function (CDF) and its complementary distribution function (CCDF). For the creep and stress relaxation master curves, the median error of modelling was less than 16 % and 16.5 % in all cases, respectively. Furthermore, it was less than 2 % for the thermosetting polymer, making this model an excellent tool for describing the creep and stress relaxation behaviour of thermosetting systems. In the case of the plasticised composites, we found a strong relationship between the material composition and the location parameter of the Gumbel model. Based on this, we developed a method that can be used to estimate the evolution of the creep compliance curve for any arbitrary OLA content (between 0 wt% and 16 wt%) at any desired temperature (between 22 °C and 70 °C).

ANALISIS HUJAN MAKSIMUM PADA DAERAH ALIRAN SUNGAI BABAK KABUPATEN LOMBOK BARAT

Hydrological analysis is the most important measurement in determining runoff in order to find ways to overcome flooding. The flow rate that is the basis for calculating flood control and protection is the planned flood flow rate obtained from the sum of the volume of rain runoff and the flow rate of the water catchment area in question during a certain return period. This research is a case study using rainfall data for 10 years, namely 2014-2023. Rainfall prediction is an estimate of the amount of rain that will fall in a river basin. There are several methods for determining annual rainfall, the Gumbel distribution method, Pearson log type III, and the normal method. After analyzing the annual rainfall data obtained at the West Lombok Regency Meteorology, Climatology and Geophysics Agency office, using three methods, namely the EJ Gumbel method, the Log Person Type III method and the Normal method, where from the three methods the maximum rainfall difference was quite large, In the following table you can see the difference in the amount of maximum rainfall in the return period T years.

Pre-Reconstruction Processing with the Cycle-Consist Generative Adversarial Network Combined with Attention Gate to Improve Image Quality in Digital Breast Tomosynthesis.

The current study proposed and evaluated "residual squeeze and excitation attention gate" (rSEAG), a novel network that can improve image quality by reducing distortion attributed to artifacts. This method was established by modifying the Cycle Generative Adversarial Network (cycleGAN)-based generator network using projection data for pre-reconstruction processing in digital breast tomosynthesis. Residual squeeze and excitation were installed in the bridge of the generator network, and the attention gate was installed in the skip connection between the encoder and decoder. Based on the radiation dose index (exposure index and division index) incident on the detector, the cases approved by the ethics committee and used for the study were classified as reference (675 projection images) and object (675 projection images). For the cases, unsupervised data containing a mixture of cases with and without masses were used. The cases were trained using cycleGAN with rSEAG and the conventional networks (ResUNet and U-Net). For testing, predictive processing was performed on cases (60 projection images) that were not used for learning. Images were generated using filtered backprojection reconstruction (kernel: Ramachandran and Lakshminarayanan) from projection data for testing data and without pre-reconstruction processing data (evaluation: in-focus plane). The distortion was evaluated using perception-based image quality evaluation (PIQE) analysis, texture analysis (feature: "Homogeneity" and "Contrast"), and a statistical model with a Gumbel distribution. PIQE has a low rSEAG value. Texture analysis showed that rSEAG and a network without cycleGAN were similar in terms of the "Contrast" feature. In dense breasts, ResUNet had the lowest "Contrast" feature and U-Net had differences between cases. The maximal variations in the Gumbel plot, rSEAG reduced the high-frequency ripple artifacts. In this study, rSEAG could improve distortion and reduce ripple artifacts.

A probabilistic approach to Lorentz balls [formula omitted]

We develop a probabilistic approach to study the volumetric and geometric properties of unit balls Bq,1n of finite-dimensional Lorentz sequence spaces ℓq,1n. More precisely, we show that the empirical distribution of a random vector X(n) uniformly distributed on its volume normalized unit ball converges weakly to a compactly supported symmetric probability distribution with explicitly given density; as a consequence we obtain a weak Poincaré-Maxwell-Borel principle for any fixed number k∈N of coordinates of X(n) as n→∞. Moreover, we prove a central limit theorem for the largest coordinate of X(n), demonstrating a quite different behavior than in the case of the ℓqn balls, where a Gumbel distribution appears in the limit. Finally, we prove a Schechtman-Schmuckenschläger type result for the asymptotic volume of intersections of volume normalized ℓq,1n and ℓpn balls.

Distribution of Rain Intensity: Daily Maximum Rainfall Data in The Province of South Sumatera and West Sumatera

The high intensity of rain in May 2024 caused flooding in the Sumatra region, especially Padang City and Palembang City. The region of West Sumatra is also known as the ethalae of disasters with frequent earthquakes, floods, and landslides. In Indonesia, many frequency analyses are conducted using the Gumbel distribution without testing data and clear hydrological reasons. It is feared that this method is considered as a routine way, thus causing the risk of unwanted deviations, resulting in many disasters and errors in the planning of water buildings. This research aims to determine and know the type of distribution that is representative for the maximum daily rainfall frequency in West Sumatra and South Sumatra Provinces as a reference for development and disaster mitigation in the region. The research location is in West Sumatra and South Sumatra. The data used for this study were taken from daily maximum rainfall data from rain stations for 35 to 40 years from 24 rain stations in West Sumatra Province, and 16 rain stations for South Sumatra region. This type of research is development research with the 4D development model (Define, Design, Develop, Disseminate). The development method consists of four different phases: Initial Investigation, Design, Realisation and Development. The data of each station was then organised into two types of data series, namely the annual maximum data series and the annual minimum data series. The results of this data series test are expected to follow one or more types of distributions commonly used in Hydrology and Drainage. The distribution results obtained for the South Sumatra region are Gama-III and LP-III, while for the West Sumatra region the distribution types are Gama-III, LP-III Normal Log and Gembel. The conclusion obtained is that West Sumatra does not have only one type of distribution. For the type of test using the SPSS.22 application, the results obtained are goodness of fit test, parametric test, Chi-Square test, Kolmogorov Smirnov test and Anderson-Darling test and Histrogram (visual) method.

Revisiting the Use of the Gumbel Distribution: A Comprehensive Statistical Analysis Regarding Modeling Extremes and Rare Events

The manuscript presents the applicability of the Gumbel distribution in the frequency analysis of extreme events in hydrology. The advantages and disadvantages of using the distribution are highlighted, as well as recommendations regarding its proper use. A literature review was also carried out regarding the methods for estimating the parameters of the Gumbel distribution in hydrology. Thus, for the verification of the methods, case studies are presented regarding the determination of the maximum annual flows and precipitations using nine methods for estimating the distribution parameters. The influence of the variability of the observed data lengths on the estimation of the statistical indicators, the estimation of the parameters, and the quantiles corresponding to the field of small exceedance probabilities (p < 1%) is also highlighted. In each case, the results are analyzed compared to those obtained with the Generalized Extreme Value distribution, the four-parameter Burr distribution, and the five-parameter Wakeby distribution estimated using the L-moments method. The results of the case studies highlight and reaffirm the statistical, mathematical, and hydrological recommendations regarding the avoidance of applying the Gumbel distribution in flood frequency analysis and its use with reservations in the case of maximum precipitation analysis, especially when the statistical indicators of the analyzed data are not close to the characteristic ones and unique to the distribution.

A new extension of the Gumbel distribution with biomedical data analysis

In the field of biomedical research, data characteristics often exhibit significant variability, challenging the applicability of classical Gumbel distribution for biomedical data modeling. To address this, this paper introduces a novel extension of the Gumbel model known as the odd beta prime Gumbel (OBP-Gum) model. Derived from the odd beta prime family, the new distribution exhibits greater kurtosis compared to the traditional Gumbel distribution. Importantly, the proposed distribution is designed to capture right-skewed, left-skewed, and nearly symmetric density functions, as well as increasing, decreasing, constant, and upside-down bathtub shapes for its hazard rate function, providing excellent curvature features for creating flexible statistical models for biomedical research. We derive the fundamental features of the OBP-Gum model, such as the quantile function, linear representations, moment generating function, moments, skewness, kurtosis, incomplete moments, and Rényi and Tsallis entropies. Parameter estimation for this new model is conducted using the maximum likelihood estimation method. A simulation study demonstrates the performance of the model parameters. The empirical findings, based on applications to two biomedical datasets, suggest that the OBP-Gum distribution outperforms existing models, particularly in handling extreme observations. Instead of relying on conventional models for decision-making, this research provides relevant stakeholders with an improved statistical distribution for more accurate biomedical data modeling.

The corrosion analysis of X80 pipeline steel welded joint using wire beam electrode and numerical simulation methods

PurposeThe purpose of this study is to characterize the galvanic corrosion behavior of a simulated X80 pipeline steel welded joint (PSWJ) reconstructed by the wire beam electrode (WBE) and numerical simulation methods.Design/methodology/approachThe galvanic corrosion of an X80 PSWJ was studied using WBE and numerical simulation methods. The microstructures of the coarse-grained heat affected zone, fine-grained heat affected zone and intercritical heat affected zone were simulated in X80 pipeline steel via Gleeble thermomechanical simulation processing.FindingsComparing the corrosion current density of coupled and isolated weld metal (WM), base metal (BM) and heat-affected zone (HAZ), the coupled WM exhibited a higher corrosion current density than isolated WM; the coupled BM and HAZ exhibited lower corrosion current densities than isolated BM and HAZ. The results exhibited that the maximum anodic galvanic current fitted the Gumbel distribution. Moreover, the numerical simulation results agreed well with the experimental data.Originality/valueThis study provides insight into corrosion evaluation of heterogeneous welded joints by a combination of experiment and simulation. The method of reconstruction of the welded joint has been proven to be a feasible approach for studying the corrosion behavior of the X80 PSWJ with high spatial resolution.

Digitizing the Williamstown, Australia Tide‐Gauge Record Back to 1872: Insights Into Changing Extremes

AbstractFor Williamstown tide gauge, at the northern‐most point of Port Phillip Bay (PPB), Melbourne, Victoria, tide registers from 1872 to 1966 and marigrams from 1950 to 1966, were digitized to extend sea‐level records back almost 100 years. Despite some vertical datum issues in the early part of the record, the data set is suitable for extreme sea‐level trend analysis after removal of the annual mean sea level. The newly digitized data was combined with the digital record to produce a combined record from 1872 to 2020. Analysis of this record revealed known problems of siltation of the tide gauge stilling well and associated reduction in tidal range at times during 1880–1895 and 1910–1940. A positive trend in tidal amplitude of 0.41 ± 0.01 mm yr−1 was found over 1966–2020, likely due to reduced hydraulic friction at the narrow entrance to PPB. Extreme sea‐level trends were examined over 1872–2020 for storm tides (the combination of storm surge and tide) after removal of the annual mean, and residuals after subtraction of the predicted tides. A non‐stationary Gumbel distribution with a time‐varying location parameter revealed statistically significant declining trends in the residuals of −0.73 ± 0.02 mm yr−1, consistent with the observed poleward movement of storm surge‐producing mid‐latitude weather systems. For storm tides a smaller declining trend of −0.40 ± 0.01 mm yr−1 was found. These trends are approximately an order of magnitude smaller than the current positive rates of mean sea level rise, meaning that storm tide hazard will continue to increase in the future. This information is relevant for future adaptation planning.

Use of multivariate techniques to regionalize rainfall patterns in semiarid Botswana

Monthly precipitation data from 58 synoptic stations throughout Botswana, spanning 1981–2016, were used in this study. The data were examined using multivariate analysis to determine regions exhibiting distinct precipitation variability patterns and regimes. To accomplish this, the T-mode of principal component analysis was applied to the correlation matrix of the data. Based on the maximum loading values of the rotational principal component scores, the T-mode indicated three separate subregions with varying precipitation patterns over time. Four clusters with distinct rainfall patterns were identified when cluster analysis was performed on the principal component scores. An assessment of the homogeneity of the clusters was performed using L-moment’s heterogeneity measure (H). Statistical analysis was employed to model annual rainfall data using five commonly used rainfall analysis probability distribution functions: normal, lognormal, gamma, Weibull, and Gumbel. The probability distributions with the greatest fit were determined based on the maximum overall score, which was calculated by adding the individual point scores of three chosen goodness-of-fit tests. Each cluster exhibited distinct probability distribution functions, with the gamma, Gumbel, lognormal, and Weibull distributions providing the most accurate descriptions.

Short-term extreme response estimation methodology for marine umbilical cables with two configuration types

This study investigates the critical engineering problem of the short-term extreme responses of marine umbilical cables, which are essential components in offshore oil and gas operations. Umbilical cables play a pivotal role in connecting subsea systems to surface platforms, necessitating a safe design to withstand external loads. This study focused on two primary umbilical cable configuration types: deep-water catenary and shallow-water lazy wave. For each configuration type, 100 3-h dynamic numerical simulations were performed, with maximum values fitted by using the Gumbel distribution, and the most probable maximum (MPM) values were set as benchmark values. The average conditional exceedance rate (ACER) method and uncertainty-incorporated ACER method were applied to study the short-term extreme values of umbilical cable responses based on limited sample data. The predicted results have been compared with the MPM values, and the accuracy and uncertainty of different efficient extreme-value analysis methods are discussed. This study provides a comprehensive methodological framework for the reliability analysis and safety design of umbilical cables in marine engineering.