Gumbel Distribution Research Articles

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.

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.

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.

Extreme value statistics of nerve transmission delay.

Delays in nerve transmission are an important topic in the field of neuroscience. Spike signals fired or received by the dendrites of a neuron travel from the axon to a presynaptic cell. The spike signal then triggers a chemical reaction at the synapse, wherein a presynaptic cell transfers neurotransmitters to the postsynaptic cell, regenerates electrical signals via a chemical reaction through ion channels, and transmits them to neighboring neurons. In the context of describing the complex physiological reaction process as a stochastic process, this study aimed to show that the distribution of the maximum time interval of spike signals follows extreme-order statistics. By considering the statistical variance in the time constant of the leaky Integrate-and-Fire model, a deterministic time evolution model for spike signals, we enabled randomness in the time interval of the spike signals. When the time constant follows an exponential distribution function, the time interval of the spike signal also follows an exponential distribution. In this case, our theory and simulations confirmed that the histogram of the maximum time interval follows the Gumbel distribution, one of the three forms of extreme-value statistics. We further confirmed that the histogram of the maximum time interval followed a Fréchet distribution when the time interval of the spike signal followed a Pareto distribution. These findings confirm that nerve transmission delay can be described using extreme value statistics and can therefore be used as a new indicator of transmission delay.

A practical method to draw from multivariate extreme value distributions

Generating random draws from multivariate extreme value (MEV) distributions plays an important role in the microsimulation of travel behaviors, which can effectively avoid heavy computational burdens from simulation based on calculated probability values, particularly in simulations for a large population or choice behaviors from a large choice set. However, there are few practical and effective methods for drawing from MEV distributions. This paper proposes a simple and computationally efficient approach for drawing from MEV distributions in the nested logit (NL), cross-nested logit (CNL), and paired combinatorial logit (PCL) models. The proposed approach to draw from the MEV distribution for a CNL model provides a new perspective to understand the underlying choice mechanism of the CNL model. To our knowledge, this is the first study to draw from an MEV distribution in the PCL model. Random draws from the proposed approach approximately follow the standard Gumbel distribution, which is the marginal distribution of NL/CNL/PCL models, and approximate correlations among alternatives well. Simulation results of NL/CNL/PCL models show that the proposed approach provides high-level accuracy in recovering model parameters with the overall mean absolute percentage bias being less than 3%. The proposed approach is computationally more efficient than similar ones because it only needs to draw from Gumbel distributions. The proposed approach can be used to simulate NL/CNL/PCL models with a large choice set or a multiple discrete-continuous generalized extreme value model in various application settings such as joint destination-mode choices, time use allocations, etc.

Cover-time Gumbel fluctuations in finite-range, symmetric, irreducible random walks on torus

In this paper, we provide the mathematical foundation for an explicit and universal feature of cover time for a large class of random work processes, which was previously observed by Chupeau et al (2015 Nat. Phys. 11 844–7). Specifically, we rigorously establish that the fluctuations of the cover time, normalized by the mean first passage time, follow a Gumbel distribution, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. The result contributes to a better understanding of cover-time behavior in random search processes, especially on the efficiency of exhaustive searches. Our approach builds upon the work of Belius (2013 Probab. Theory Relat. Fields 157 635–89) on cover times for simple random walks, leveraging a strong coupling between the random walk and random interlacements.

A robust index for metal artifact quantification in computed tomography.

Objective assessment of metal artifact strength and the effectiveness of metal artifact reduction algorithms in computed tomography requires a quantitative metric. Metrics described in the literature are typically employed to compare the artifact strength in images reconstructed from the same raw data, but their robustness against varying scan conditions and repeated scans over time as it occurs in periodic quality assurance has not been investigated. A new robust metric for quantifying metal artifacts in computed-tomography images is proposed and compared to other commonly used metrics. The proposed artifact metric is based on the location parameter of the Gumbel distribution, described previously in the literature, but normalized to the location parameter in a background region-of-interest to obtain a noise-independent artifact metric. The metric was compared to three other quantitative metal artifact metrics (artifact-index, contrast-to-noise ratio, Gumbel-evaluation method) by evaluating metals artifacts in phantom scans and in clinical images. Robustness of the artifact metrics was evaluated using repeated scans with varying noise and against small variations in the selected regions-of-interest. The proposed artifact metric was independent of the underlying image noise and could be reproduced more consistently under slight changes of the region-of-interest within the artifact than the other investigated methods. The coefficient-of-variation was 5.7% on average with varying regions-of-interest in phantom scans and 2.5% in patient scans compared to 9.2% in phantoms scans and 9.9% in patient scans for the next-best performing noise-independent metric. Setup reproducibility was better than 5% and was comparable to the other metrics. The new metric correlated linearly with the artifact strength. The contrast-to-noise ratio, although often used in artifact quantification, was found to be an inadequate metric due to its lack of robustness against minute changes in the position, size, and pixel values of the region-of-interest chosen for calculating the metric and because it showed no correlation with the artifact strength. A new metal artifact metric has been proposed that is robust under changing scan conditions and less sensitive to user-dependent choices of the region-of-interest than other metrics. The new metric is straightforward to calculate and simple to implement in software commonly used for evaluation of medical imaging systems.

Three-parameters Gumbel Distribution: Formulation and Parameter Estimation

Modeling extreme value theories is really gaining interest in the world with scientist working to improve the flexibility of the distributions by adding parameter(s). Extreme value distributions are always described to include families of Gumbel, Weibull and Frechet distributions. Of the three distributions, Gumbel distribution is the most commonly used in the extreme value theory analysis. Existing literature has shown that the addition of parameter to a distribution makes it robust and/or more flexible hence the study intends to improve the existing two parameters Gumbel distribution using the Marshall and Olkin proposed method for introducing a new estimator/parameter to an existing distribution. The developed distribution will be important to the applications in some life time studies like high temperature, earthquakes, network designs, horse racing, queues in supermarket, insurance, winds, risk management, ozone concentration, flood, engineering and financial concepts. The parameters for the introduced distribution was estimated using Maximum Likelihood Estimation method. The introduced three parameters Gumbel distribution is a probability distribution function which can be used in modelling statistical data. The maximum likelihood estimates for the three parameters namely shape, location and dispersion are efficient, sufficient and consistent and this makes the function more flexible and better for application. The three parameters Gumbel distribution can be used in modeling and analysis of normal data, skewed data and extreme data since it will provide efficient, sufficient and consistent estimates.