Gumbel Distribution Research Articles

Stochastic Model of the Annual Maximum Rainfall Series Using Probability Distributions

Rainfall is a natural process that is often characterized by significant variability and uncertainty. Stochastic models of rainfall typically involve the use of probability distributions to describe the likelihood of different outcomes occurring. This study aimed to model the annual maximum of daily rainfall in Makassar City, Indonesia for the period 1980–2022, specifically focusing on the rainy season (November to April) using probability distributions to estimate return periods. The study used the Generalized Extreme Value (GEVD) and Gumbel distributions. The Kolmogorov-Smirnov test was used to determine the suitability of each distribution, and the likelihood ratio test was employed to determine the best-fit model. The Mann-Kendall test was used to detect any trends in the data. The results indicated that the Gumbel distribution was the best-fit model for data in November, December, January, March, and April, while GEV was appropriate for February. No trends were observed in any of the months. The study then estimated the maximum rainfall for various return periods. January produced the highest maximum rainfall estimates for the 2, 3, and 5-year return periods, while February produced the highest maximum rainfall estimates for the 10 and 20-year return periods. Information about maximum rainfall can be valuable for the government and other stakeholders in developing flood prevention strategies and mitigating the effects of heavy rainfall, particularly during the peak months of the rainy season in Makassar City, which are December, January, and February.

Multiscale Coupling and the Maximum of {mathcal {P}}(phi )_2 Models on the Torus

We establish a coupling between the {mathcal {P}}(phi )_2 measure and the Gaussian free field on the two-dimensional unit torus at all spatial scales, quantified by probabilistic regularity estimates on the difference field. Our result includes the well-studied phi ^4_2 measure. The proof uses an exact correspondence between the Polchinski renormalisation group approach, which is used to define the coupling, and the Boué–Dupuis stochastic control representation for {mathcal {P}}(phi )_2. More precisely, we show that the difference field is obtained from a specific minimiser of the variational problem. This allows to transfer regularity estimates for the small-scales of minimisers, obtained using discrete harmonic analysis tools, to the difference field.As an application of the coupling, we prove that the maximum of the {mathcal {P}}(phi )_2 field on the discretised torus with mesh-size epsilon > 0 converges in distribution to a randomly shifted Gumbel distribution as epsilon rightarrow 0.

Extreme Value Prediction of Traffic Loads Using the Average Conditional Exceedance Rate Method

An efficient prediction of the extreme value of traffic loads is crucial for the structural design, reliability evaluation, maintenance planning, and further life-cycle cost analysis of bridges. In this work, a novel method is proposed for predicting the appropriate extreme traffic load distribution. Specifically, the average conditional exceedance rate (ACER) statistical model is estimated from the historical traffic loads which was collected through a weigh-in-motion system installed in toll stations. The basic idea of the ACER approach lies in the introduction of a cascade of conditioning approximations and the average exceedance rate to capture the dependence effects and obtain the data tail, the trend features of which are fitted with a similar Gumbel distribution function and extrapolated to the concerned level. An illustration case dealing with traffic loads using the ACER strategy is presented, the extreme value and confidence interval (CI) in any return period can be predicted by application of this approach. Furthermore, the peaks-over-threshold (POT) method based on the asymptotic extreme theory is also applied to illustrate the advantages of the ACER method. The ACER method has advantages in analyzing extreme traffic loads, with good robustness and the ability to handle extreme value prediction for different sampling strategies, it also can produce more accurate confidence intervals and predicts consistent extreme values. The study results are expected to help accurately determine traffic loads and ensure safety in bridge engineering.

ASSESSMENT OF MAXIMUM HAIL SIZES AND THEIR VALUE AT A 100-YEAR RETURN PERIOD

Hailstorms are relatively rare phenomena in Viet Nam, and understanding their characteristics is crucial for assessing the potential impact of hail events on various sectors. This study explores the maximum hailstone sizes observed in Viet Nam from 1961 to 2021 and uses the Gumbel distribution to estimate hailstone sizes at different return periods. Hail events in Viet Nam occur infrequently, with a common occurrence rate of less than once per year. However, some regions in the North, especially in the mountainous areas of the NorthWestern region, experience hail events approximately 1 - 2 times per year. The study records the largest hailstone sizes observed during the research period, with notable examples including 12 cm at the Bac Ha station and 10 cm at the Chiem Hoa station. These sizes exceed the calculated values for a 100-year return period, emphasizing the need for further research and preparedness. Despite the limited dataset of hailstone measurements, the Gumbel distribution is found to be a useful tool for estimating hailstone sizes at a 100-year return period. The theoretical estimates closely align with the observed values. At a 15-year return period, the most common maximum hailstone sizes range from approximately 1 - 3 cm, while at a 100-year return period, prevalent maximum hailstone sizes vary from 2.5 to 5.0 cm.

Prediction of Long-Term Offshore Structural Responses Based on Nonlinear Wave Modeling

Wind-generated random wave loads are the dominant loads to consider for maintaining the reliability of fixed offshore structures. Based on probabilistic techniques, the inherent randomness of the wave loading can be used to predict extreme offshore structural response, which is Gaussian in nature. However, researchers have found that the hydrodynamic component and structural dynamics substantially impact the frequency spectrum, leading to a non-Gaussian stochastic offshore structural response. A finite-memory non-linear system (FMNSNL) has been proven to be an efficient approach to evaluate the non-Gaussian stochastic offshore structural response compared to the conventional method, Monte Carlo time simulation. However, the analysis has been conducted based on short-term distribution only. The most satisfactory analysis is based on long-term distribution. Hence, further investigation in this paper will evaluate the long-term probability distribution of extreme offshore structural responses. As a result, a 100-year extreme offshore structural response prediction achieves 80% to 96% accuracy compared to the Monte Carlo time simulation. The probability distribution has been evaluated using the Gumbel distribution function throughout this investigation. Still, there is a little deviation at the tail end of the distribution between the simulated response values and the fitted line. A different distribution function, such as the Generalised Extreme Value (GEV) distribution, is advised for future work.

Detection of Common Bacterial And Viral Pathogens Causing Community Acquired Pneumonia Using Rt-pcr And Its Correlation with Some Clinical And Demographic Parameters

Dohuk city is hydrologically located in a relatively narrow watershed bounded by mountain ridges with heavy rainfall seasons. Due to its topographical characteristics and rapid urban development, the city is prone to floods during intense rainfall events or storms and in need of a proper storm water drainage system and flood control hydraulic structures for managing intense storm rainfalls, avoiding urban flooding, property loss, and most importantly human casualties. Nevertheless, the local civil engineers are in need of having up-to-date hydrological formulas, relationships, and information in order to be able to adequately design the flood control hydraulic structures. One of these important relationships is IDF curves which correlates the Intensity, Duration, and Frequency of an observed rainfall event. The authors have been able to develop updated IDF curves and empirical intensity formulas for Dohuk city by converting maximum annual rainfall readings for 21 consecutive years (2000-2020) into sub-daily rainfall records for durations of 10, 20, 30, 60, 120, 180, 360, 720, 1440 minutes with recurrence intervals of 2, 5, 10, 25, 50, 100 years using Gumbel and Log Pearson distribution (LPT III). The accuracy and reliability of the results were statistically confirmed through applying coefficient of determination with the value of (R2=1) and Goodness of fit test (Chi-square) with the confidence degree of (95%)

Estimation of cumulative amplitude distributions of miniature postsynaptic currents allows characterising their multimodality, quantal size and variability

A miniature postsynaptic current (mPSC) is a small, rare, and highly variable spontaneous synaptic event that is generally caused by the spontaneous release of single vesicles. The amplitude and variability of mPSCs are key measures of the postsynaptic processes and are taken as the main characteristics of an elementary unit (quantal size) in traditional quantal analysis of synaptic transmission. Due to different sources of biological and measurement noise, recordings of mPSCs exhibit high trial-to-trial heterogeneity, and experimental measurements of mPSCs are usually noisy and scarce, making their analysis demanding. Here, we present a sequential procedure for precise analysis of mPSC amplitude distributions for the range of small currents. To illustrate the developed approach, we chose previously obtained experimental data on the effect of the extracellular matrix on synaptic plasticity. The proposed statistical technique allowed us to identify previously unnoticed additional modality in the mPSC amplitude distributions, indicating the formation of new immature synapses upon ECM attenuation. We show that our approach can reliably detect multimodality in the distributions of mPSC amplitude, allowing for accurate determination of the size and variability of the quantal synaptic response. Thus, the proposed method can significantly expand the informativeness of both existing and newly obtained experimental data. We also demonstrated that mPSC amplitudes around the threshold of microcurrent excitation follow the Gumbel distribution rather than the binomial statistics traditionally used for a wide range of currents, either for a single synapse or when taking into consideration small influences of the adjacent synapses. Such behaviour is argued to originate from the theory of extreme processes. Specifically, recorded mPSCs represent instant random current fluctuations, among which there are relatively larger spikes (extreme events). They required more level of coherence that can be provided by different mechanisms of network or system level activation including neuron circuit signalling and extrasynaptic processes.

Aggregated wind power characteristic curves and artificial intelligence for the regional wind power infeed estimation

The wind power generation is highly dependent on current weather conditions. In the course of the energy transition, the generation levels from volatile wind energy are constantly increasing. Accordingly, the prediction of regional wind power generation is a particularly important and challenging task due to the highly distributed installations. This paper presents a study on the role of regional wind power infeed estimation and proposes a multi-aggregated wind power characteristics model based on three scaled Gumbel distribution functions. Multi-levels of wind turbines and their allocation are investigated for the regional aggregated wind power. Relative peak power performance and full load hours are compared for the proposed model and the real measurement obtained from a local distribution system operator. Furthermore, artificial intelligence technologies using neural networks, such as Long Short-Term Memory (LSTM), stacked LSTM and CNN–LSTM, are investigated by using different historical measurement as input data. The results show that the suggested stacked LSTM performs stably and reliably in regional power prediction.

Alibeyköy Havzası İçin En Uygun Taşkın Olasılık Dağılımı

Determination of peak flow rates is crucial in reducing the economic and social impact of flood. Therefore, the investigation of various methods for estimating flood is of paramount importance. Flood frequency analysis can be used as a practical method in predicting the peak flow values as the flood peaks have return periods that are typically much larger than the recording length. In this study, 7 different probability distributions (normal (N), two-parameter lognormal (LN2), three-parameter lognormal (LN3), extreme value type I (Gumbel), generalized extreme value (GEV), Pearson Type III (P3) and Log-Pearson Type III (LP3)) are employed for flood frequency analysis of Alibeyköy Watershed using 44-years of measured annual maximum flow. K-S and PPCC tests are applied to determine the most suitable distributions to estimate the flood flow rate. Based on these tests, GEV and Gumbel distributions appear to be the most preferable distributions in flood flow estimation.

Design Wave Height Estimation under the Influence of Typhoon Frequency, Distance, and Intensity

The extreme sea conditions caused by typhoons pose a threat to the design safety of marine and coastal engineering structures. In the past, design wave height calculation models that only considered the frequency of typhoons ignored the influence of other hazard factors of typhoons, resulting in lower design standards. In this paper, typhoon frequency, intensity, and distance are selected, and dimensional influences of different factors are eliminated through standardization processing. Based on the correlation between different hazard factors, we have obtained a multi-dimensional discrete joint probability distribution of typhoon hazard factors and constructed a new design wave height that considers the comprehensive effects of typhoon frequency, intensity, and distance. Our results show that the design wave height values of the 50-year, 100-year, and 200-year events are 12.59%, 8.10%, and 3.14% higher than the Gumbel distribution, which is more in line with the distribution of the wave height of waves under severe typhoon conditions. The new model can fully reflect the impact of typhoons on wave height, which can provide a reference for the design safety of marine engineering in the South China Sea.

Reckoning flood frequency and susceptibility area in the lower Brahmaputra floodplain using geospatial and hydrological approach

AbstractClimate change has remarkably intensified the occurrence of floods around the globe. Flooding causes loss of life and property. Flood frequency analysis (FFA) is an important investigation and plays a key role in flood‐related studies. Geographically, the study area is confined in the lower Brahmaputra floodplains, flat slope, and rivers are braided in nature. Because of Heavy rainfall, major rivers in the area carry huge influxes of surged water during the summer period. Hence, disastrous flooding can be seen every year in the study region. The present study aims to model flood frequency using the hydrological data to understand the effects within the area. FFA approaches like Gumbel, Log Pearson type 3 (LP‐3), and Log‐Normal (LN) were used, and comparative analyses were done using water level data for the Manas, Aie, and Brahmaputra Rivers. Moreover, remote sensing and the geographic information system (GIS) environment were used to generate FFA‐based flood predictive inundation map at 5, 10, 50, 100, and 200 years of return periods. Here, Gumbel's distribution has found the best fit for all the rivers among the three. The distribution reveals that at a 200‐year return period, the highest water level would be increased by 1.45, 2.41, and 4 m for the Manas, Aie, and Brahmaputra Rivers, respectively. The study shows that almost 493.54 and 673.72 km2 of areas are expected to be submerged at 5 and 200‐year return periods according to Gumbel's distribution; LP‐3 distribution predicted 493.01 and 555.66 km2, and the log‐normal distribution method predicted 432.51 and 555.74 km2 of flood‐sensitive areas at 5 and 200‐year return periods, respectively. The FFA highlighted spatio‐temporal effects on the expansion of submerged areas. We hope that the findings of the present study will aid in the different flood hazard management strategies for future endeavors.

Overall Design of Multi-module Offshore Floating Bridge and a Method for Random Analysis of Anchor Tension

This paper proposes a multi-module floating bridge design for offshore applications and establishes a coupled motion response analysis method for the anchoring system of the floating bridge. Time-domain coupled analysis is performed under the combined effects of wind, wave, and current loads, resulting in the maximum tension of the anchor chain in random waves. Using the typical island and reef sea area of Zhoushan, China as an example, the randomness of waves is considered, and a random sampling method is adopted. The main configuration, connection structure, and mooring system design schemes are presented, and the maximum tension of the anchoring system is numerically calculated and analyzed. The random distribution of the maximum anchor chain tension is obtained, the goodness-of-fit test is conducted for the fitting results, and the main degree-of-freedom motion response is analyzed under different exceedance probabilities. The results show that the multi-module offshore floating bridge meets the design requirements, and the statistical characteristics of the maximum tension of the anchor chain under random waves conform to the Gumbel distribution. The mode of the random distribution of the maximum anchor chain tension increases by 7.65% when fully loaded compared to the unloaded condition. In different exceedance probability scenarios, there are significant differences in the maximum tension of the anchor chain and the motion response of the main degrees of freedom of the floating bridge.

Reliability of Extreme Wind Speeds Predicted by Extreme-Value Analysis

The reliability of extreme wind speed predictions at large mean recurrence intervals (MRI) is assessed by bootstrapping samples from representative known distributions. The classical asymptotic generalized extreme value distribution (GEV) and the generalized Pareto (GPD) distribution are compared with a contemporary sub-asymptotic Gumbel distribution that accounts for incomplete convergence to the correct asymptote. The sub-asymptotic model is implemented through a modified Gringorten method for epoch maxima and through the XIMIS method for peak-over-threshold values. The mean bias error is shown to be minimal in all cases, so that the variability expressed by the standard error becomes the principal reliability metric. Peak-over-threshold (POT) methods are shown to always be more reliable than epoch methods due to the additional sub-epoch data. The generalized asymptotic methods are shown to always be less reliable than the sub-asymptotic methods by a factor that increases with MRI. This study reinforces the previously published theory-based arguments that GEV and GPD are unsuitable models for extreme wind speeds by showing that they also provide the least reliable predictions in practice. A new two-step Weibull-XIMIS hybrid method is shown to have superior reliability.

Assessment of Peak Flood Entering Hirakud Reservoir on River Mahanadi of Odisha, India

-Estimating the Peak Flood Discharge for a desired return time is necessary for the planning, design, and management of hydraulic infrastructure like barrages, dams, spillways, and bridges. This paper presents the results of a study that examined the frequency of floods entering Hirakud reservoir on river Mahanadi of Odisha state, India using the Gumbel's distribution and Log Pearson type III techniques, two frequently used models in hydrological studies for the probability distribution to mimic stream flows. The method has been used to forecast yearly maximum discharge data of Mahanadi river for a time interval of thirty years (1988-2017). The research also demonstrates a positive connection between the observed and expected peak discharges for the Log normal, Log Pearson type III (LP3) and Gumbel’s (EV1) models (R2 = 0.989), (R2 = 0.989), and R2 = 0.998 respectively. Peak flood values for periods 2, 10, 25, 50, 100, 200 and 400year return periods were also determined using the same approach, which will help for flood management through the reservoir. Key Words:flood frequency, Gumbel's distribution, Log Pearson type III, Log normal, flood management, R 2

Regional distribution of intensity–duration–frequency (IDF) relationships in Sultanate of Oman

Analysis of extreme rainfall parameters including rainfall intensities is a fundamental requisite in safe planning, designing, and operating various hydrologic and water engineering projects against storms and floods. In arid and semi-arid regions, such as Oman, sufficient long-term rainfall data with short aggregation are usually not available in most locations across the country. This paper presents the development of intensity duration frequency (IDF) curves using the available rainfall data from 65 meteorological stations situated at different elevations and regions throughout Oman. Gumbel distribution was fitted to the observed data and rainfall intensities were found for various return periods. Rainfall analysis showed the average annual rainfall of 109.21 mm with a standard deviation of 92.82 mm, Skewness coefficient of 1.62 and Kurtosis coefficient of 3.08 for all the studied stations from 1977 to 2017. The statistical analysis showed that the estimated rainfall intensities for various return periods are high in the mountainous region compared to the desert or interior region, and the coastal region of the country. Also, the empirical parameters of IDF formula for all studied stations were established using non-linear regression. Finally, the contour maps for all the parameters were drawn which could be used to determine the IDF relationships for ungauged locations. This study will be useful for the decision makers and practicing hydrologists for planning and design of water resources systems in Oman.

Hydraulic Planning in Insular Urban Territories: The Case of Madeira Island—Ribeira Brava, Tabua

This study’s primary goal was to conduct an analysis of the flood propensity of the Tabua (Ribeira Brava) drainage basin’s main watercourse. In addition to that, this study also recommends two different methodologies in order to mitigate flood impacts, namely by dimensioning a detention basin and adjusting the riverbed roughness coefficient. Regarding the study on the flood propensity, it was necessary to resort to geomorphological data, which were obtained when characterizing the watershed; these data were crucial to determining the expected peak flow rate, according to the Gumbel distribution methodology and considering a 100-year return period, and to perform necessary tasks in the SIG ArcGIS 10.5 software. Lastly, the drainage capacity of this drainage basin’s river mouth was also analyzed in order to conclude whether it would have the capacity to drain the total volume of rainwater if an extreme flood event were to happen. Indeed, the main results show that this watershed’s river mouth does not have the necessary drainage capacity to cope with an extreme event for the return period that was considered. As a consequence, the two aforementioned mitigation measures were developed considering the Tabua (Ribeira Brava) drainage basin’s specific features. The size of the detention basin was estimated through the Dutch method and the simplified triangular hydrograph method, while the adjustment of the roughness coefficient was considered a valid solution to enhance the drainage capacity of this river mouth.

Best Fitting of Probability Distribution for Monthly and Annual Maximum Rainfall Prediction in Junagadh Region (Gujarat-India)

Rain is a meager and crucial hydrological variable in arid and semi-arid region. Junagadh (Gujarat-India) reels under monsoon rainfall uncertainties and thereby the agriculture and other water resources management activities suffer. Therefore, urgent attention is needed to address water resources conservation and crop damage issues due to deficits or excess rainfall. Water resources development of any locality depends on amount of runoff generated and rainfall received. Appropriate probability distributions need to be selected and fitted to the historical time series of rainfall for better frequency analysis and forecasting of the rainfall. The daily rainfall data was collected for a period of 38 years i.e., from 1984 to 2021. This research attempts to fit eightdifferent theoretical probability distributions to the monthly and annual maximum rainfall for one to five consecutive days to select the best one for the better prediction of maximum rainfall. For determination of goodness of fit Chi-Square and Nash-Sutcliffe Efficiency were carried out by comparing the expected values with the observed values. The results indicated that the Gumbel distribution emerged to be the best fit for the prediction of monthly and annual maximum rainfall of Junagadh Region.

Extreme Sea Levels and Their Attribution for the Canadian Pacific Coast From a Baroclinic Regional Ocean Model and Tide‐Gauge Data

AbstractThe existing operational storm surge model for the Canadian Pacific coast is barotropic without coupling with tides. Here a baroclinic ocean model, which includes tides, storm surges, intra‐seasonal, seasonal, and interannual variations, is developed to reproduce coastal sea levels in the northeast Pacific from 1993 to 2020. The model results are validated against observations at tide‐gauge sites, demonstrating overall good skills for tides and non‐tidal sea levels. The Gumbel distribution is applied to derive extreme sea levels from hourly sea level anomalies. The model‐based extreme sea levels agree approximately with the observation‐based ones. For the 100‐year return period, modeled total extreme sea levels range from 2.14 m at Victoria Harbour to 4.43 m at Queen Charlotte City, and modeled non‐tidal extreme sea levels from 1.02 m at Bella Bella to 1.78 m at Queen Charlotte City. On average, the model‐based values underestimate the 100‐year return level by 0.18 m for total extreme sea levels and by 0.23 m for non‐tidal extreme sea levels. Both the model‐ and observation‐based results reveal that the storm surge at periods less than 10 d accounts for 60% of the 100‐year non‐tidal extreme sea level, and the intra‐seasonal, seasonal, and interannual variations account for 40% of it. The present study improves coastal sea level modeling and model‐based extreme level estimates. Moreover, this study also demonstrates for the first time that the intra‐seasonal to interannual changes contribute substantially to extreme sea levels and that El Niño can enhance this contribution on the Canadian Pacific coast.

Learning Noise-Induced Reward Functions for Surpassing Demonstrations in Imitation Learning

Imitation learning (IL) has recently shown impressive performance in training a reinforcement learning agent with human demonstrations, eliminating the difficulty of designing elaborate reward functions in complex environments. However, most IL methods work under the assumption of the optimality of the demonstrations and thus cannot learn policies to surpass the demonstrators. Some methods have been investigated to obtain better-than-demonstration (BD) performance with inner human feedback or preference labels. In this paper, we propose a method to learn rewards from suboptimal demonstrations via a weighted preference learning technique (LERP). Specifically, we first formulate the suboptimality of demonstrations as the inaccurate estimation of rewards. The inaccuracy is modeled with a reward noise random variable following the Gumbel distribution. Moreover, we derive an upper bound of the expected return with different noise coefficients and propose a theorem to surpass the demonstrations. Unlike existing literature, our analysis does not depend on the linear reward constraint. Consequently, we develop a BD model with a weighted preference learning technique. Experimental results on continuous control and high-dimensional discrete control tasks show the superiority of our LERP method over other state-of-the-art BD methods.

Reducing Sentiment Bias in Pre-trained Sentiment Classification via Adaptive Gumbel Attack

Pre-trained language models (PLMs) have recently enabled rapid progress on sentiment classification under the pre-train and fine-tune paradigm, where the fine-tuning phase aims to transfer the factual knowledge learned by PLMs to sentiment classification. However, current fine-tuning methods ignore the risk that PLMs cause the problem of sentiment bias, that is, PLMs tend to inject positive or negative sentiment from the contextual information of certain entities (or aspects) into their word embeddings, leading them to establish spurious correlations with labels. In this paper, we propose an adaptive Gumbel-attacked classifier that immunes sentiment bias from an adversarial-attack perspective. Due to the complexity and diversity of sentiment bias, we construct multiple Gumbel-attack expert networks to generate various noises from mixed Gumbel distribution constrained by mutual information minimization, and design an adaptive training framework to synthesize complex noise by confidence-guided controlling the number of expert networks. Finally, we capture these noises that effectively simulate sentiment bias based on the feedback of the classifier, and then propose a multi-channel parameter updating algorithm to strengthen the classifier to recognize these noises by fusing the parameters between the classifier and each expert network. Experimental results illustrate that our method significantly reduced sentiment bias and improved the performance of sentiment classification.