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Examples of Gumbel (γ = 0 in solid line), Fréchet (for γ = 1 in dashed line) and Reversed Weibull (for γ = −1 in dotted line) cdfs.
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... cdf G γ (x) is known as the generalized extreme value or as the extreme value cdf in the von Mises form, and the parameter γ is called the extreme value index. Figure 3 gives examples of Gumbel, Fréchet and Reversed Weibull distributions. Now, we will present the sketch of the Theorem's proof, following the approach of Beirlant et al. (2004b) which transfers the convergence in distribution to the convergence of expectations for the class of real, bounded and continuous functions (see Helly-Bray Theorem in Billingsley (1995)). ...
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... Therefore, numerous distributions are examined by hydrologists in different parts of the world [40]. Although there are different theoretical distributions to fit the extreme data series, the generalized extreme value distribution (GEVD) is the most applied technique in rainfall frequency analysis [2,41]. The GEVD is the collective of three statistical distributions that are commonly applied for flood hazard analysis. ...
Climate change impacts have the potential to alter the design rainfall estimates around the world. Decreasing trends in the summer and winter rainfall in New South Wales (NSW), Australia have already been observed due to climate variability and change. The derivation of design rainfall from historical rainfall, which is required for the design of stormwater management infrastructure, may be ineffective and costly. It is essential to consider climate change impacts in estimating design rainfall for the successful design of stormwater management infrastructure. In this study, the probability of the occurrence of daily extreme rainfall has been assessed under climate change conditions. The assessment was performed using data from 29 meteorological stations in NSW, Australia. For the evaluation of future design rainfall, the probability of the occurrence of extreme rainfall for different recurrence intervals was developed from daily extreme rainfall for the periods of 2020 to 2099 and compared with the current Australian Bureau of Meteorology (BoM) design rainfall estimates. The historical mean extreme rainfall across NSW varied from 37.71 mm to 147.3 mm, indicating the topographic and climatic influences on extreme rainfall. The outcomes of the study suggested that the future design rainfall will be significantly different from the current BoM estimates for most of the studied stations. The comparison of the results showed that future rainfall in NSW will change from −4.7% to +60% for a 100-year recurrence interval. However, for a 2-year recurrence interval, the potential design rainfall change varies from an approximately 8% increase to a 40% decrease. This study revealed that the currently designed stormwater management infrastructure will be idle in the changing climate.
... However, evidence in favor of alternative domains does exist, suggesting the field is still open to exploration. Adaptive walks with Weibull-distributed DFEs are characterized by fewer large-effect beneficial mutations compared to Gumbel EVDs [25,Fig 1]. Rokyta et al. [16] observed a Weibull distribution in the ID11 ssDNA phage, hinting at an upper limit on the size of beneficial fitness effects. ...
... Further, stabilizing selection can generate a Weibull-distributed EVD by limiting the size of beneficial mutations as populations approach the optimum [18]. On the other hand, Fréchet EVDs have more frequent beneficial mutations [25,Fig 1]. Schenk et al. [26] observed a Fréchet EVD in Escherichia coli adapting to antibiotics. ...
The tempo and mode of adaptation depends on the availability of beneficial alleles. Genetic interactions arising from gene networks can restrict this availability. However, the extent to which networks affect adaptation remains largely unknown. Current models of evolution consider additive genotype-phenotype relationships while often ignoring the contribution of gene interactions to phenotypic variance. In this study, we model a quantitative trait as the product of a simple gene regulatory network, the negative autoregulation motif. Using forward-time genetic simulations, we measure adaptive walks towards a phenotypic optimum in both additive and network models. A key expectation from adaptive walk theory is that the distribution of fitness effects of new beneficial mutations is exponential. We found that both models instead harbored distributions with fewer large-effect beneficial alleles than expected. The network model also had a complex and bimodal distribution of fitness effects among all mutations, with a considerable density at deleterious selection coefficients. This behavior is reminiscent of the cost of complexity, where correlations among traits constrain adaptation. Our results suggest that the interactions emerging from genetic networks can generate complex and multimodal distributions of fitness effects.
... By the Fisher-Tippett-Gnedenko theorem (see, e.g., Ref. [21]), there exist only three families of extreme value distributions G ξ (x)-the Fréchet, Gumbel and reversed Weibull families-each characterized by the tail index ξ determining the shape of the distribution. Regularly varying distributions (both continuous and their discrete counterparts) form the MDA of the Fréchet distribution, for which ξ > 0. The powerlaw exponent of any regularly varying distribution in the MDA of the Fréchet distribution can be directly inferred from the tail index, ...
Distinguishing power-law distributions from other heavy-tailed distributions is challenging, and this task is often further complicated by subsampling effects. In this work, we evaluate the performance of two commonly used methods for detecting power-law distributions—the maximum likelihood method of Clauset and the extreme value method of Voitalov —in distinguishing subsampled power laws from two other heavy-tailed distributions, the lognormal and the stretched exponential distributions. We focus on a random subsampling method commonly applied in network science and biological sciences. In this subsampling scheme, we are ultimately interested in the frequency distribution of elements with a certain number of constituent parts—for example, species with k individuals or nodes with k connections—and each part is selected to the subsample with an equal probability. We investigate how well the results obtained from low-subsampling-depth subsamples generalize to the original distribution. Our results show that the power-law exponent of the original distribution can be estimated fairly accurately from subsamples, but classifying the distribution correctly is more challenging. The maximum likelihood method falsely rejects the power-law hypothesis for a large fraction of subsamples from power-law distributions. While the extreme value method correctly recognizes subsampled power-law distributions with all tested subsampling depths, its capacity to distinguish power laws from the heavy-tailed alternatives is limited. However, these false positives tend to result not from the subsampling itself but from the estimators' inability to classify the original sample correctly. In fact, we show that the extreme value method can sometimes be expected to perform better on subsamples than on the original samples from the lognormal and the stretched exponential distributions, while the contrary is true for the main tests included in the maximum likelihood method.
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... Another future research direction is the focused investigation of extreme values applying, for example, Extreme Value Theory, which suggests a different set of distributions to capture extremely rare but impactful events (Charras-Garrido and Lezaud, 2013). Similarly, scholars may consider Laplace distributions for more complex cases when entrepreneurial performance can take on both positive and negative outcomes (e.g., return on investment, profitability), which implies the relevance of two tails for extreme phenomena in natural and social sciences (Gel, 2010). ...
This study extends emerging theories of star performers to digital platforms, an increasingly prevalent entrepreneurial context. It hypothesizes that the unique characteristics of many digital platforms (e.g., low marginal costs, feedback loops, and network effects) produce heavy-tailed performance distributions, indicating the existence of star entrepreneurs. Using longitudinal data from an online learning platform, proportional differentiation is identified as the most likely generative mechanism and lognormal distribution as the most likely shape for distributions of entrepreneurial performance in digital contexts. This study contributes theory and empirical evidence for non-normal entrepreneurial performance with implications for scholars and practitioners of digital entrepreneurship. Executive summary The performance of 'star' entrepreneurs on digital platforms can be 100-or 1000-fold that of their average competitors. When performance is plotted as a distribution, star performers reside in the tails of these distributions. The assumption of a normal distribution of performance in the bulk of entrepreneurship research implies that most performance observations are clustered around the average. Instead, most entrepreneurs on digital platforms exhibit sub-par performance, while a minority captures a major fraction of the generated value. This paper argues that the unique characteristics of digital contexts-nearly zero marginal costs, feedback loops, and network effects-drive such extreme performance. Using data from Udemy, a digital platform where independent producers (entrepreneurs) offer educational videos (digital products) to a large pool of potential customers, we provide evidence that entre-preneurial performance is lognormally rather than normally distributed. We further identify proportional differentiation as the underlying generative mechanism. Thus, star performance on digital platforms is not driven only by the rich-get-richer effect. Instead, both the initial value of performance and the rate at which it is accumulated play important roles in explaining extreme performance outcomes. This discovery has important implications for entrepreneurship theory and practice. Our findings, for example, signal that some late entrants who successfully pursue high customer accumulation rates in domains with high knowledge intensity can become star entrepreneurs.
... Teply (2012) opined that one of the earliest studies on operational risk management was carried out by Embrechts et al in 1997 in which they did the modelling of extreme events for insurance and finance. According to Garrido and Lezaud (2013), extreme value theory primarily aims to predict the occurrence of rare events that are not within the range of the available data and is one of the standard approaches to studying risks; it is a branch of statistics that deal with the extreme deviations from the median of probability distributions i.e. based on the language of probability theory and thus the first question to ask is whether a probability approach applies to the studied risk. As highlighted in the work of Adegbie and David (2020), Extreme value theory is a tool used to determine the probabilities (risks) associated with extreme events and it helps in promoting the assessment and management of extreme financial risks. ...
... The Extreme Value Theory (EVT) proposes a more robust framework for the prediction of extremes, which are modelled with a Pareto distribution [9]. Few publications have dealt with EVT in the context of renewable production forecasting, with the exceptions of [5], [6] and [10] who propose EVT forecasts of extremal quantiles of VRE production. ...
... Pickland's theorem [9] stipulates that the distribution of the maxima of the i.i.d variable of y * above the threshold u converges towards a Generalized Pareto Distribution (GPD). The GPD is defined by its parameter vector θ : ={u, σ, γ}, where γ defines the overall shape of the distribution of extremes and σ quantifies the spread of extreme values. ...
Virtual power plants aggregating multiple renewable energy sources such as Photovoltaics and Wind are promising candidates for the provision of balancing ancillary services. A requisite for the provision of these services is that forecasts of aggregated production need to be highly reliable in order to minimize the risk of not providing the service. Yet, a reliability greater than 99% is unattainable for standard forecasting models. This work proposes alternative models for the day-ahead prediction of the lowest quantiles (0.1% to 0.9 %) of renewable Virtual power plant production. The proposed approaches derive conditional quantile forecasts of aggregated Wind/PV/Hydro production, obtained from tailored parametric models and machine learning models, including a Convolutional Neural Network architecture for predicting extremes. Reliability deviation is reduced up to 50 % and probabilistic skill score up to 18% compared to Quantile Regression Forest. Forecasting models are subsequently applied to the provision of downward reserve capacity by a renewable Virtual power plant. Increased forecasting reliability leads to a higher reliability of the reserve capacity, but reduces the average reserve volume offered by the renewable aggregation.
... In order to be able to provide reliable information, design rainfall estimates have to be based on sufficiently long time-series of rainfall observations from climate stations at a high temporal resolution (e.g. Charras-Garrido and Lezaud, 2013). Especially for the estimates of rare events (Tr>=100a) this restricts the analyses usually to a rather limited number of precipitation stations, hence requiring substantial spatial interpolation efforts in order to regionalize the information. ...
Spatially explicit quantification on design storms are essential for flood risk assessment and planning. Since the limited temporal data availability from weather radar data, design storms are usually estimated on the basis of rainfall records of a few precipitation stations having a substantially long time coverage. To achieve a regional picture these station based estimates are spatially interpolated, incorporating a large source of uncertainty due to the typical low station density, in particular for short event durations. In this study we present a method to estimate spatially explicit design storms with a return period of up to 100 years on the basis of statistically extended weather radar precipitation estimates based on the ideas of regional frequency analyses and subsequent bias correction. Associated uncertainties are quantified using an ensemble-sampling approach and event-based bootstrapping. With the resulting dataset, we compile spatially explicit design storms for various return periods and event durations for the federal state of Baden Württemberg, Germany. We compare our findings with two reference datasets based on interpolated station estimates. We find that the transition in the spatial patterns of the design storms from a rather random short duration events, 15 minute) to a more structured, orographically influenced pattern (long duration events, 24 hours) seems to be much more realistic in the weather radar based product. However, the absolute magnitude of the design storms, although bias-corrected, is still generally lower in the weather radar product, which should be addressed in future studies in more detail.
... Extreme-Value Theory (EVT) is a statistical approach for analyzing extreme events (Charras-Garrido and Lezaud, 2013). EVT has been used to analyze the frequency of extreme environmental events, especially floods (Quintela-del-Río and Francisco-Fernández, 2018), but also droughts (Xu et al., 2011). ...
Many studies that investigate mitigation strategies of greenhouse-gas (GHG) emissions from farming systems often build farm typologies from average data from multiple farms. Results from farm typologies are useful for general purposes but fail to represent variability in farm characteristics due to management practices or climate conditions, particularly when considering consequences of extreme environmental events. This limitation raises the issue of better distinguishing, within datasets of farms, farms that have average characteristics from those that deviate from average trends, in order to improve assessment of how climate variability influences farm performance. We applied the statistical method called Extreme Value Theory (EVT) to identify dairy farms that produced "extreme" amounts of forage. Applying EVT to a dataset of dairy farms from Normandy, Lorraine and Nord-Pas-de-Calais (France) identified subsamples of 10-30% of dairy farms with the smallest or largest amounts of grass from pastures or maize silage in each region. Characteristics of farms with extreme amounts of each forage often differed among regions due to the influence of geography and climate. Farms with the largest amounts of grass or the smallest amounts of maize silage had a variety of cow breeds in Normandy and Lorraine but had only Holstein cows in Nord-Pas-de-Calais. Conversely, most farms with the smallest amounts of grass or the largest amounts of maize silage had Holstein cows, regardless of region. The region also influenced whether farms were oriented more toward producing milk with higher fat and protein contents (Normandy and Lorraine) or toward producing larger amounts of milk (Nord-Pas-de-Calais). As the amount of a given forage changed from smallest to largest, a significant increase or decrease in the amount of milk produced usually changed GHG and enteric methane (CH4) emissions per farm in the same direction as the amount of milk produced. For instance, an extreme increase in the amount of grass fed on farms (1314 vs. 5093 kg/livestock unit/year, respectively) in Normandy was associated with decreased mean milk production (8236 vs. 5834 l/cow/year, respectively) and GHG (7117 vs. 5587 kg CO2 eq./farm/year) and enteric CH4 (3870 vs. 3296 kg CO2 eq./farm/year, respectively) emissions.
... The POT approach is considered more advantageous in fitting natural phenomena as it includes all independent events above a prescribed threshold, whereas the alternative BM approach is based on selection of a single maxima per equidistant time segments, ignoring other key rare events occurring within the same time segment [20]. By virtue, The BM method has often been considered a wasteful approach to EVA if other data on extremes are available [25]. Hence this study uses the POT approach. ...
This paper provides an Extreme Value Analysis (EVA) of the hourly water level record at Fort Denison dating back to 1915 to understand the statistical likelihood of the combination of high predicted tides and the more dynamic influences that can drive ocean water levels higher at the coast. The analysis is based on the Peaks-Over-Threshold (POT) method using a fitted Generalised Pareto Distribution (GPD) function to estimate extreme hourly heights above mean sea level. The analysis highlights the impact of the 1974 East Coast Low event and rarity of the associated measured water level above mean sea level at Sydney, with an estimated return period exceeding 1000 years. Extreme hourly predictions are integrated with future projections of sea level rise to provide estimates of relevant still water levels at 2050, 2070 and 2100 for a range of return periods (1 to 1000 years) for use in coastal zone management, design, and sea level rise adaptation planning along the NSW coastline. The analytical procedures described provide a step-by-step guide for practitioners on how to develop similar baseline information from any long tide gauge record and the associated limitations and key sensitivities that must be understood and appreciated in applying EVA.
... However, given the lack of sufficient (or any) anomalous data, learning is not possible. We advocate the use of extreme value theory (EVT) [10] to learn a surrogate for . The core idea is to assume that the anomalous observations are the extreme values of . ...
... EVT [10] is the study of extremes of data distributions. The foundations were laid by Fisher and Tippett [18] and Gnedenko [28] who demonstrated the closed forms of the distributions of the extreme values of i.i.d. ...
Data‐driven anomaly detection methods typically build a model for the normal behavior of the target system, and score each data instance with respect to this model. A threshold is invariably needed to identify data instances with high (or low) scores as anomalies. This presents a practical limitation on the applicability of such methods, since most methods are sensitive to the choice of the threshold, and it is challenging to set optimal thresholds. The issue is exacerbated in a streaming scenario, where the optimal thresholds vary with time. We present a probabilistic framework to explicitly model the normal and anomalous behaviors and probabilistically reason about the data. An extreme value theory based formulation is proposed to model the anomalous behavior as the extremes of the normal behavior. As a specific instantiation, a joint nonparametric clustering and anomaly detection algorithm (INCAD) is proposed that models the normal behavior as a Dirichlet process mixture model. Results on a variety of datasets, including streaming data, show that the proposed method provides effective and simultaneous clustering and anomaly detection without requiring strong initialization and threshold parameters.
