Extreme Value Model Research Articles

Discrete Choice Prox-Functions on the Simplex

We derive new prox-functions on the simplex from additive random utility models of discrete choice. They are convex conjugates of the corresponding surplus functions. In particular, we explicitly derive the convexity parameter of discrete choice prox-functions associated with generalized extreme value models, and specifically with generalized nested logit models. Incorporated into subgradient schemes, discrete choice prox-functions lead to a probabilistic interpretations of the iteration steps. As illustration, we discuss an economic application of discrete choice prox-functions in consumer theory. The dual averaging scheme from convex programming adjusts demand within a consumption cycle.

Non‐stationary frequency analysis of extreme streamflow disturbance in a typical ecological function reserve of China under a changing climate

AbstractExtreme hydro‐meteorological events occur more frequently with global warming, which have a great impact on ecology, environment and hydrology. Hydro‐meteorological variables often exhibit disturbances of non‐stationary characteristics. In this paper, we take a typical ecological function reserve (Poyang Lake Basin) as an instance to study the trend and probability characters of extreme streamflow based on stationary and non‐stationary generalized extreme value (GEV) models. Time and large‐scale climate factors such as Southern Oscillation (SO), North Atlantic Oscillation (NAO), Arctic Oscillation (AO), Pacific Decadal Oscillation (PDO), Atlantic multidecadal oscillation (AMO), NINO3 SST (NINO3) and NINO3.4 SST (NINO3.4) are incorporated as covariates to establish non‐stationary model. The main results are as follows: (1) Downward trends prevail for maximum streamflow series, but upward trends dominate for minimum streamflow series. (2) The tested optimal models of some maximum streamflow series are non‐stationary, while the optimal models of all minimum streamflow series are non‐stationary. In the 1970s and before (1980s and thereafter), the return periods of optimal model are greater (less) than those of stationary model for most minimum series, and the risks of river flooding and drying up without calculating by optimal model are underestimated (overestimated). (3) The estimated quantiles tend to be larger during La Niña than during El Niño in this study area. Other climate indices also have obvious effects on the estimated quantiles. This study indicates that the influence of climate factors on the extreme streamflow should be paid more attention and non‐stationary modelling will benefit ecological management under climate change.

Application of Bivariate Extreme Value models to describe the joint behavior of temporal and speed related surrogate measures of safety

Limitations of historical crash data and the advantages of surrogate measures of safety have already been stressed by various authors. To describe nearness-to-collision, mostly time-based indicators are applied, and there is a consensus among researchers that speed-related indicators are needed to account for the severity dimension. There have been attempts to fit bivariate Extreme Value models to predict the number of crashes, however modeling crash frequency together with severity has received much less attention.The aim of this paper is to apply Extreme Value models to various pairs of temporal and speed-related indicators in order to investigate the dependence structure between them as well as to construct probability based risk levels and examine them in relation to severity levels. Bivariate threshold excess models were fitted to a dataset of left-turning and straight moving vehicle interactions recorded in a signalized intersection. The dependence structure between variable pairs were thoroughly investigated; it was concluded that temporal and speed related indicators were found independent, which means that road users getting close to each other in time does not necessarily come with high speeds. Therefore these indicators should be combined in order to properly predict severity; a temporal indicator on its own is not enough to make inferences about the severity of events. Using the calculated joint probability of events risk levels were constructed illustrating the events of equal probability.

Validating the Bayesian hierarchical extreme value model for traffic conflict-based crash estimation on freeway segments with site-level factors

This study validates the Bayesian hierarchical extreme value model that is developed for estimating crashes from traffic conflicts. The model consists of a generalized extreme value distribution that characterizes the behavior of block maxima extremes and a Bayesian hierarchical structure that incorporates the non-stationarity and unobserved heterogeneity into the extreme analysis. In addition to the block-level factors, the site-level factors are also included in the model development for the first time. The model was applied to data of lane change conflicts collected from 11 basic freeway segments in Guangdong Province, China. Block-level factors such as traffic volume per 10 min, number of lane change events per 10 min, and proportion of oversized vehicles per 10 min and site-level factors such as segment length, curvature, and grade were considered. Two types of Bayesian hierarchical extreme value models were developed, including models without site-level factors and models with site-level factors. These models were also compared to at-site models that were developed for 11 segments separately. The results show that Bayesian hierarchical extreme value models significantly outperform the at-site models in terms of crash estimation accuracy and precision. As well, including site-level factors further improves the model performance in terms of goodness-of-fit. This demonstrates the validity of the Bayesian hierarchical extreme value model. The results also show that number of lane change events, segment length, and grade are significant factors which have adverse effect on the safety of lane changes on freeway segments.

A bivariate extreme value model for estimating crash frequency by severity using traffic conflicts

Estimating crash frequency by severity levels using traffic conflicts remains relatively unexplored in conflict-based traffic safety assessment, limiting its application scope and appeal compared to traditional methods. No studies to date have predicted the frequency of severe and non-severe crashes utilizing traffic conflicts. This study aims to address this critical gap and stimulate discussion and development in this critical area. The study estimates the frequency of severe crashes and non-severe crashes by jointly modeling the indicators of crash frequency, namely, Time to Collision (TTC) and Modified Time to Collision (MTTC), and crash severity, namely, predicted post-collision change in velocity (Delta-V or Δv), using bivariate Extreme Value. Severe crashes here are defined as crashes with a Maximum Abbreviated Injury Scale rating of greater than or equal to 3. Rear-end conflict data (TTC ≤ 3.0 s) were collected for two days (12 h each day) from two four-legged signalized intersections in Brisbane, Australia. Bivariate peak-over-threshold models for both TTC and MTTC indicators, combined with Delta-V, were estimated. Alternatively, another univariate approach was also attempted where the probability of crash occurrence was estimated using the univariate peak-over-threshold model with TTC (or MTTC) and then multiplied with the injury probability estimated from Delta-V to estimate the frequencies of severe and non-severe injury crashes. The study results demonstrate that the bivariate approach is more advantageous than the univariate approach due to a superior statistical fit to the data and more precise estimations of crash frequencies by severity levels. Both TTC and MTTC indicators, in combination with Delta-V, provide comparable results using the bivariate approach owing to the weak asymptotic dependence between the frequency and severity indicators. Comparing the combined dataset model of the two intersections with the intersection-based models shows that sharing information between similar traffic sites improves the accuracy and precision of prediction.

Extreme Value Modeling of Daily Maximum Temperature with the r-Largest Order Statistics

Extreme Value Modeling of Daily Maximum Temperature with the r-Largest Order Statistics

Estimating concurrent climate extremes: A conditional approach

Simultaneous concurrence of extreme values across multiple climate variables can result in large societal and environmental impacts. Therefore, there is growing interest in understanding these concurrent extremes. In many applications, not only the frequency but also the magnitude of concurrent extremes are of interest. One way to approach this problem is to study the distribution of one climate variable given that another is extreme. In this work we develop a statistical framework for estimating bivariate concurrent extremes via a conditional approach, where univariate extreme value modeling is combined with dependence modeling of the conditional tail distribution using techniques from quantile regression and extreme value analysis to quantify concurrent extremes. We focus on the distribution of daily wind speed conditioned on daily precipitation taking its seasonal maximum. The Canadian Regional Climate Model large ensemble is used to assess the performance of the proposed framework both via a simulation study with specified dependence structure and via an analysis of the climate model-simulated dependence structure.

Exploring Multiple\u2010discreteness in Freight Transport. A Multiple Discrete Extreme Value Model Application for Grain Consolidators in Argentina

There are some examples where freight choices may be of a multiple discrete nature, especially the ones at more tactical levels of planning. Nevertheless, this has not been investigated in the literature, although several discrete-continuous models for mode/vehicle type and shipment size choice have been developed in freight transport. In this work, we propose that the decision of port and mode of the grain consolidators in Argentina is of a discrete-continuous nature, where they can choose more than one alternative and how much of their production to send by each mode. The Multiple Discrete Extreme Value Model (MDCEV) framework was applied to a stated preference data set with a response variable that allowed this multiple-discreteness. To our knowledge, this is the only application of the MDCEV in regional freight context. Free alongside ship price, freight transport cost, lead-time and travel time were included in the utility function and observed and random heterogeneity was captured by the interaction with the consolidator’s characteristics and random coefficients. In addition, different discrete choice models were used to compare the forecasting performance, willingness to pay measures and structure of the utility function against.

Return Period Evaluation of the Largest Possible Earthquake Magnitudes in Mainland China Based on Extreme Value Theory.

The largest possible earthquake magnitude based on geographical characteristics for a selected return period is required in earthquake engineering, disaster management, and insurance. Ground-based observations combined with statistical analyses may offer new insights into earthquake prediction. In this study, to investigate the seismic characteristics of different geographical regions in detail, clustering was used to provide earthquake zoning for Mainland China based on the geographical features of earthquake events. In combination with geospatial methods, statistical extreme value models and the right-truncated Gutenberg–Richter model were used to analyze the earthquake magnitudes of Mainland China under both clustering and non-clustering. The results demonstrate that the right-truncated peaks-over-threshold model is the relatively optimal statistical model compared with classical extreme value theory models, the estimated return level of which is very close to that of the geographical-based right-truncated Gutenberg–Richter model. Such statistical models can provide a quantitative analysis of the probability of future earthquake risks in China, and geographical information can be integrated to locate the earthquake risk accurately.

A new empirical distribution for the design wave heights under the impact of typhoons

In the South China Sea, where storm surges are frequent, accurate calculation of the design wave heights is crucial to the coastal structures and subsequent disaster prevention and relief. The data sampling methods have developed from annual extreme value sampling to over threshold value sampling to use the precious observed wave heights more effectively. However, the problems still exist, resulting in "sharp peak" and "thick tail" for the over threshold data. And traditional design wave heights estimation models cannot fully fit the tail of the data. The strong typhoon also makes the variation patterns of associated wave height factors highly uncertain. To address these issues, this study proposes a Poisson—Gumbel-Pareto distribution model based on the cumulative distribution functions and the combined extreme value distribution theory. The new model not only takes the impacts of typhoon frequency on wave heights into consideration when the wave heights are over the threshold, it can better represent the peak of the data, and also makes the tail of the probability density function curves gradually descend, that is, the tail data can be fully utilized and fitted using this new method. Taking the measured wave and typhoon data from the Naozhou Ocean Observatory in the South China Sea from 1990 to 2016, the design wave heights in the western Guangdong waters have been calculated on the level of once-in-many-years, the design wave heights for 100-year and 200-year are 9.51m and 10.11m respectively. The analysis shows that the new empirical distribution improves the previous extreme value models' disadvantages, with reasonable and stable projections in the high return periods, which is especially favorable when dealing with extreme sea conditions.

Random parameters Bayesian hierarchical modeling of traffic conflict extremes for crash estimation

The use of Extreme Value Theory (EVT) models for traffic conflict-based crash estimation is becoming increasingly popular with considerable recent advances achieved. The latest advances include developing EVT models that combine several conflict indicators and the use of data from several sites to increase the sample size of conflict extremes. Nevertheless, one important issue while developing EVT models is accounting for the unobserved heterogeneity across different conflict observation sites and road user behaviours which can lead to biased and inefficient parameter estimates and erroneous inferences. This study proposes a random parameters (RP) Bayesian hierarchical extreme value modeling approach to account for the unobserved heterogeneity. The proposed approach is applied to estimate rear-end crashes from traffic conflicts collected from four signalized intersections in the city of Surrey, British Columbia. Traffic conflicts were characterized by four indicators: time to collision (TTC), modified TTC (MTTC), post-encroachment time (PET), and deceleration rate to avoid a crash (DRAC). MTTC was used to fit the generalized extreme value distribution, while the other three conflict indicators were treated as covariates. Six covariates including TTC, PET, DRAC, traffic volume, shock wave area, and platoon ratio were considered to account for non-stationarity in conflict extremes. Several RP, random intercepts (RI), and fixed parameters (FP) Bayesian hierarchical univariate extreme value models were developed. The results indicate that the RP model outperforms both the RI model and the FP model in terms of crash estimation accuracy and precision. Such superiority may be due to the ability of the RP model to better account for the unobserved heterogeneity.

Conditional modelling approach to multivariate extreme value distributions: application to extreme rainfall events in South Africa

Multivariate extreme value models are used to investigate the combined behaviour of several weather variables. To investigate joint dependence of extreme rainfall events, a multivariate conditional modelling approach was considered to analyse the behaviour of joint extremes of rainfall events at selected weather stations in South Africa. The results showed that the multivariate conditional modelling fitted to daily maximum rainfall events provided apparent benefits in terms of improved precision in the estimation of the marginal parameters of generalised Pareto distribution. The conditional modelling approach provided all forms of dependence using Laplace marginal transformations, for which all weather stations are not extreme equally. Bootstrap sampling was also employed to account for models uncertainty in computing the prediction standard errors, and compared with the prediction obtained from the conditional model fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, and the extremal dependence structure described consistently for extreme daily maximum rainfall events between weather stations. The current study contributes towards understanding the salient features on the extremal dependence of rainfall extremes which are associated with e.g., flash floods and landslides. This knowledge has practical applications in disaster risk preparedness by responsible authorities.

A novel data-driven method for structural health monitoring under ambient vibration and high-dimensional features by robust multidimensional scaling

Dealing with the problem of large volumes of high-dimensional features and detecting damage under ambient vibration are critical to structural health monitoring. To address these challenges, this article proposes a novel data-driven method for early damage detection of civil engineering structures by robust multidimensional scaling. The proposed method consists of some simple but effective computational parts including a segmentation process, a pairwise distance calculation, an iterative algorithm regarding robust multidimensional scaling, a matrix vectorization procedure, and a Euclidean norm computation. AutoRegressive Moving Average models are fitted to vibration time-domain responses caused by ambient excitations to extract the model residuals as high-dimensional features. In order to increase the reliability of damage detection and avoid any false alarm, the extreme value theory is considered to determine a reliable threshold limit. However, the selection of an appropriate extreme value distribution is crucial and troublesome. To cope with this limitation, this article introduces the generalized extreme value distribution and its shape parameter for choosing the best extreme value model among Gumbel, Fréchet, and Weibull distributions. The main contributions of this article include developing a novel data-driven strategy for early damage detection and addressing the limitation of using high-dimensional features. Experimental data sets of two well-known civil structures are utilized to validate the proposed method along with some comparative studies. Results demonstrate that the proposed data-driven method in conjunction with the extreme value theory is highly able to detect damage under ambient vibration and high-dimensional features.

A day in the life with an automated vehicle: Empirical analysis of data from an interactive stated activity-travel survey

Fully Automated Vehicles (AVs) have been widely expected to revolutionise the future travel experience. Recent studies have shown that their impact may also reach beyond the travel episode, and lead their users to alter other activities performed during the day – their daily lifestyles. This study is among the first to empirically investigate the changes that travellers expect in their daily activities with AVs. To this aim, we created an interactive stated activity-travel survey, in which respondents designed their current daily schedule and, following that, redesigned it while imagining that their most frequently used travel mode is replaced with an AV. We administered the survey to 509 commuters in the Netherlands and analysed (changes in) on-board and stationary activity patterns using the multiple discrete-continuous extreme value (MDCEV) model. Results show a clear increase in the prevalence of various on-board activities in the AV compared to current modes, and even stronger increase for the high income and higher educated groups. Changes in stationary activities are less pronounced: no changes in the aggregate, but some changes within particular socio-demographic groups. Specific changes in stationary activities were associated with specific changes in on-board activities for the higher educated respondents: switching to AVs, they were more likely than others to add on-board work, meals, and leisure to their trips and more likely to add a getting ready activity to their stationary schedules. This study contributes to the growing body of literature that recognises and models on-board activities as an integral part of daily schedules.

Statistical Modelling of Extreme Rainfall Indices using Multivariate Extreme Value Distributions

Multivariate extreme value models are used to investigate the combined behaviour of several weather variables. To investigate joint dependence of extreme rainfall events, a multivariate conditional modelling approach was considered to analyse the behaviour of joint extremes of rainfall events at selected weather stations in South Africa. Moreover, 1-day to 5-day indices of rainfall events were constructed to investigate the frequencies and intensities of rainfall events for selected weather stations. Then, the conditional multivariate modelling was fitted to investigate dependence between series of extreme rainfall events. The conditional multivariate modelling has provided all forms of dependence, using Laplace marginal transformations, for which all weather stations are not equally extreme. Bootstrap sampling was also employed to account for models uncertainty in computing the prediction standard errors and compared with the prediction obtained from the conditional modelling that was fitted to extreme data. The results obtained from predictions reflected both the marginal and the dependence features, as well as the extremal dependence structure described consistently for indices of rainfall events between weather stations. The modelling framework and results of this study contribute towards understanding the salient features on the extremal dependence of rainfall extremes which are associated with, e.g. flash floods and landslides. This knowledge has practical applications in disaster risk preparedness by communities.

A new closed form multiple discrete-continuous extreme value (MDCEV) choice model with multiple linear constraints

Traditional multiple-discrete continuous choice models that have been formulated and applied in recent years consider a single linear resource constraint, which, when combined with consumer preferences, determines the optimal consumption point. However, in reality, consumers may face multiple resource constraints, such as those associated with time, money, and storage capacity. Ignoring such multiple constraints and instead using a single constraint can, and in general will, lead to poor data fit and inconsistent preference estimation, which can then have a serious negative downstream effect on forecasting and welfare/policy analysis. Unlike earlier attempts to address this multiple constraint situation, we formulate a new multiple-constraint (MC) multiple discrete continuous extreme value (MDCEV) model (or the MC-MDCEV model) that retains a closed-form probability structure and is as simple to estimate as the MDCEV model with one constraint. We achieve this by assuming a type-I extreme value distribution for the error term in its minimization form in the baseline utility preference of each good rather than a maximization form as in Bhat's (2005; 2008) original MDCEV formulation. The statistical foundation of the proposed model is based on the fact that the difference between a minimal type-I extreme value random variable with scale σ and the weighted sum of the exponential of standardized minimal type-I extreme value random variables (scaled up by σ) leads to an apparently new multivariate distribution that has an elegant and closed-form survival distribution function. Results from a simulation experiment show that our proposed model substantially outperforms single-constraint models; the results also highlight the serious mis-estimation that is likely to occur if only a subset of active constraints is used. The proposed model is applied to a case of week-long activity participation where individuals are assumed to maximize their utility from time-use subject to time and money budgets.It is hoped that our proposed simple closed-form multi-constraint MDCEV model will contribute to a new direction of application possibilities and to new research into situations where consumers face multiple constraints within a multiple discrete-continuous choice context.

Influence of Pacific Decadal Oscillation on global precipitation extremes

While the influence of the Pacific Decadal Oscillation (PDO) on precipitation has been extensively studied, its influence on precipitation extremes remains not well-known. Based on a series of generalized extreme value modeling experiments, this paper demonstrates some distinct regional patterns of the PDO’s influence on precipitation extremes worldwide. In eastern China, the well-known ‘southern flood (drought) and northern drought (flood)’ pattern in summer corresponds well to the positive (negative) phase of the PDO. In Australia, there tends to be a tripole pattern, with positive correlation between precipitation extremes and the PDO in the central region, and negative correlation in both eastern and western Australia. The precipitation extremes in northwestern Europe and western Russia roughly hold positive correlation with the PDO. These regional patterns of the PDO’s influences are explained via comparative analyses of the atmospheric circulation conditions between cold and warm PDO phases. Certain precipitation extremes tend to be missed or happen more than once during different phases of the PDO at more than 2/3 of stations in a typical region. The cold phase tends to exert more consistent influences than the warm phase in these typical regions. These findings not only indicate different risks of extreme precipitation for the typical regions during different phases of the PDO, but also have important implications for the near-term projection of variable regional climate extremes under global warming.

Computationally efficient forecasting procedures for Kuhn-Tucker consumer demand model systems: Application to residential energy consumption analysis

This paper proposes simple and computationally efficient forecasting algorithms for random utility maximization-based multiple discrete-continuous (MDC) choice models with additively separable utility functions, such as the Multiple Discrete-Continuous Extreme Value (MDCEV) model. The algorithms build on simple yet insightful, analytical explorations with the Karush Kuhn-Tucker (KKT) conditions of optimality that shed new light on the properties of the models. The MDCEV model and the forecasting algorithms proposed in this paper are applied to a household-level energy consumption dataset to analyze residential energy consumption patterns in the United States. Further, simulation experiments are undertaken to assess the computational performance of the proposed and existing KT demand forecasting algorithms for a range of choice situations with small and large choice sets.

Transport Networking Companies Demand and Flow Estimation in New York City

Given the burgeoning growth in transport networking companies (TNC)-based ride hailing systems and their growing adoption for trip making, it is important to develop modeling frameworks to understand TNC ride hailing demand flows at the system level. Two choice dimensions are identified: (1) a demand component that estimates origin level TNC demand at the taxi zone level and (2) a distribution component that analyzes how these trips from an origin are distributed across the region. The origin level demand is analyzed using linear mixed models while flows from origin to multiple destinations is analyzed using a multiple discrete-continuous extreme value (MDCEV) model. The data for the analysis is drawn from New York City Taxi and Limousine Commission for 12 months from January through December 2018. For this analysis, weekday morning peak hour demand and distribution patterns are examined. The model components are developed using a comprehensive set of independent variables. The model estimation results offer very intuitive results for origin demand and distribution of flows across destinations. The model was validated by predicting trips to destination taxi zones and it was found that predicted model performs well in identifying high preference destination zones. In addition, elasticity effects are computed by evaluating the percentage change in baseline marginal utility in response to increasing the value of exogenous variables by 10%, 25% and 50%, respectively.

An examination of factors affecting residential energy consumption using a multiple discrete continuous approach

Residential energy use has become an important source of global energy demand growth and carbon emissions growth. Residential building energy usage accounts for about 22% of the total energy use in the United States. In the current study, we address residential energy usage by addressing two decisions: (1) source of energy (such as electric and natural gas) and (2) consumption by energy source for various purposes. A Multiple Discrete Continuous Extreme Value model that allows us to analyze the source and consumption decisions in an integrated framework is developed. The model is estimated using data drawn from the 2015 Residential Energy Consumption Survey that provides energy use details for electricity, natural gas, fuel oil and Liquefied Petroleum Gas for residential units across United States. An exhaustive set of independent variables including location characteristics, household characteristics, housing characteristics, appliance use and climatic characteristics were employed in the model estimation. The model estimation results are augmented with a comprehensive policy analysis to illustrate how various independent variables affect energy use by source. A comparison of energy use between urban and rural regions, by varying household size and housing unit size are examined.