Multivariate Extreme Value Models Research Articles

The SPAR Model: A New Paradigm for Multivariate Extremes: Application to Joint Distributions of Metocean Variables

Abstract This paper presents the application of a new multivariate extreme value model for the estimation of metocean variables. The model requires fewer assumptions about the forms of the marginal distributions and dependence structure compared to existing approaches, and provides a flexible and rigorous framework for modeling multivariate extremes. The method involves a transformation of variables to polar coordinates. The tail of the radial variable is then modeled using the generalized Pareto distribution, with parameters conditional on angle, providing a natural extension of univariate theory to multivariate problems. The resulting model is referred to as the semi-parametric angular-radial (SPAR) model. We consider the estimation of the joint distributions of (1) wave height and wave period, and (2) wave height and wind speed. We show that the SPAR model provides a good fit to the observations in terms of both the marginal distributions and dependence structures. The use of the SPAR model for estimating long-term extreme responses of offshore structures is discussed, using some simple response functions for floating structures and an offshore wind turbine with monopile foundation. We show that the SPAR model is able to accurately reproduce response distributions, and provides a realistic quantification of uncertainty.

A note on statistical tests for homogeneities in multivariate extreme value models for block maxima

AbstractMathematical theory suggests to model annual or seasonal maxima by the generalized extreme value distribution. In environmental applications like hydrology, record lengths are typically small, whence respective parameter estimators typically exhibit a large variance. The variance may be decreased by pooling observations from different sites or variables, but this requires to check the validity of the inherent homogeneity assumption. The present paper provides an overview of (partly new) respective asymptotic significance tests. It is found that the tests' levels are often violated in typical finite‐sample situations, whence a parametric bootstrap approach based on max‐stable process models is proposed to obtain more accurate critical values. As a side product, we present an overview of asymptotic results on a variety of common estimators for GEV parameters in a multisample situation of varying record lengths.

Asymptotic Dependence Modelling of the BRICS Stock Markets

With the use of empirical data, this paper focuses on solving financial and investment issues involving extremal dependence of 10 pairwise combinations of the 5 BRICS (Brazil, Russia, India, China, and South Africa) stock markets. Daily closing equity indices from 5 January 2010 to 6 August 2018 are used in the study. Unlike previous literature, we use bivariate point process and conditional multivariate extreme value models to investigate the extremal dependence of the stock market returns. However, it is observed that the point process was able to model many more extreme observations or exceedances that contribute to the likelihood estimation. It gives more information than the threshold excess method of the CMEV model. This study shows varying levels of low extremal dependence structure whose outcomes are highly beneficial to investors, portfolio managers and other market participants interested in maximising investment returns and financial gains.

Higher-dimensional spatial extremes via single-site conditioning

Currently available models for spatial extremes suffer either from inflexibility in the dependence structures that they can capture, lack of scalability to high dimensions, or in most cases, both of these. We present an approach to spatial extreme value theory based on the conditional multivariate extreme value model, whereby the limit theory is formed through conditioning upon the value at a particular site being extreme. The ensuing methodology allows for a flexible class of dependence structures, as well as models that can be fitted in high dimensions. To overcome issues of conditioning on a single site, we suggest a joint inference scheme based on all observation locations, and implement an importance sampling algorithm to provide spatial realizations and estimates of quantities conditioning upon the process being extreme at any of one of an arbitrary set of locations. The modelling approach is applied to Australian summer temperature extremes, permitting assessment of the spatial extent of high temperature events over the continent.

Transferability of multivariate extreme value models for safety assessment by applying artificial intelligence-based video analytics

Traffic conflict techniques represent the state-of-the-art for road safety assessments. However, the lack of research on transferability of conflict-based crash risk models, which refers to applying the developed crash risk estimation models to a set of external sites, can reduce their appeal for large-scale traffic safety evaluations. Therefore, this study investigates the transferability of multivariate peak-over threshold models for estimating crash frequency-by-severity. In particular, the study proposes two transferability approaches: (i) an uncalibrated approach involving a direct application of the uncalibrated base model to the target sites and (ii) a threshold calibration approach involving calibration of conflict thresholds of the conflict indicators. In the latter approach, the conflict thresholds of the Modified Time-To-Collision (MTTC) and Delta-V indicators were calibrated using local data from the target sites. Finally, the two transferability approaches were compared with a complete re-estimation approach where all the model parameters were estimated using local data. All three approaches were tested for a target set of signalized intersections in Southeast Queensland, Australia. Traffic movements at the target intersections were observed using video cameras for two days (12 h each day). The road user trajectories and rear-end conflicts were extracted using an automated artificial intelligence-based algorithm utilizing state-of-the-art Computer Vision methods. The base models developed in an earlier study were then transferred to the target sites using the two transferability approaches and the local data from the target sites. Results show that the threshold calibration approach provides the most accurate and precise predictions of crash frequency-by-severity for target sites. Thus, for peak-over threshold models, the threshold parameter is the most important, and its calibration improves the performance of the base models. The complete re-estimation of models for individual target sites yields inferior fits and less precise crash estimates than the two transferability approaches since they utilize fewer traffic conflict extremes in their development than the larger dataset utilized in base model development. Therefore, the study results can significantly advance the applicability of traffic conflict models for crash risk estimation at transport facilities.

Modeling Concurrent Hydroclimatic Extremes With Parametric Multivariate Extreme Value Models

AbstractEstimating the dependence structure of concurrent extremes is a fundamental issue for accurate assessment of their occurrence probabilities. Identifying the extremal dependence behavior is also crucial for scientific understanding of interactions between the variables of a multidimensional environmental process. This study investigates the suitability of parametric multivariate extreme value models to correctly represent and estimate the dependence structure of concurrent extremes. Probabilistic aspects of multivariate extreme value theory with point process representation are discussed and illustrated with application to the concurrence of rainfall deficits, soil moisture deficits, and high temperatures. Application is concerned with the investigation of extremal behavior and risk assessment in Marathwada, a drought‐prone region of Maharashtra state, India. To characterize the multivariate extremes, marginal distributions are specified first and transformed into unit Fréchet margins. Standardized distributions are represented by a Poisson point process and coordinates of data points are transformed to pseudo‐polar coordinates to make the dependence form more explicit. The extremal dependence structure is described through angular densities on the unit simplex. Strong dependence is observed between soil moisture deficits and high temperatures, whereas rainfall deficits are mildly dependent on these two variables. Overall, a weak dependence is observed between the variables considered. Estimated extremal dependence is further used to compute probabilities of a few critical extreme combinations. Results demonstrate the ability of parametric multivariate models to characterize the complex dependence structure of concurrent extremes. These models can provide a powerful new perspective for appropriate statistical analysis of dependent hydroclimatic extremes in higher dimensions.

Ride to the hills, ride to your school: Physical effort and mode choice

• New insights into trade-offs between energy expenditure and other variables. • Improved understanding on substitution between means of transport. • Analysis of elasticities of energy input and social welfare. • Economic model of planning policies for commute-to-school. Altering the network of public facilities like schools is likely to affect travel choices, in particular route and mode choice. Traveling to school is not only a major contributor to public transport demand in peak periods, but is in many instances also the trip purpose where young people are engaged in bicycling. However, in contrast to public transport, an active travel mode like bicycling needs (substantial) own physical activity as input into the ‘production’ of a trip. The objective of this paper is to improve our understanding of the substitution between bicycling and public transport in school travel focusing on an underrated determinant in the literature – the personal effort (e.g. energy input in terms of kcal) of students when traveling by bike (or walk). It is shown that a local government aiming at maximizing social welfare when making decisions on the density of public school facilities would be well advised to take this cross effect into account. We use a large unique data set of travel-to-school mode choice in the city of Dresden (Germany). We estimate a series of multivariate extreme value models and derive the elasticity of bike usage and the cross-elasticity of public transport demand with respect to the physical effort. The results reveal that effort elasticity of bike usage is significantly negative, whereas the effort cross-elasticity of public transport demand is significantly positive. In both cases, the responses are highest for school travel distances between 1 and 3 km, but relatively inelastic, meaning that a widespread adoption of pedelecs (or even e-bikes) in school travel could have only limited impacts on peak-period public transport demand.

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.

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.

Ride to the Hills, Ride to Your School: Physical Effort and Mode Choice

Altering the network of public facilities like schools is likely to affect travel choices, in particular route and mode choice. Traveling to school is not only a major contributor to public transport demand in peak periods, but is in many instances also the trip purpose where young people are engaged in bicycling. However, in contrast to public transport, an active travel mode like bicycling needs (substantial) own physical activity as input into the ‘production’ of a trip. The objective of this paper is to improve our understanding of the substitution between bicycling and public transport in school travel focusing on an underrated determinant in the literature – the personal effort (e.g. energy input in terms of kcal) of students when traveling by bike (or walk). It is shown that a local government aiming at maximizing social welfare when making decisions on the density of public school facilities would be well advised to take this cross effect into account. We use a large unique data set of travel-to-school mode choice in the city of Dresden (Germany). We estimate a series of multivariate extreme value models and derive the elasticity of bike usage and the cross-elasticity of public transport demand with respect to the physical effort. The results reveal that effort elasticity of bike usage is significantly negative, whereas the effort cross- elasticity of public transport demand is significantly positive. In both cases, the responses are highest for school travel distances between 1 and 3 kilometers, but relatively inelastic, meaning that a widespread adoption of pedelecs (or even e-bikes) in school travel could have only limited impacts on peak-period public transport demand.

Multivariate Bayesian hierarchical modeling of the non-stationary traffic conflict extremes for crash estimation

Recent studies have developed univariate and bivariate Bayesian hierarchical extreme value models to improve traffic conflict-based crash estimation through addressing the issues of non-stationarity of conflict extremes and individual conflict indicators representing only partial severity aspects of a traffic event. Although the use of bivariate models showed considerable improvements compared to univariate models, two traffic conflict indicators may not be sufficient to cover all severity aspects of a traffic event. Therefore, a high dimensional multivariate Bayesian hierarchical extreme value model based on more than two conflict indicators merits further research. This study develops a multivariate Bayesian hierarchical extreme value modeling approach, which comprises a multivariate extreme value model and a Bayesian hierarchical structure. The former integrates several traffic conflict indicators in a unified framework, while the latter combines traffic conflicts from different sites, incorporating several covariates and site-specific unobserved heterogeneity. A model estimation approach for the multivariate Bayesian hierarchical extreme value model is proposed and applied to estimate rear-end crashes from four signalized intersections in the city of Surrey, British Columbia. The modified time to collision (MTTC), post encroachment time (PET), and deceleration rate to avoid a crash (DRAC) were employed as conflict indicators. Three covariates including traffic volume, shock wave area, and platoon ratio were considered to account for non-stationarity in conflict extremes. The results show that in terms of crash estimation accuracy, the high dimensional multivariate (e.g., trivariate) Bayesian hierarchical extreme value model outperforms both the low dimensional multivariate (e.g., bivariate) and univariate Bayesian hierarchical extreme value models. This is likely due to the multivariate model having the capability of sharing information across several different conflict indicators.

Estimation of extreme wind pressure coefficient in a zone by multivariate extreme value theory

Knowledge on the design value of extreme wind pressure coefficients (EWPC) of a specific zone of buildings is essential for the wind-resistant capacity of claddings. This paper presents a method to estimate the representative EWPC introducing the multivariate extreme value model. The spatial correlations of the extreme wind pressures at different locations can be consider through the multivariate extreme value. The moving average method is also adopted in this method, so that the measured point pressure can be converted to wind pressure of an area. The proposed method is applied to wind tunnel test results of a large flat roof building. Comparison with existing methods shows that it can give a good estimation for all target zones with different sizes.

Causal Mechanism of Extreme River Discharges in the Upper Danube Basin Network

SummaryExtreme hydrological events in the Danube river basin may severely impact human populations, aquatic organisms and economic activity. One often characterizes the joint structure of extreme events by using the theory of multivariate and spatial extremes and its asymptotically justified models. There is interest, however, in cascading extreme events and whether one event causes another. We argue that an improved understanding of the mechanism underlying severe events is achieved by combining extreme value modelling and causal discovery. We construct a causal inference method relying on the notion of the Kolmogorov complexity of extreme conditional quantiles. Tail quantities are derived by using multivariate extreme value models, and causal-induced asymmetries in the data are explored through the minimum description length principle. Our method CausEV for causality for extreme values uncovers causal relationships between summer extreme river discharges in the upper Danube basin and finds significant causal links between the Danube and its Alpine tributary Lech.

An Algorithm for Assortment Optimization Under Parametric Discrete Choice Models

This work concerns the assortment optimization problem that refers to selecting a subset of items that maximizes the expected revenue in the presence of the substitution behavior of consumers specified by a parametric choice model. The key challenge lies in the computational difficulty of finding the best subset solution, which often requires exhaustive search. The literature on constrained assortment optimization lacks a practically efficient method which that is general to deal with different types of parametric choice models (e.g., the multinomial logit, mixed logit or general multivariate extreme value models). In this paper, we propose a new approach that allows to address this issue. The idea is that, under a general parametric choice model, we formulate the problem into a binary nonlinear programming model, and use an iterative algorithm to find a binary solution. At each iteration, we propose a way to approximate the objective (expected revenue) by a linear function, and a polynomial-time algorithm to find a candidate solution using this approximate function. We also develop a greedy local search algorithm to further improve the solutions. We test our algorithm on instances of different sizes under various parametric choice model structures and show that our algorithm dominates existing exact and heuristic approaches in the literature, in terms of solution quality and computing cost.

Estimation of Analog/RF Parametric Test Metrics Based on a Multivariate Extreme Value Model

Analog/RF built-in test (BIT) techniques are essential for reducing the very high costs of specification-based tests and for high-safety applications. The adoption of a BIT technique needs to be decided at the design stage, and this can be facilitated by estimating the test quality in terms of errors such as test escapes ( ${T_{E}}$ ) and yield loss ( ${Y_{L}}$ ). Test quality estimation at the design stage has been traditionally very difficult for analog/RF circuits due to the lack of fault models that properly cover parametric faulty behavior. In recent years, statistical simulation has been considered in combination with learning techniques for the estimation of parametric test metrics. Extreme value theory (EVT) has provided a rigorous tool for the computation of parametric test metrics. However, test metrics estimation has been limited to the use of a univariate model. In this paper, we extend this approach by using a multivariate extreme value model. We illustrate this for the evaluation of an RF LNA BIT technique using a bivariate model.

Multidimensional extremal dependence coefficients

Extreme value modeling has been attracting the attention of researchers in diverse areas such as the environment, engineering, and finance. Multivariate extreme value distributions are particularly suitable to model the tails of multidimensional phenomena. The analysis of the dependence among multivariate maxima is useful to evaluate risk. Here we present new multivariate extreme value models, as well as, coefficients to assess multivariate extremal dependence.

A dynamic programming approach for quickly estimating large network-based MEV models

We propose a way to estimate a family of static Multivariate Extreme Value (MEV) models with large choice sets in short computational time. The resulting model is also straightforward and fast to use for prediction. Following Daly and Bierlaire (2006), the correlation structure is defined by a rooted, directed graph where each node without successor is an alternative. We formulate a family of MEV models as dynamic discrete choice models on graphs of correlation structures and show that the dynamic models are consistent with MEV theory and generalize the network MEV model (Daly and Bierlaire, 2006). Moreover, we show that these models can be estimated quickly using the concept of network flows and the nested fixed point algorithm (Rust, 1987). The main reason for the short computational time is that the new formulation allows to benefit from existing efficient solution algorithms for sparse linear systems of equations.We present numerical results based on simulated data with varying number of alternatives and nesting structures. We estimate large models, for example, a cross-nested model with 200 nests and 500,000 alternatives and 210 parameters that needs between 100–200 iterations to converge (4.3 h on an Intel(R) 3.2 GHz machine using a non-parallelized code). We also show that our approach allows to estimate a cross-nested logit model of 111 nests with a real data set of more than 100,000 observations in 14 h.

A method of integrating correlation structures for a generalized recursive route choice model

We propose a way to estimate a generalized recursive route choice model. The model generalizes other existing recursive models in the literature, i.e., (Fosgerau et al., 2013b; Mai et al., 2015c), while being more flexible since it allows the choice at each stage to be any member of the network multivariate extreme value (network MEV) model (Daly and Bierlaire, 2006). The estimation of the generalized model requires defining a contraction mapping and performing contraction iterations to solve the Bellman’s equation. Given the fact that the contraction mapping is defined based on the choice probability generating functions (CPGF) (Fosgerau et al., 2013b) generated by the network MEV models, and these CPGFs are complicated, the generalized model becomes difficult to estimate. We deal with this challenge by proposing a novel method where the network of correlation structures and the structure parameters given by the network MEV models are integrated into the transport network. The approach allows to simplify the contraction mapping and to make the estimation practical on real data.We apply the new method on real data by proposing a recursive cross-nested logit (RCNL) model, a member of the generalized model, where the choice model at each stage is a cross-nested logit. We report estimation results and a prediction study based on a real network. The results show that the RCNL model performs significantly better than the other recursive models in fit and prediction.

Sampling of Alternatives in Random Regret Minimization Models

Sampling of alternatives is often required in discrete choice models to reduce the computational burden and to avoid describing a large number of attributes. This approach has been used in many areas, including modeling of route choice, vehicle ownership, trip destination, residential location, and activity scheduling. The need for sampling of alternatives is accentuated for random regret minimization (RRM) models because, unlike random utility models, the regret function for each alternative depends on all of the alternatives in the choice-set. In this paper we develop and test a method to achieve consistency, asymptotic normality, and relative efficiency of the estimators while sampling alternatives in a class of models that includes RRM. The proposed method can be seen as an extension of the approach used to address sampling of alternatives in multivariate extreme value models. We illustrate the methodology using Monte Carlo experimentation and a case study with real data. Experiments show that the proposed method is practical, performs better than a truncated model, and results in finite-sample estimates that provide a good approximation of those obtained with a model considering all of the alternatives.

National scale multivariate extreme value modelling of waves, winds and sea levels

It has long been recognised that extreme coastal flooding can arise from the joint occurrence of extreme waves, winds and sea levels. There has been significant development in statistical methods that are capable of appropriately accounting for the dependence between the variables. The standard simplified approach used in England and Wales can result in an underestimation of flood risk unless correction factors are applied. This paper describes the application of a state-of-the-art multivariate extreme value model, Heffernan and Tawn (2004), to offshore winds, waves and sea levels around the coast of England. The methodology overcomes the limitations of the standard method. The output of the statistical analysis is a Monte-Carlo simulation of offshore extreme events and it is necessary to translate these into peak overtopping rates for use as input to flood risk assessments. Due to non-linearities inherent in wave transformation and overtopping processes, it is not possible to identify the events which give the most extreme overtopping rates; indeed the same overtopping rate can be achieved from a wide range of offshore wave, wind and water level combinations. It was computationally impractical to transform all of these MC events from the offshore to the nearshore and through to wave overtopping rates using numerical models. A series of emulators of the SWAN wave transformation model have therefore been constructed. The emulators translate the offshore extreme events through to the nearshore in a computationally efficient manner. The outcome of the process is a national set of extreme events at a 1km resolution in the nearshore region. These events can subsquently be processed into overtopping rates for coastal flood risk assessment. The SWAN 1D model has been used for the nearshore transformation and a neural network model, BAYONET, used to calculate wave overtopping rates. Whilst the methodology has been applied for national flood risk assessment, it has the potential to be implemented for wider use, including climate change impact assessment, nearshore wave climates for detailed local assessments and coastal flood forecasting. This paper explains the limitations of the existing approach and shows outputs from the new methods, demonstrating the improvement in the results that are possible.