Multivariate Extreme Value Research Articles

Extreme rainfall prediction using spatial extreme value by Max Stable Process (MSP) Smith model approach

This research covers Spatial Extreme Value method application with Max-Stable Process (MSP) approach that will be used to analysis Extreme Rainfall in Semarang city. Extreme value sample are selected by Block Maxima methods, it will be estimated into Spatial Extreme Value form by including location factors. Then it transform to Frechet distribution because it has a Heavy tail pattern. Max Stable Process (MSP) is an extension of the multivariate extreme value distribution into infinite dimension of the Extreme Value Theory. MSP using Generalized Extreme Value (GEV) approach, it has three forms distribution that is Gumbel, Frechet and Weibull. Extreme Value Samples that are selected by Block Maxima will follow GEV distribution. The best model that obtained based on Max-Stable with Smith model can be used to predict Extreme Rainfall by count Return Level. Then can be further analysis of various aspects, so it can provide a suggestion about the potential of natural disasters which are caused by extreme rainfall. Extreme rainfall prediction using the Smith model in the return period of the next two years, at the Semarang City Climatology, Tanjung Mas and Ahmad Yani Station is predicted to be a maximum of 100.7539 mm, 100.1052 mm, and 109.9379 mm.

Practical Method to Simulate Multiple Discrete-Continuous Generalized Extreme Value Model: Application to Examine Substitution Patterns of Household Transportation Expenditures

The multiple discrete-continuous generalized extreme value (MDCGEV) model has been derived from multivariate extreme value (MEV)-based stochastic specifications to relax the independence assumption in the multiple discrete-continuous extreme value (MDCEV) model. It is analogous to the situation where a generalized extreme value (GEV) model relaxes the same assumption in a multinomial logit (MNL) model. However, unlike the case of single discrete choice model where substitution patterns can be understood based on elasticity expressions for a change in the value of an explanatory variable, the MDCEV and its variants do not offer closed-form elasticity expressions. The predictions must be compared explicitly under the base case and policy case scenarios. To perform a prediction exercise with MDCEV or its variants, random samples have to be drawn from the relevant stochastic distributions, which is actually not a straightforward task. In this paper, a practical method is proposed for drawing from an MEV distribution and the method is demonstrated to examine substitution patterns in an MDCGEV model for household transportation expenditures. The empirical results show that the cross-elasticities of explanatory variables in the MDCGEV model exhibit more variations than those in MDCEV and multiple discrete-continuous nested extreme value (MDCNEV) models.

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.

The tail dependograph

All characterizations of non-degenerate multivariate tail dependence structures are both functional and infinite-dimensional. Taking advantage of the Hoeffding–Sobol decomposition, we derive new indices to measure and summarize the strength of dependence in a multivariate extreme value analysis. The tail superset importance coefficients provide a pairwise ordering of the asymptotic dependence structure. We then define the tail dependograph, which visually ranks the extremal dependence between the components of the random vector of interest. For the purpose of inference, a rank-based statistic is derived and its asymptotic behavior is stated. These new concepts are illustrated with both theoretical models and real data, showing that our methodology performs well in practice.

Regression‐type models for extremal dependence

AbstractWe propose a vector generalized additive modeling framework for taking into account the effect of covariates on angular density functions in a multivariate extreme value context. The proposed methods are tailored for settings where the dependence between extreme values may change according to covariates. We devise a maximum penalized log‐likelihood estimator, discuss details of the estimation procedure, and derive its consistency and asymptotic normality. The simulation study suggests that the proposed methods perform well in a wealth of simulation scenarios by accurately recovering the true covariate‐adjusted angular density. Our empirical analysis reveals relevant dynamics of the dependence between extreme air temperatures in two alpine resorts during the winter season.

An evaluation of the consistency of extremes in gridded precipitation data sets

Noting a strong imperative to understand precipitation extremes, and that considerable uncertainty affects observational data sets, this paper compares the representation of extremes in a number of widely used daily gridded products, derived from rain gauge data, satellite retrieval and reanalysis for the conterminous United States. Analysis is based upon the concept of “tail dependence” arising in multivariate extreme value theory, and we infer the level of temporal dependence in the joint tail of the precipitation probability distribution for pairwise comparisons of products. In this way, we consider the range of products more like an ensemble and examine the relationships between members, and do not attempt to define, or compare products to, some ground truth. Linear correlation between products is also computed. Considerable discrepancy between groups of products, both annually and seasonally, is linked to source data and complex terrain. In particular, products based on rain gauge data showed remarkable similarity, but differed considerably, showing almost total loss of extremal dependence during DJF in mountainous regions, when compared with satellite products. Additionally, simulated re-forecasts revealed reasonable temporal agreement with large scale generated extremes. The diversity and extent of discrepancies identified across all products raises important questions about their use, and we urge caution, particularly for products derived from satellite data.

Is Bitcoin the New Digital Gold? Evidence From Extreme Price Movements in Financial Markets

Is bitcoin the new digital gold? To answer this question, we investigate the potential benefits of bitcoin during extremely volatile periods. We use the multivariate extreme value theory, which is the appropriate statistical approach to model the tail dependence structure of the return distribution. First, considering positions in equity markets, we find - similarly to previous studies - that the correlation of extreme returns increases during stock market crashes and decreases during stock market booms. Second, by combining each equity market with bitcoin, we find that the correlation of extreme returns sharply decreases during both market booms and crashes, indicating that bitcoin could provide the sought-after benefits of diversification during turbulent times. A similar result is obtained for gold, confirming its well-recognized status as a safe haven when a crisis occurs. Finally, we find a low extreme correlation between bitcoin and gold, which implies that both assets can be used together in times of turbulence in financial markets to protect equity positions. From a portfolio management perspective, we show that the introduction of bitcoin (along with gold) substantially improves the performance of equity positions under tail risk constraints. Such evidence indicates that bitcoin can be considered the new digital gold. However, gold itself can still play an important role in portfolio risk management.

Spatial Cluster Detection in Mobility Networks: A Copula Approach

Summary In mobility network capacity planning, characterizing the mobility network traffic is one of the most challenging tasks. Besides the growth trend and multiple periodic temporal patterns for normal traffic, the problem is complicated by the occasionally intense traffic for special events and its dynamic spatial relationships. Identifying the areas that have different traffic patterns compared with their neighbouring areas is a problem of spatial hotspot detection. In the paper, a copula-based method is proposed: using a multivariate extreme value copula, the upper tail dependence of the traffic distributions of neighbouring cell towers is evaluated, and then a cluster of multiple time series (i.e. multiple cell towers) with high upper tail dependence is detected. The method proposed is validated by using synthetic data as well as real mobility traffic data.

ABC model selection for spatial extremes models applied to South Australian maximum temperature data

Max-stable processes are a common choice for modelling spatial extreme data as they arise naturally as the infinite-dimensional generalisation of multivariate extreme value theory. Statistical inference for such models is complicated by the intractability of the multivariate density function. Nonparametric, composite likelihood-based, and Bayesian approaches have been proposed to address this difficulty. More recently, a simulation-based approach using approximate Bayesian computation (ABC) has been employed for estimating parameters of max-stable models. ABC algorithms rely on the evaluation of discrepancies between model simulations and the observed data rather than explicit evaluations of computationally expensive or intractable likelihood functions. The use of an ABC method to perform model selection for max-stable models is explored. Three max-stable models are regarded: the extremal-t model with either a Whittle–Matérn or a powered exponential covariance function, and the Brown–Resnick model with power variogram. In addition, the non-extremal Student-t copula model with a Whittle–Matérn or a powered exponential covariance function is also considered. The method is applied to annual maximum temperature data from 25 weather stations dispersed around South Australia.

Estimation of intensity–duration–frequency curves using max-stable processes

We present an approach to estimate intensity–duration–frequency (IDF) curves based on max-stable processes. The proposed method has been inspired by the seminal study of Nadarajah et al. (J R Stat Soc B 60(2):473–496, 1998), who used a multivariate extreme value distribution (MEVD) to estimate (IDF) curves from rainfall records. Max-stable processes are continuous extensions of MEVD (i.e. the marginal distributions of rainfall maxima at different durations are generalized extreme valued), which are more flexible, allow for extreme rainfall estimation at any arbitrary duration d (i.e. not just a discrete set, as is the case of MEVD), while preserving asymptotic dependencies. The latter characteristic of IDF estimates results from the combined effect of the statistical structure of rainfall (i.e. temporal dependencies), as well as the IDF construction process, which involves averaging of the original series to obtain rainfall maxima at different temporal resolutions. We apply the method to hourly precipitation data, and compare it to empirical estimates and the results produced by a semiparametric approach. Our findings indicate that max-stable processes fit well the statistical structure and inter-dependencies of annual rainfall maxima at different durations, produce slightly more conservative estimates relative to semiparametric methods, while allowing for extrapolations to durations and return periods beyond the range of the available data. The proposed statistical model is fully parametric and likelihood based, providing a theoretically consistent basis in solving the problem at hand.

Testing extreme dependence in financial time series

Financial interdependence indicates a process through which transmission of shock originating in the financial market of one economy spreads to others. This paper provides a new idea of Residual and Recurrence Times of high or low values for bivariate time series to detect extreme dependence or contagion. In presence of financial extreme dependence, the distributions of residual and recurrence times are not the same. We examine the equality of two distributions using the permutation test. In comparison to other methods in multivariate extreme value theory, our proposed method does not need the i.i.d. assumption. Our method can handle the situation where the extremes for different components do not occur at the same time. We justify our methods in two ways: first using thorough simulation studies and then applying the proposed method to real data on weekly stock indices from sixteen markets.

Extreme Correlation in Cryptocurrency Markets

In this paper, we study the contemporaneous tail dependence structure in a pairwise comparison of the ten largest cryptocurrencies, namely Bitcoin, Dash, Dogecoin, Ethereum, Litecoin, Monero, Namecoin, Novacoin, Peercoin, and Ripple. We apply multivariate extreme value theory and we estimate a bias-corrected extreme correlation coefficient. Our findings reveal clear patterns of significantly high bivariate dependency in the distribution tails of some of the most basic and widespread cryptocurrencies, primarily over various downside constraints. This means that extreme correlation is not related to cryptocurrency market volatility per se, but to the trend of the cryptocurrency market. Therefore, extreme correlation increases in bear markets, but not in bull markets for these pairs. Interestingly, there is also a significant number of pairs which exhibit a weak level of dependency in distribution tails.

Bivariate Extreme Value Statistics of Offshore Jacket Support Stresses in Bohai Bay

This paper presents a generic Monte Carlo-based approach for bivariate extreme response prediction for fixed offshore structures, particularly jacket type. The bivariate analysis of extremes is often poorly understood and generally not adequately considered in most practical measurements/situations; that is why it is important to utilize the recently developed bivariate average conditional exceedance rate (ACER) method. According to the current literature study, there is not yet a direct application of the bivariate ACER method to coupled offshore jacket stresses. This study aims at being first to apply bivariate ACER method to jacket critical stresses, aiming at contributing to safety and reliability studies for a wide class of fixed offshore structures. An operating jacket located in the Bohai bay was taken as an example to demonstrate the proposed methodology. Satellite measured global wave statistics was used to obtain realistic wave scatter diagram in the jacket location area. Second-order wave load effects were taken into account, while simulating jacket structural response. An accurate finite element ANSYS model was used to model jacket response dynamics, subject to nonlinear hydrodynamic wave and sea current loads. Offshore structure design values are often based on univariate statistical analysis, while actually multivariate statistics is more appropriate for modeling the whole structure. This paper studies extreme stresses that are simultaneously measured/simulated at two different jacket locations. Due to less than full correlation between stresses in different critical jacket locations, application of the multivariate (or at least bivariate) extreme value theory is of practical engineering interest.

A general approach to generate random variates for multivariate copulae

SummaryWe suggest two methods for simulating from a multivariate copula in an arbitrary dimension. Although our main emphasis in this paper is on multivariate extreme value distributions, the proposed methods can be applied to any copula. The basic idea is to approximate the (unknown) density of the copula by a distribution that has a piece‐wise constant (histogram) density. This is achieved by partitioning the support of a given copula C into a large number of hyper‐rectangles and using them to generate random variates from an approximation of the copula. We suggest two methods for finding this approximation which correspond to either finding hyper‐rectangles which have equal probability mass with respect to C, or determining a partition using hyper‐squares of equal volume and finding the corresponding probability mass of each hyper‐square. We also discuss how the generated random variates can be used as proposals in a Metropolis–Hastings algorithm, when C is an absolutely continuous distribution function, to generate a sequence of random variates from C. An implementation of the proposed methodologies is provided for the statistical computing and graphics environment R in our package called SimCop.

The efficacy of life insurance company general account equity asset allocations: a safety-first perspective using vine copulas

AbstractWe study the asset allocation decision of a life insurance company’s general account with respect to the possibility of large negative economic shocks and examine how this account is affected by policyholder investment decisions in the company’s separate account. This is accomplished using a performance metric that incorporates downside risk measured using univariate and multivariate extreme value distributions. Because of its well-known price volatility, diversification attributes, and significant weight in the combined general and separate accounts, our primary focus is the company’s equity investments. Although industry asset allocations have varied over the past two decades, we find that the actual allocations to equity in the general account are close to the allocation percentages suggested by our extreme value metrics and both are far below the maximum values indicated by the relevant regulatory bodies.

Measuring systemic risk across financial market infrastructures

We measure systemic risk in the network of financial market infrastructures (FMIs) as the probability that two or more FMIs have a large credit risk exposure to a common FMI participant. We construct indicators of credit risk exposures in three main Canadian FMIs and use multivariate extreme value methods to estimate this probability. We find large differences in the levels of systemic risk across participants. Conditional on the participant being distressed, we re-estimate these probabilities and find that some participants create large exposures to FMIs, resulting in a larger level of systemic risk than the rest of the participants. Our results suggest that an appropriate oversight of FMIs may benefit from an in-depth system-wide analysis, which may have useful implications for the macroprudential regulation of the financial system.

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 joint probability distribution model of directional extreme wind speeds based on the t-Copula function

The probabilistic information of directional extreme wind speeds is important for precisely estimating the design wind loads on structures. A new joint probability distribution model of directional extreme wind speeds is established based on observed wind-speed data using multivariate extreme value theory with the t-Copula function in the present study. At first, the theoretical deficiencies of the Gaussian-Copula and Gumbel-Copula models proposed by previous researchers for the joint probability distribution of directional extreme wind speeds are analysed. Then, the t-Copula model is adopted to solve this deficiency. Next, these three types of Copula models are discussed and evaluated with Spearman\'s rho, the parametric bootstrap test and the selection criteria based on the empirical Copula. Finally, the extreme wind speeds for a given return period are predicted by the t-Copula model with observed wind-speed records from several areas and the influence of dependence among directional extreme wind speeds on the predicted results is discussed.

Sparse representation of multivariate extremes with applications to anomaly detection

Capturing the dependence structure of multivariate extreme events is a major concern in many fields involving the management of risks stemming from multiple sources, e.g., portfolio monitoring, insurance, environmental risk management and anomaly detection. One convenient (nonparametric) characterization of extreme dependence in the framework of multivariate Extreme Value Theory (EVT) is the angular measure, which provides direct information about the probable “directions” of extremes, i.e., the relative contribution of each feature/coordinate of the largest observations. Modeling the angular measure in high-dimensional problems is a major challenge for the multivariate analysis of rare events. The present paper proposes a novel methodology aiming at exhibiting a particular kind of sparsity within the dependence structure of extremes. This is achieved by estimating the amount of mass spread by the angular measure on representative sets of directions corresponding to specific sub-cones of R + d . This dimension reduction technique paves the way towards scaling up existing multivariate EVT methods. Beyond a non-asymptotic study providing a theoretical validity framework for our method, we propose as a direct application a first anomaly detection algorithm based on multivariate EVT. This algorithm builds a sparse normal profile of extreme behaviors, to be confronted with new (possibly abnormal) extreme observations. Illustrative experimental results provide strong empirical evidence of the relevance of our approach.

Robust multivairiate extreme value at risk allocation

Recent research in economy, especially in finance, is using the assumption that the returns distribution is normal for determining optimal allocation for different models in finance optimization like Markowitz model. However this assumption is not already exact because the problem of the asymmetric distribution has a big impact. Our paper proposes a Multivariate Extreme Value at Risk (MEVaR) approach to find the optimal allocation of a portfolio based on the extreme value theory. A detailed procedure and implementation on two different portfolio (the first, for an emerging market as Morocco and the second on a Canadian portfolio of a very liquid market. Are given for demonstrating the consistency of the new approach? Results are compared with the Worst-Case-Value at Risk (WCVaR) proposed by El Ghaoui (2003), and Partitioned Value at Risk (PVaR) approach Joel Goh et al. (2011). They establish that the MEVaR performs well and is used in prediction model.