Using extreme value theory to measure value-at-risk for daily electricity spot prices

The recent deregulation in electricity markets worldwide has heightened the importance of risk management in energy markets. Assessing Value-at-Risk (VaR) in electricity markets is arguably more difficult than in traditional financial markets because the distinctive features of the former result in a highly unusual distribution of returns-electricity returns are highly volatile, display seasonalities in both their mean and volatility, exhibit leverage effects and clustering in volatility, and feature extreme levels of skewness and kurtosis. With electricity applications in mind, this paper proposes a model that accommodates autoregression and weekly seasonals in both the conditional mean and conditional volatility of returns, as well as leverage effects via an EGARCH specification. In addition, extreme value theory (EVT) is adopted to explicitly model the tails of the return distribution. Compared to a number of other parametric models and simple historical simulation based approaches, the proposed EVT-based model performs well in forecasting out-of-sample VaR. In addition, statistical tests show that the proposed model provides appropriate interval coverage in both unconditional and, more importantly, conditional contexts. Overall, the results are encouraging in suggesting that the proposed EVT-based model is a useful technique in forecasting VaR in electricity markets. (c) 2005 International Institute of Forecasters. Published by Elsevier B.V. All rights reserved.

Trimmed Likelihood Estimation of the Parameters of the Generalized Extreme Value Distributions: a Monte-Carlo Study

2000 Mathematics Subject Classification: Primary 62F35; Secondary 62P99

Extreme value statistics in Raman fiber lasers

We present the numerical study of the statistical properties of the partially coherent quasi-CW high-Q cavity Raman fiber laser. The statistical properties are different for the radiation generated at the spectrum center or spectral wings. It is found that rare extreme events are generated at the far spectral wings at one pass only. The mechanism of the extreme events generation is a turbulent-like four-wave mixing of numerous longitudinal generation modes. The similar mechanism of extreme waves appearance during the laser generation could be important in other types of fiber lasers. © 2012 Copyright Society of Photo-Optical Instrumentation Engineers (SPIE).

Extreme value statistics in quasi-CW Raman fiber lasers

It is found that rare extreme events are generated in a Raman fiber laser. The mechanism of the extreme events generation is a turbulent-like four-wave mixing of numerous longitudinal generation modes. © 2012 OSA.

Extreme value statistics in Raman fiber lasers

We present the numerical study of the statistical properties of the partially coherent quasi-CW high-Q cavity Raman fiber laser. The statistical properties are different for the radiation generated at one or many cavity passes. It is found that rare extreme events are generated at the far spectral wings of the spectrum. The mechanism of the extreme events generation is a turbulent-like four-wave mixing of numerous longitudinal generation modes. © 2011 Optical Society of America.

The transformed-stationary approach: a generic and simplified methodology for non-stationary extreme value analysis

Statistical approaches to study extreme events require, by definition, long time series of data. In many scientific disciplines, these series are often subject to variations at different temporal scales that affect the frequency and intensity of their extremes. Therefore, the assumption of stationarity is violated and alternative methods to conventional stationary extreme value analysis (EVA) must be adopted. Using the example of environmental variables subject to climate change, in this study we introduce the transformed-stationary (TS) methodology for non-stationary EVA. This approach consists of (i) transforming a non-stationary time series into a stationary one, to which the stationary EVA theory can be applied, and (ii) reverse transforming the result into a non-stationary extreme value distribution. As a transformation, we propose and discuss a simple time-varying normalization of the signal and show that it enables a comprehensive formulation of non-stationary generalized extreme value (GEV) and generalized Pareto distribution (GPD) models with a constant shape parameter. A validation of the methodology is carried out on time series of significant wave height, residual water level, and river discharge, which show varying degrees of long-term and seasonal variability. The results from the proposed approach are comparable with the results from (a) a stationary EVA on quasi-stationary slices of non-stationary series and (b) the established method for non-stationary EVA. However, the proposed technique comes with advantages in both cases. For example, in contrast to (a), the proposed technique uses the whole time horizon of the series for the estimation of the extremes, allowing for a more accurate estimation of large return levels. Furthermore, with respect to (b), it decouples the detection of non-stationary patterns from the fitting of the extreme value distribution. As a result, the steps of the analysis are simplified and intermediate diagnostics are possible. In particular, the transformation can be carried out by means of simple statistical techniques such as low-pass filters based on the running mean and the standard deviation, and the fitting procedure is a stationary one with a few degrees of freedom and is easy to implement and control. An open-source MAT-LAB toolbox has been developed to cover this methodology, which is available at https://github.com/menta78/tsEva/(Mentaschi et al., 2016).

Inundações urbanas: desafios ao ordenamento do território. O caso da Cidade da Praia (Cabo Verde)

Pela sua posição geográfica e particulares condições climáticas, devidas à sua inserção na faixa saheliana com caraterísticas de marcada aridez, Cabo Verde é um arquipélago com condições naturais adversas, pautado principalmente pela seca prolongada. Muitas vezes esta conjuntura é interpolada por curtos períodos de fortes chuvadas que podem originar cheias e inundações nos principais centros urbanos. Os eventos ocorridos revelam consequências graves, desde prejuízos na agricultura, perda de animais, destruição de infra-estruturas, perda de bens materiais e, mesmo, vítimas humanas mortais. Este estudo tem como objetivos principais: (i) perceber os problemas e desafios que se colocam à cidade da Praia (maior centro urbano do país, com forte crescimento e expansão urbana) perante situações de inundação; (ii) contribuir para o maior conhecimento das causas e consequências dessas inundações; (iii) definir quais as áreas de maior suscetibilidade às cheias e quais as que possuem um maior risco potencial. Optou-se por uma metodologia integrada, através do levantamento bibliográfico, cartográfico, numérico e percetivo (com base em entrevistas e inquéritos). Para o estudo das bacias hidrográficas foram calculados índices morfométricos, definidas classes de permeabilidade do substrato geológico e aplicado o método multicritério de Reis (2011) para a definição das áreas suscetíveis às cheias. Analisaram-se as precipitações máximas diárias anuais e respetivos períodos de retorno, com a aplicação do método de Gumbel. A análise de notícias de jornais, referentes ao período compreendido entre 1980 e 2011, foi fundamental para o conhecimento da distribuição espácio-temporal dos eventos perigosos de inundação em Cabo Verde e na cidade da Praia. Os resultados obtidos revelam um significativo grau de suscetibilidade às cheias na cidade da Praia. As áreas de maior risco potencial às inundações encontram-se no setor central da cidade, resultante da conjugação da convergência do escoamento das três ribeiras principais, da elevada densidade populacional e de construção desordenada nos leitos de cheia e nas áreas deprimidas, onde se acumulam as águas.

Inundações Urbanas - Desafios ao Ordenamento do Território - O

Pela sua posição geográfica e particulares condições climáticas, devidas à sua inserção na faixa saheliana com caraterísticas de marcada aridez, Cabo Verde é um arquipélago com condições naturais adversas, pautado principalmente pela seca prolongada. Muitas vezes esta conjuntura é interpolada por curtos períodos de fortes chuvadas que podem originar cheias e inundações nos principais centros urbanos. Os eventos ocorridos revelam consequências graves, desde prejuízos na agricultura, perda de animais, destruição de infra-estruturas, perda de bens materiais e, mesmo, vítimas humanas mortais. Este estudo tem como objetivos principais: (i) perceber os problemas e desafios que se colocam à cidade da Praia (maior centro urbano do país, com forte crescimento e expansão urbana) perante situações de inundação; (ii) contribuir para o maior conhecimento das causas e consequências dessas inundações; (iii) definir quais as áreas de maior suscetibilidade às cheias e quais as que possuem um maior risco potencial. Optou-se por uma metodologia integrada, através do levantamento bibliográfico, cartográfico, numérico e percetivo (com base em entrevistas e inquéritos). Para o estudo das bacias hidrográficas foram calculados índices morfométricos, definidas classes de permeabilidade do substrato geológico e aplicado o método multicritério de Reis (2011) para a definição das áreas suscetíveis às cheias. Analisaram-se as precipitações máximas diárias anuais e respetivos períodos de retorno, com a aplicação do método de Gumbel. A análise de notícias de jornais, referentes ao período compreendido entre 1980 e 2011, foi fundamental para o conhecimento da distribuição espácio-temporal dos eventos perigosos de inundação em Cabo Verde e na cidade da Praia. Os resultados obtidos revelam um significativo grau de suscetibilidade às cheias na cidade da Praia. As áreas de maior risco potencial às inundações encontram-se no setor central da cidade, resultante da conjugação da convergência do escoamento das três ribeiras principais, da elevada densidade populacional e de construção desordenada nos leitos de cheia e nas áreas deprimidas, onde se acumulam as águas.

Statistical modelling of extreme rainfall in Madeira Island

Extreme rainfall events have triggered a significant number of flash floods in Madeira Island along its past and recent history. Madeira is a volcanic island where the spatial rainfall distribution is strongly affected by its rugged topography. In this thesis, annual maximum of daily rainfall data from 25 rain gauge stations located in Madeira Island were modelled by the generalised extreme value distribution. Also, the hypothesis of a Gumbel distribution was tested by two methods and the existence of a linear trend in both distributions parameters was analysed. Estimates for the 50– and 100–year return levels were also obtained. Still in an univariate context, the assumption that a distribution function belongs to the domain of attraction of an extreme value distribution for monthly maximum rainfall data was tested for the rainy season. The available data was then analysed in order to find the most suitable domain of attraction for the sampled distribution. In a different approach, a search for thresholds was also performed for daily rainfall values through a graphical analysis. In a multivariate context, a study was made on the dependence between extreme rainfall values from the considered stations based on Kendall’s τ measure. This study suggests the influence of factors such as altitude, slope orientation, distance between stations and their proximity of the sea on the spatial distribution of extreme rainfall. Groups of three pairwise associated stations were also obtained and an adjustment was made to a family of extreme value copulas involving the Marshall–Olkin family, whose parameters can be written as a function of Kendall’s τ association measures of the obtained pairs.

Probabilistic structure of an annual extreme rainfall series of a coastal area of the State of São Paulo, Brazil

This study aimed to describe the probabilistic structure of the annual series of extreme daily rainfall (Preabs), available from the weather station of Ubatuba, State of São Paulo, Brazil (1935-2009), by using the general distribution of extreme value (GEV). The autocorrelation function, the Mann-Kendall test, and the wavelet analysis were used in order to evaluate the presence of serial correlations, trends, and periodical components. Considering the results obtained using these three statistical methods, it was possible to assume the hypothesis that this temporal series is free from persistence, trends, and periodicals components. Based on quantitative and qualitative adhesion tests, it was found that the GEV may be used in order to quantify the probabilities of the Preabs data. The best results of GEV were obtained when the parameters of this function were estimated using the method of maximum likelihood. The method of L-moments has also shown satisfactory results.

Uncertainties In Future Projections Of Extreme Rainfall: The Role Of Climate Model, Emission Scenario And Randomness

Climate change has resulted in substantial variations in annual extreme rainfall quantiles in different durations and return periods. Predicting the future changes in extreme rainfall quantiles is essential for various water resources design, assessment, and decision making purposes. Current Predictions of future rainfall extremes, however, exhibit large uncertainties. According to extreme value theory, rainfall extremes are rather random variables, with changing distributions around different return periods; therefore there are uncertainties even under current climate conditions. Regarding future condition, our large-scale knowledge is obtained using global climate models, forced with certain emission scenarios. There are widely known deficiencies with climate models, particularly with respect to precipitation projections. There is also recognition of the limitations of emission scenarios in representing the future global change. Apart from these large-scale uncertainties, the downscaling methods also add uncertainty into estimates of future extreme rainfall when they convert the larger-scale projections into local scale. The aim of this research is to address these uncertainties in future projections of extreme rainfall of different durations and return periods. We plugged 3 emission scenarios with 2 global climate models and used LARS-WG, a well-known weather generator, to stochastically downscale daily climate models’ projections for the city of Saskatoon, Canada, by 2100. The downscaled projections were further disaggregated into hourly resolution using our new stochastic and non-parametric rainfall disaggregator. The extreme rainfall quantiles can be consequently identified for different durations (1-hour, 2-hour, 4-hour, 6-hour, 12-hour, 18-hour and 24-hour) and return periods (2-year, 10-year, 25-year, 50-year, 100-year) using Generalized Extreme Value (GEV) distribution. By providing multiple realizations of future rainfall, we attempt to measure the extent of total predictive uncertainty, which is contributed by climate models, emission scenarios, and downscaling/disaggregation procedures. The results show different proportions of these contributors in different durations and return periods.