Properties of extremal dependence models built on bivariate max-linearity

Bivariate max-linear models provide a core building block for characterizing bivariate max-stable distributions. The limiting distribution of marginally normalized component-wise maxima of bivariate max-linear models can be dependent (asymptotically dependent) or independent (asymptotically independent). However, for modeling bivariate extremes they have weaknesses in that they are exactly max-stable with no penultimate form of convergence to asymptotic dependence, and asymptotic independence arises if and only if the bivariate max-linear model is independent. In this work we present more realistic structures for describing bivariate extremes. We show that these models are built on bivariate max-linearity but are much more general. In particular, we present models that are dependent but asymptotically independent and others that are asymptotically dependent but have penultimate forms. We characterize the limiting behavior of these models using two new different angular measures in a radial-angular representation that reveal more structure than existing measures

Assessing and modelling extremal dependence in spatial extremes

Offshore structures, such as oil platforms and vessels, must be built such that they can withstand extreme environmental conditions (e.g., high waves and strong winds) that may occur during their lifetime. This means that it is essential to quantify probabilities of the occurrence of such extreme events. However, a difficulty arises in that there are very limited data available at these levels. The statistical field of extreme value theory provides asymptotically motivated models for extreme events, hence allowing extrapolation to very rare events. In addition to the risk to a single site, we are also interested in the joint risk of multiple offshore platforms being affected by the same extreme event. In order to understand joint extremal behaviour for two or more locations, the spatial dependence between the different locations must be considered. Extremal dependence between two locations can be of two types: asymptotic independence (AI) when the extremes at the two sites are unlikely to occur together, and asymptotic dependence (AD) when it is possible for both sites to be affected simultaneously. For finite samples it is often difficult to determine which type of dependence the data are more consistent with. In a large ocean basin it is reasonable to expect both of these features to be present, with some close by locations AD, with the dependence decreasing with distance, and some far apart locations AI. In this thesis we develop new diagnostic tools for distinguishing between AD and AI and illustrate these on North Sea wave height data. We also investigate how extremal dependence changes with direction and find evidence for spatial anisotropy in our data set. The most widely used spatial models assume asymptotic dependence or perfect independence between sites, which is often unrealistic in practice. Models that attempt to capture both AD and AI exist, but they are difficult to implement in practice due to their complexity and they are restricted in the forms of AD and AI they can model. In this thesis we introduce a family of bivariate distributions that exhibits all the required features of short, medium and long range extremal dependence required for pairwise dependence modelling in spatial applications

The Mathematics and Statistics of Quantitative Risk Management

It was the aim of this workshop to gather a multidisclipinary and international group of scientists at the forefront of research in econometrics, financial time series analysis, extreme value theory, financial mathematics, insurance mathematics and quantitative risk management. The heterogeneous composition of this group of researchers allowed one to discuss different facets of the mathematics and statistics of quantitative risk management, to communicate the state-of-the-art in the different areas, and to point towards new directions of research

Stochastic physical models for wind fields and precipitation extremes

A major goal of this thesis is to introduce stochastic, physically consistent models for precipitation extremes based on the moisture budget. The moisture budget describes the moisture flux convergence and is essential for the generation of precipitation and in particular extreme precipitation. The introduced models are used to extensively study to which extent the budget equation can account for characteristics of precipitation extremes. An important question in this respect is under which conditions the budget equation generates a heavy-tailed behavior. A further point is to understand whether the spatial structure of the humidity transport is essential in generating precipitation extremes. It is demonstrated that the humidity budget equation does not allow for the emergence of heavy-tailed precipitation distributions from light-tailed distribution for wind and humidity. At the same time finite sample approximations of the models suggest that asymptotic properties may be of very limited practical relevance. The models considered here show a remarkable stability to the correlation of wind and humidity. We prove the convergence of a precipitation model to its max-stable limit, which yields asymptotic spatial independence of precipitation extremes. Further, there is no prominent difference between precipitation extremes in purely rotational or purely divergent flow. The budget equation reveals a strong sensitivity to the marginal distributions of wind and humidity and further assumptions, which shows the need for well-established distributional assumptions for these variables. In order to model moisture flux convergence spatially consistent a multivariate Gaussian random field formulation is introduced. It represents the differential relations of a wind field and related variables such as the streamfunction, velocity potential, vorticity, and divergence. The covariance model of the Gaussian random field is based on a flexible bivariate Mat´ern covariance function for the streamfunction and velocity potential. It allows for different variances in the potentials, nonzero correlations between them, anisotropy, and a flexible smoothness parameter. The joint covariance function of the related variables is derived analytically. Further, it is shown that a consistent model with nonzero correlations between the potentials and positive definite covariance function is possible, rebutting a claim of Obukhov (1954). The statistical model is fitted to forecasts of the horizontal wind fields of a mesoscale numerical weather prediction system. Parameter uncertainty is assessed by a parametric bootstrap method. The estimates reveal only physically negligible correlations between the potentials. The covariance model provides opportunity for a wealth of applications in data assimilation

A geometric investigation into the tail dependence of vine copulas

Vine copulas are a type of multivariate dependence model, composed of a collection of bivariate copulas that are combined according to a specific underlying graphical structure. Their flexibility and practicality in moderate and high dimensions have contributed to the popularity of vine copulas, but relatively little attention has been paid to their extremal properties. To address this issue, we present results on the tail dependence properties of some of the most widely studied vine copula classes. We focus our study on the coefficient of tail dependence and the asymptotic shape of the sample cloud, which we calculate using the geometric approach of Nolde (2014). We offer new insights by presenting results for trivariate vine copulas constructed from asymptotically dependent and asymptotically independent bivariate copulas, focusing on bivariate extreme value and inverted extreme value copulas, with additional detail provided for logistic and inverted logistic examples. We also present new theory for a class of higher dimensional vine copulas, constructed from bivariate inverted extreme value copulas

Extreme Rainfall Events: Incorporating Temporal and Spatial Dependence to Improve Statistical Models

The proper design of protective measurements against floods related to heavy precipitation has long been a question of interest in many fields of study. A crucial component for such design is the analysis of extreme historical rainfall using Extreme Value Theory (EVT) methods, which provide information about the frequency and magnitude of possible future events. Characterizing an entire basin or geographical catchment requires the extension of univariate EVT methods to capture the spatial variability of the data. This extension requires that the similarity of the data for nearby stations be included in the model, resulting in more efficient use of the data. This dissertation focuses on using statistical models incorporating spatial dependence for modeling annual rainfall maxima. Additionally, we present ways of adapting the models to capture the dependence between rainfall of different time scales. These models are used in order to pursue two aims. The first aim is to improve our understanding of the mechanisms that lead to dependence on extreme rainfall. The second aim is to improve the resulting estimates when incorporating the dependence into the models. Two published studies make up the main findings of this dissertation. The models used in both studies involve the use of Brown-Resnick max-stable processes, allowing the models to explicitly account for the dependence on either the temporal or the spatial domain. These conditional models are compared for both cases to a model that ignores the dependence, allowing us to determine the impact of the dependence in both situations. Contributions to three other studies using the concept of dependence are also summarized. In the first study, we assess the impact of including the dependence between rainfall series of different aggregation durations when estimating Intensity-Duration-Frequency curves. This assessment was done in a case study for the Wupper catchment in Germany. This study found that including the dependence in the model had a positive effect on the prediction accuracy when focusing on rainfall with short durations (d <= 10h) and large probabilities of non-exceedance. Therefore, we recommend using max-stable processes when a study focuses on short-duration rainfall. In the second study, we investigate how the spatial dependence of extreme rainfall in Berlin-Brandenburg changes seasonally and how this change could impact the estimates from a max-stable model that includes this dependence. The seasonality was determined by estimating the parameters of a summer and winter semi-annual block maxima model. The results from this study showed that, for the summer maxima, the dependence structure was adequately captured by an isotropic Brown-Resnick model. On the contrary, the same model performed poorly for the winter maxima, suggesting that a change in the assumptions is needed when dealing with typical winter events, typically frontal or stratiform for this region. These results show that accounting for the meteorological properties of the rainfall-generating processes can provide useful information for the design of the models. Overall, our findings show that including meteorological knowledge in statistical models can improve their resulting estimations. In particular, we find that, under certain conditions, using statistical dependence to incorporate knowledge about the differences in temporal and spatial scales of rainfall-generating mechanisms can lead to a positive impact in the models.Die richtige Auslegung von Schutzmaßnahmen gegen Überschwemmungen im Zusammenhang mit Starkniederschlägen ist seit langem eine Frage, die in vielen Studienbereichen von Interesse ist. Eine entscheidende Komponente für eine solche Planung ist die Analyse extremer historischer Niederschläge mit Methoden der Extremwertstatistik, die Informationen über die Häufigkeit und das Ausmaß möglicher künftiger Ereignisse liefern. Die Charakterisierung eines ganzen Einzugsgebiets oder einer geografischen Einheit erfordert die Erweiterung der univariaten Extremwerstatistik-Methoden, um die räumliche Variabilität der Daten zu erfassen. Diese Erweiterung erfordert, dass die Ähnlichkeit der Daten für nahe gelegene Stationen in das Modell einbezogen wird, was zu einer effizienteren Nutzung der Daten führt. Diese Dissertation konzentriert sich auf die Verwendung statistischer Modelle, die die räumliche Abhängigkeit bei der Modellierung von jährlichen Niederschlagsmaxima berücksichtigen. Darüber hinaus werden Möglichkeiten zur Anpassung der Modelle vorgestellt, um die Abhängigkeit zwischen Niederschlägen auf verschiedenen Zeitskalen zu erfassen. Diese Modelle werden zur Verfolgung zweier Ziele eingesetzt. Das erste Ziel besteht darin, unser Verständnis der Mechanismen zu verbessern, die zur Abhängigkeit von extremen Niederschlägen führen. Das zweite Ziel besteht darin, die resultierenden Schätzungen zu verbessern, wenn die Abhängigkeit in die Modelle einbezogen wird. Zwei veröffentlichte Studien bilden die wichtigsten Ergebnisse dieser Dissertation. Die in beiden Studien verwendeten Modelle basieren auf max-stabilen Brown-Resnick-Prozessen, die es den Modellen ermöglichen, die Abhängigkeit entweder auf der zeitlichen oder auf der räumlichen Ebene ausdrücklich zu berücksichtigen. Diese bedingten Modelle werden für beide Fälle mit einem Modell verglichen, das die Abhängigkeit ignoriert, so dass wir die Auswirkungen der Abhängigkeit in beiden Situationen bestimmen können. Es werden auch Beiträge zu drei anderen Studien zusammengefasst, die das Konzept der Abhängigkeit verwenden. In der ersten Studie bewerten wir die Auswirkungen der Einbeziehung der Abhängigkeit zwischen Niederschlagsreihen unterschiedlicher Aggregationsdauern bei der Schätzung von Intensitäts-Dauer-Frequenz-Kurven. Diese Bewertung wurde in einer Fallstudie für das Einzugsgebiet der Wupper in Deutschland durchgeführt. Diese Studie ergab, dass sich die Einbeziehung der Abhängigkeit in das Modell positiv auf die Vorhersagegenauigkeit auswirkt, wenn man sich auf Niederschläge mit kurzen Dauern (d <= 10 h) und großer Nichtüberschreitungwahrscheinlichkeit konzentriert. Daher empfehlen wir die Verwendung von max-stabilen Prozessen, wenn sich eine Studie auf Regenfälle von kurzer Dauer konzentriert. In der zweiten Studie untersuchen wir, wie sich die räumliche Abhängigkeit von Extremniederschlägen in Berlin-Brandenburg saisonal verändert und wie sich diese Veränderung auf die Schätzungen eines max-stabilen Modells auswirken könnte, das diese Abhängigkeit berücksichtigt. Die Saisonalität wurde durch die Schätzung der Parameter eines halbjährlichen Sommer- und Winter-Blockmaxima-Modells bestimmt. Die Ergebnisse dieser Studie zeigten, dass die Abhängigkeitsstruktur für die Sommermaxima durch ein isotropes Brown-Resnick-Modell angemessen erfasst wurde. Im Gegensatz dazu zeigte dasselbe Modell eine schlechte Leistung für die Wintermaxima, was darauf hindeutet, dass eine Änderung der Annahmen erforderlich ist, wenn es um typische Winterereignisse geht, die in dieser Region typischerweise frontal oder stratiförmig sind. Diese Ergebnisse zeigen, dass die Berücksichtigung der meteorologischen Eigenschaften der Niederschlagsprozesse nützliche Informationen für die Gestaltung der Modelle liefern kann. Insgesamt zeigen unsere Ergebnisse, dass die Einbeziehung von meteorologischem Wissen in statistische Modelle die daraus resultierenden Schätzungen verbessern kann. Insbesondere stellen wir fest, dass unter bestimmten Bedingungen die Nutzung der statistischen Abhängigkeit zur Einbeziehung von Wissen über die Unterschiede in den zeitlichen und räumlichen Skalen der regenerzeugenden Mechanismen zu einer positiven Wirkung in den Modellen führen kann

Multivariate extreme storm surge flooding events on the UK’s east coast

In the United Kingdom (UK), floods, and specifically coastal flooding, are a hazard that is commonly thought likely to increase due to the impacts of climate change and the results of development in areas at risk. East coast storm surges have been extremely devastating in the recent past, such as the events of 1953 or the winter of 2013/14. The challenge is to analysis the risk of widespread, concurrent and clustered coastal flooding in a regional scale. It is widely accepted that extreme value analysis (EVA) is an important tool for studying coastal flood risk, but it requires the estimation of a threshold to define extreme events and has to cope with the problems of missing values within the dataset. The main areas of research discussed in this thesis involve making improvements to the way that extreme thresholds are selected and providing an alternative approach for multivariate missing values. By applying an automated threshold selection method to the data, more plausible and less subjective results can be yielded over the traditional manual approach. The alternative multivariate analysis at regional scale considers the statistical dependences between locations and which possible combination of events to take into account in order to handle missing values within time series dataset. Both areas of research provide developments to existing extreme value methodologies, hence enhancing the predicted future storm surge coastal flood modelling. An application of this research is to analysis the potential impacts of proposed nuclear power stations considering the increase likelihood of occurrence of extreme storm surge events. This research undertakes EVA with the statistical programming language R. However, R provides a range of functions embedded in different R packages, it was necessary to create new functions, scripts and commands to improve the analysis of extremes in order to undertake the threshold selection and cope with missing values. This research selects, as a case study, fourteen tide gauges along the East Coast of the UK from Lerwick to Dover. The main measure is skew surge due to be an independent and identically distributed variable and all phase differences in the calculations are removed. The multivariate model provides the likelihood of future significant storm surge flooding events along the East Coast of the UK. Results show that return levels for 50, 100 and 250 years estimates higher impact of ≈1m in Felixstowe, Sheerness, Immingham, Cromer and Lowestoft, while the northern gauges show an increment of ≈0.5m. Moreover, due to the overdispersion of the dataset, high predicted values are estimated in Lowestoft, Felixstowe and Dover where currently nuclear power sites are generating energy and new sites will be built in the future. In summary, the main aim of this research is to undertake a multivariate extreme model to analysis the potential impacts of future storm surge coastal flooding at a regional scale. By analysing extreme skew surge events at a regional level, a more complex storm surge coastal flooding model can be elaborated, and therefore, better results can be obtained. The multivariate extreme model requires how to select extreme events and how to handle missing values within the dataset. Hence, the proposed Automated Graphic Threshold Selection (AGTS) method provides a mathematical and computational tool to select extreme threshold, and moreover, the Multivariate Extreme Missing Value Approach (MEMVA) handles the missing values in time series dataset. The multivariate extreme model has the potential to improve the regional risk assessment of widespread, concurrent and clustered coastal flooding events

Some applications of copulae to finance

The aim of this thesis is to extend theory and to develop practical applications of copulae in finance. A copula is a dependence function that links random variables - expressed through their marginal distributions - to their joint or multivariate distribution

Hypothesis Testing Using Spatially Dependent Heavy-Tailed Multisensor Data

The detection of spatially dependent heavy-tailed signals is considered in this dissertation. While the central limit theorem, and its implication of asymptotic normality of interacting random processes, is generally useful for the theoretical characterization of a wide variety of natural and man-made signals, sensor data from many different applications, in fact, are characterized by non-Gaussian distributions. A common characteristic observed in non-Gaussian data is the presence of heavy-tails or fat tails. For such data, the probability density function (p.d.f.) of extreme values decay at a slower-than-exponential rate, implying that extreme events occur with greater probability. When these events are observed simultaneously by several sensors, their observations are also spatially dependent. In this dissertation, we develop the theory of detection for such data, obtained through heterogeneous sensors. In order to validate our theoretical results and proposed algorithms, we collect and analyze the behavior of indoor footstep data using a linear array of seismic sensors. We characterize the inter-sensor dependence using copula theory. Copulas are parametric functions which bind univariate p.d.f. s, to generate a valid joint p.d.f. We model the heavy-tailed data using the class of alpha-stable distributions. We consider a two-sided test in the Neyman-Pearson framework and present an asymptotic analysis of the generalized likelihood test (GLRT). Both, nested and non-nested models are considered in the analysis. We also use a likelihood maximization-based copula selection scheme as an integral part of the detection process. Since many types of copula functions are available in the literature, selecting the appropriate copula becomes an important component of the detection problem. The performance of the proposed scheme is evaluated numerically on simulated data, as well as using indoor seismic data. With appropriately selected models, our results demonstrate that a high probability of detection can be achieved for false alarm probabilities of the order of 10^-4. These results, using dependent alpha-stable signals, are presented for a two-sensor case. We identify the computational challenges associated with dependent alpha-stable modeling and propose alternative schemes to extend the detector design to a multisensor (multivariate) setting. We use a hierarchical tree based approach, called vines, to model the multivariate copulas, i.e., model the spatial dependence between multiple sensors. The performance of the proposed detectors under the vine-based scheme are evaluated on the indoor footstep data, and significant improvement is observed when compared against the case when only two sensors are deployed. Some open research issues are identified and discussed