M. I. Ortego

Extremes from scarce data: The role of Bayesian and scaling techniques in reducing uncertainty

This paper deals with the analysis of extreme wave heights and their uncertainties. The main purpose is to assess confidence intervals using a conventional extreme value, and a Bayesian approach. It is shown how the introduction of an a priori information helps to bound the upper confidence limit. The analysis is performed with wave-height data recorded off the Spanish Catalan coast (NW Mediterranean) and wave-height data from the Dutch coast (North Sea). An analysis with natural-scale and log-transformed wave-height time series has been performed. This scale selection is proven to be advantageous for naturally bounded variables and also better captures some distribution features. The paper ends with a discussion on how the different techniques can be used to select a statistically robust threshold for an extreme event definition. This affects the evaluation of risk in low-lying coastal areas, associated to variables controlling flooding and erosion risks.

Assessing wavestorm hazard evolution in the NW Mediterranean with hindcast and buoy data

It has been suggested that climate change might modify the occurrence rate of large storms and their magnitude, due to a higher availability of energy in the atmosphere-ocean system. Forecasting physical models are commonly used to assess the effects. No one expects the physical model forecasts for one specific day to be accurate; we consider them to be good if they adequately describe the statistical characteristics of the climate. The Peak-Over-Threshold (POT) method is a common way to statistically treat the occurrence and magnitude of hazardous events: here, occurrence is modelled as a Poisson process and magnitude over a given threshold is assumed to follow a Generalized Pareto Distribution (GPD). We restrict our attention to Weibull-related GPDs, which exhibit an upper bound, to comply with the fact that any physical process has a finite upper limit. This contribution uses this framework to model time series of log-significant wave-height constructed joining quasi-collocated hindcast data and buoy measurements. Two of the POT model parameters (inhomogeneous Poisson rate and logarithm of the GPD shape parameter are considered to be a combination of a linear function of time and a series indicator function. The third parameter, logarithm of the GPD upper bound, is considered to have only a series indicator component. The resulting parameters are estimated using Bayesian methods. Using hincast and buoy series, the time span of the data set is extended, enhancing the precision of statistical results about potential linear changes. Simultaneously the statistical behaviour of hincast and buoy series are compared. At the same time, the step function allows to calibrate the statistical reproduction of storms by hindcasting.

Multivariate statistical modelling of future marine storms

Extreme events, such as wave-storms, need to be characterized for coastal infrastructure design purposes. Such description should contain information on both the univariate behaviour and the joint-dependence of storm-variables. These two aspects have been here addressed through generalized Pareto distributions and hierarchical Archimedean copulas. A non-stationary model has been used to highlight the relationship between these extreme events and non-stationary climate. It has been applied to a Representative Concentration Pathway 8.5 Climate-Change scenario, for a fetch-limited environment (Catalan Coast). In the non-stationary model, all considered variables decrease in time, except for storm-duration at the northern part of the Catalan Coast. The joint distribution of storm variables presents cyclical fluctuations, with a stronger influence of climate dynamics than of climate itself.

The distribution of extremes in the degree sequence: A Gumbel distribution approach

In this work, an approximation of the asymptotics of the distribution for the maximum and minimum of the degree sequence defined over a random graph is determined using a new approach in terms of the Gumbel distribution for extremes.

Compositional data for global monitoring: The case of drinking water and sanitation

INTRODUCTION At a global level, access to safe drinking water and sanitation has been monitored by the Joint Monitoring Programme (JMP) of WHO and UNICEF. The methods employed are based on analysis of data from household surveys and linear regression modelling of these results over time. However, there is evidence of non-linearity in the JMP data. In addition, the compositional nature of these data is not taken into consideration. This article seeks to address these two previous shortcomings in order to produce more accurate estimates. METHODS We employed an isometric log-ratio transformation designed for compositional data. We applied linear and non-linear time regressions to both the original and the transformed data. Specifically, different modelling alternatives for non-linear trajectories were analysed, all of which are based on a generalized additive model (GAM). RESULTS AND DISCUSSION Non-linear methods, such as GAM, may be used for modelling non-linear trajectories in the JMP data. This projection method is particularly suited for data-rich countries. Moreover, the ilr transformation of compositional data is conceptually sound and fairly simple to implement. It helps improve the performance of both linear and non-linear regression models, specifically in the occurrence of extreme data points, i.e. when coverage rates are near either 0% or 100%.

Modeling Extremal Dependence Using Copulas. Application to Rainfall Data

Copula distributions are a useful tool to describe dependence between two or more random variables. Sklar’s theorem allows to treat separately the dependence structure (the copula) and the marginal distributions. In the bivariate case, where is the bivariate cumulative distribution function, are the marginal cdfs and is the copula linking the marginals. When the marginal distributions are continuous the copula that links the marginal distributions is unique. There are several copula families, representing different dependence structures. A new copula family, CrEnC copulas, has been studied. This copula family is flexible enough to describe different types of dependence. An application to the spatial dependence of precipitation of two near locations is presented. A data set of 30 years of daily precipitation recorded at two nearby rain gauges located in the Valencia Region, Eastern Spain, has been studied. For each rain gauge, the logarithm of precipitation is modeled using a Generalized Pareto Distribution (GPD). The CrEnC copula family is used to model dependence between both extremal precipitation variables. Bayesian methods are applied to obtain estimations of the model parameters: marginal GPD and CrEnC copula parameters.