Scale effect in hazard assessment - application to daily rainfall

International audienceDaily precipitation is recorded as the total amount of water collected by a rain-gauge in 24h. Events are modelled as a Poisson process and the 24h precipitation by a Generalized Pareto Distribution (GPD) of excesses. Hazard assessment is complete when estimates of the Poisson rate and the distribution parameters, together with a measure of their uncertainty, are obtained. The shape parameter of the GPD determines the support of the variable: Weibull domain of attraction (DA) corresponds to finite support variables, as should be for natural phenomena. However, Fréchet DA has been reported for daily precipitation, which implies an infinite support and a heavy-tailed distribution. We use the fact that a log-scale is better suited to the type of variable analyzed to overcome this inconsistency, thus showing that using the appropriate natural scale can be extremely important for proper hazard assessment. The approach is illustrated with precipitation data from the Eastern coast of the Iberian Peninsula affected by severe convective precipitation. The estimation is carried out by using Bayesian techniques

The normal distribution in some constrained sample spaces

Phenomena with a constrained sample space appear frequently in practice. This is the case, for example, with strictly positive data, or with compositional data, such as percentages or proportions. If the natural measure of difference is not the absolute one, simple algebraic properties show that it is more convenient to work with a geometry different from the usual Euclidean geometry in real space, and with a measure different from the usual Lebesgue measure, leading to alternative models that better fit the phenomenon under study. The general approach is presented and illustrated using the normal distribution, both on the positive real line and on the D-part simplex. The original ideas of McAlister in his introduction to the lognormal distribution in 1879, are recovered and updated

Vulnerability models for environmental risk assessment. Application to a nuclear power plant containment building

Environmental risk management consists of making decisions on human activities or construction designs that are affected by the environment and/or have consequences or impacts on it. In these cases, decisions are made such that risk is minimized. In this regard, the forthcoming paper develops a close form that relates risk with cost, hazard, and vulnerability; and then focuses on vulnerability. The vulnerability of a system under an external action can be described by the conditional probability of the degrees of damage after an event. This vulnerability model can be obtained by a simplicial regression of those outputs, as a response variable, on explanatory variables. After a theoretical explanation, the authors present the case study of a nuclear power plant containment building. Once a given overpressure is registered inside the containment building, three possible outputs are to be considered: serviceability, breakdown, and collapse. The study consists of three steps: (a) modelling the containment building using the finite element method; (b) given an overpressure, simulating uncertain parameters related to material constitutive equations in order to obtain the corresponding proportions; (c) performing a simplicial regression to obtain a meaningful vulnerability model. The simulation provides normalized-to-unity outputs under the overpressure conditions. The obtained vulnerability model is in definite correspondence with previous results in nuclear power plant safety analysis reports.Spain. Ministerio de Educación, Cultura y Deporte (BOE n. 190, August 9th, 2012) to collaborate with the Department of Applied Mathematics III at UPC-BarcelonaTech from September 2012 to June 2013)Spain. Ministerio de Ciencia y Tecnología (project ‘Ingenio Mathematica (i-MATH)’ (Ref. No. CSD2006-00032))Spain. Ministerio de Ciencia y Tecnología (project ‘CODARSS’ (Ref. MTM2009-13272))Spain. Ministerio de Economía y Competitividad (project ‘METRICS’ (Ref. MTM2012- 33236))Catalonia (Spain). Agencia de Gestio d'Ajuts Universitaris i de Recerca (project Ref. 2009SGR424

The effect of scale in daily precipitation hazard assessment

Daily precipitation is recorded as the total amount of water collected by a rain-gauge in 24 h. Events are modelled as a Poisson process and the 24 h precipitation by a Generalised Pareto Distribution (GPD) of excesses. Hazard assessment is complete when estimates of the Poisson rate and the distribution parameters, together with a measure of their uncertainty, are obtained. The shape parameter of the GPD determines the support of the variable: Weibull domain of attraction (DA) corresponds to finite support variables as should be for natural phenomena. However, Fréchet DA has been reported for daily precipitation, which implies an infinite support and a heavy-tailed distribution. Bayesian techniques are used to estimate the parameters. The approach is illustrated with precipitation data from the Eastern coast of the Iberian Peninsula affected by severe convective precipitation. The estimated GPD is mainly in the Fréchet DA, something incompatible with the common sense assumption of that precipitation is a bounded phenomenon. The bounded character of precipitation is then taken as a priori hypothesis. Consistency of this hypothesis with the data is checked in two cases: using the raw-data (in mm) and using log-transformed data. As expected, a Bayesian model checking clearly rejects the model in the raw-data case. However, log-transformed data seem to be consistent with the model. This fact may be due to the adequacy of the log-scale to represent positive measurements for which differences are better relative than absolute

Evidence functions : a compositional approach to information

The discrete case of Bayes' formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes' formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection