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International audience; In the framework of the LIBRIS project (Contribution to seismic risk assessment in Lebanon), a work package is devoted to the use of broken or unbroken speleothems (stalagmites and stalactites) to better characterize the local seismic hazard. This kind of study is particularly interesting in the long return period range, where historical information is far from being sufficient. The proposed approach is based on a previous study by Lacave et al. (2004). Two pilot caves, located north of Beirut, along or close to a major fault, were first selected. Then, in situ measurements of speleothems were done using a 3D laser scan. The data were processed to obtain a detailed sampling of the broken and unbroken speleothem population (number, precise shape). These data are finally used to compute, in a statistical approach, the probability of exceeding, or non exceeding, of a certain level of seismic acceleration in the study area.

Within the special geometry of the simplex, the sample space of compositional data, compositional orthonormal coordinates allow the application of any multi- variate statistical approach. The search for meaningful coordinates has suggested balances (between two groups of parts)|based on a sequential binary partition of a D-part composition|and a representation in form of a CoDa-dendrogram. Projected samples are represented in a dendrogram-like graph showing: (a) the way of grouping parts; (b) the explanatory role of subcompositions generated in the partition process; (c) the decomposition of the variance; (d) the center and quantiles of each balance. The representation is useful for the interpretation of balances and to describe the sample in a single diagram despite the number of parts. Also, samples of two or more populations, as well as several samples from the same population, can be represented in the same graph, as long as they have the same parts registered. The approach is illustrated by an example of food consumption in Europe.

The longitudinal studies has increased the importance of statistical methods for time-to event data that can incorporate time-dependent covariates. Cox proportional regression model(Cox, 1972) is one such method that is widely used. It is a statistical technique for exploring the relationship between the survival of a patient and several explanatory variables. The follow-up time may range from few weeks to many years.

Este trabajo ha sido financiado por la Comisión Asesora de Investigación, Ciencia y Tecnología, Proyectos números 3184/83 y 1151/84 y por el Comité Cientifico Hispano-Norteamericano, proyecto número CCB-8604/015

A compositional approach to air quality index (AQI) based on a log-contrast of pollutants over non-polluted air fraction was recently developed. This approach let to three open questions about consistency and formalization of such index. First question is to find out the conditions under which the approximation of the log-contrast by the logarithm of the geometric mean of pollutants concentrations, are reliable. Second, is to examine a way to define a reasonable scale for express air quality, taking into account the effects of air pollution on population health and on social or economic activity. The third one is to find out the properties of this kind of index with respect to the addition of a new set of air pollutants in the composition. To deal with these open questions, a decision making tree should be designed in order to select suitable weights on the air pollutants. A preliminary application study of air pollutants composition distribution in some cities is presented. In addition, the approach of the log-contrast by the logarithm of the weighted geometric mean is described for these cities.

The estimation of occurrence probabilities of extremal quantities is essential in the study of hazards associated with natural phenomena. The extremal quantities of interest usually correspond to phenomena characterized by two or more magnitudes, often showing dependence among them. In order to better characterize situations that could be dangerous, the magnitudes that describe the phenomenon should be jointly described. A Poisson-GPD model, which describes the occurrence of extremal events and their marginal sizes, has been established: the occurrence of the extremal events is represented by means of a Poisson process, and each event is characterized by a size modelled by a Generalized Pareto Distribution, GPD. The dependence between events is modelled through copula functions: a family of Gumbel copulas, suitable for the type of data treated, and a new type of copula that is introduced, the CrEnC copula. The CrEnC copula minimizes the mutual information in situations in which only partial information in the form of restrictions is available, such as marginal models or joint moments of the variables. In this context, data are often scarce, and the uncertainty in the estimation of the model will be great. A Bayesian estimation process that takes into account this uncertainty has been established. Goodness-of-fit of some aspects of the model (GPD goodness-of-fit, GPD-Weibull hypothesis and global goodness-of-fit) has been checked using a selection of Bayesian p-values, which incorporate the uncertainty of the parameter estimation. Once the model has been estimated, a post-process of information has been performed to obtain a posteriori quantities of interest, such as exceedance probabilities of reference values or return periods of events of a certain size. The proposed model is applied to three datasets, with different characteristics. The results obtained are good: the introduced CrEnC copulas correctly represent the dependence in situations in which only partial information is available, and the Bayesian estimation of the parameters of the model gives added value to the results, because it allows the uncertainty of the posterior estimates, such as hazard and dependence parameters, to be evaluated., La estimación de probabilidades de ocurrencia de cantidades extremales es imprescindible en el estudio de la peligrosidad de fenómenos naturales. Las cantidades extremales de interés suelen corresponder a fenómenos caracterizados por dos o más magnitudes, que en muchos casos son dependientes entre sí. Por tanto, para poder caracterizar mejor las situaciones que pudieran resultar peligrosas, se deben estudiar conjuntamente las magnitudes que describen el fenómeno. Se ha establecido un modelo Poisson-GPD que permite describir la ocurrencia de los sucesos extremales y sus tamaños marginales: la ocurrencia de los sucesos extremales se representa mediante un proceso de Poisson y cada suceso se caracteriza por un tamaño modelado según una distribución generalizada de Pareto, GPD. La dependencia entre sucesos se modeliza mediante funciones cópula: se utiliza una familia de cópulas Gumbel, adecuada al tipo de datos, y se introduce un nuevo tipo de cópula, la cópula CrEnC. La cópula CrEnC minimiza la información mutua en situaciones donde se dispone de información parcial en forma de restricciones, tales como los modelos marginales o momentos conjuntos de las variables. La representación de estas cópulas en R^2 permite mejorar tanto su estima como la apreciación de la bondad de ajuste a los datos. Se proporciona un algoritmo de estimación de cópulas CrEnC, que incluye una aproximación de las funciones normalizadoras mediante el método Montecarlo. En este contexto los datos suelen ser escasos, por lo que la incertidumbre en la estimación del modelo será elevada. Se ha establecido un proceso de estimación bayesiana de los parámetros, la cual permite tener en cuenta esta incertidumbre. La bondad de ajuste de diversos aspectos del modelo (bondad de ajuste GPD, hipótesis GPD-Weibull y bondad de ajuste global) se ha valorado mediante una selección de p-valores bayesianos, los cuales incorporan la incertidumbre de la estimación de los parámetros. Una vez estimado el modelo, se realiza un post-proceso de la información, donde se obtienen cantidades a posteriori de interés, como probabilidades de excedencia de valores de referencia o periodos de retorno de sucesos de un tamaño determinado. El modelo propuesto se aplica a tres conjuntos de datos de características diferentes. Se obtienen buenos resultados: las cópulas CrEnC introducidas representan correctamente la dependencia en situaciones en las que sólo se dispone de información parcial y la estimación bayesiana de los parámetros del modelo proporciona valor añadido a los resultados, ya que permite evaluar la incertidumbre de las estimaciones y tenerla en cuenta al obtener las cantidades a posteriori deseadas