Dependence properties of spatial rainfall extremes and areal reduction factors

Areal reduction factors (ARFs) transform an estimate of extreme rainfall at a point to an estimate of extreme rainfall over a spatial domain, and are commonly used in flood risk estimation. For applications such as the design of large infrastructure, dam safety and land use planning, ARFs are needed to estimate flood risk for very rare events that are often larger than the biggest historical events. The nature of the relationship between ARFs and frequency for long return periods is unclear as it depends on the asymptotic dependence structure of rainfall over a region, i.e., the extent to which rainfall from a surrounding region is extreme as rainfall at a point becomes more extreme. Miscalculating this for very rare events could lead to poor design of infrastructure. To investigate this, spatial rainfall processes are simulated using asymptotically dependent and independent models, and the implications for ARFs of the asymptotic assumptions are explored in a synthetic study. The models are then applied to a case study in Victoria, Australia, using 88 daily rainfall gauges with 50 years of data. The analysis shows that the observed data follow the behaviour of an asymptotically independent process, leading to ARFs that decrease with increasing return period. The study demonstrates that the use of inverted max-stable process models to simulate ARFs can provide a rigorous alternative to empirical approaches, particularly for long return periods requiring significant extrapolation from the data.

Un modèle factoriel dynamique pour séries temporelles

Andrei Zenide

This work deals with factorial models for multiple time series. Its core content puts it at the interface between statistics and finance. After a brief description of the historical link between the two sciences, it reviews the literature on factorial models that are close to the model introduced in this work called The dynamical factor analysis model for time series. This model makes the hypothesis that the observed time series are influenced by a common factor, difficult to define and impossible to measure. No a priori structure is put on the factor, at each point time the value of the factor is considered as a new parameter that has to be estimated. As a consequence of this fact, the number of parameters is large and it is not possible to provide the usual asimptotic properties of the estimations by letting the number of periods tend to infi- nity. Asymptotic results in our context have been obtained by increasing the number of time series. The model makes the hypothesis that there is a linear dependence between the time series and the factor with coefficients, that are not constant over time, but rather follow a smooth random walk. This mean that the linear structure of the model is evolving slowly from a period to another. All the information which is not contained in the factor and the coefficients is considered as white noise. Using the normal distribution makes the estimation more easy and opens a toolbox of statistical methods that have been developed for this kind of data. The model is a part of the family of state space models and the Kalman filter is an essential ingredient of our estimator. The effort was concentrated on the elaboration of the structure of the model. Its complexity was constrained by the difficulty of estimation. The final shape of the model does not allow an analytical solution of the optimization problem introduced by the maximum likelihood estimation of the parameters. Numerical solutions have been found and compared with the parameters for simulated data. Some others models have been developed as simpler versions of the dynamical factor model for time series. The case where the factor can be observed has been studied and a new method for the estimation has been provided and compared with the existing methods. A second study considers a latent factor model without noise. Two methods for the estimation of the factor have been provided. The last chapter contains a detailed description of the main statistical tools used during this work. The links with the previous chapters are highlighted followed by comments.

EPFL2004

Bayesian risk analysis of financial time series

Christophe Osinski

This thesis is a contribution to financial statistics. One of the principal concerns of investors is the evaluation of portfolio risk. The notion of risk is vague, but in finance it is always linked to possible losses. In this thesis, we present some measures allowing the valuation of risk with the help of Bayesian methods. An exploratory analysis of data is presented to describe the sampling properties of financial time series. This analysis allows us to understand the origins of the daily returns studied in this thesis. Moreover, a discussion of different models is presented. These models make strong assumptions on investor behaviour, which are not always satisfied. This exploratory analysis shows some differences between the behaviour anticipated under equilibrium models, and that of real data. The Bayesian approach has been chosen because it allows one to incorporate all the variability, in particular that associated with model choice. The models studied in this thesis allow one to take heteroskedasticity into account, as well as particular shapes of the tails of returns. ARCH type models and models based on extreme value theory are studied. One original aspect of this thesis is its use of Bayesian analysis to detect change points in financial time series. We suppose that a market has two phases, and that it switches from a state to the other at random. Another new contribution is a model integrating heteroskedasticity and time dependence of extreme values, by superposition of the model proposed by Bortot and Coles (2003) and a GARCH process. This thesis uses simulation intensively for the estimation of risk measures. The drawback of simulation is the amount of time needed to obtain accurate estimates. However, simulation allows one to produce results when direct calculation is not feasible. For example, simulation allows one to compute risk estimates for time horizons greater than one day. The methods presented in this thesis are illustrated on simulated data, and on real data from European and American markets. This thesis involved the construction of a library containing C and S code to perform risk analysis using GARCH and extreme value theory models. The results show that model uncertainty can be incorporated, and that risk measures for time horizons greater than one can be obtained by simulation. The methods presented in this thesis have a natural representation involving conditioning. Thus, they permit the computation of both conditional and unconditional risk estimates. Three methods are described: the GARCH method; the two-state GARCH method; and the HBC method. Unconditional risk estimation using the GARCH method is satisfactory on data which seem stationary, but not reliable on data which are non-stationary, such as data with change points. The two-state GARCH model does a little better, but gives very satisfactory results when the risk is estimated conditionally on time. The HBC method does not give satisfactory results.

EPFL2006

Spatial analysis of karst conduit networks and determination of parameters controlling the speleogenesis along preferential lithostratigraphic horizons

Marco Filipponi

The main objective of this thesis is to improve the understanding of the position and characteristics of karst conduits within a rock massif. Such a characterisation is an important issue in civil engineering and in hydrogeology. Today in practice dissolution voids are considered as random in most cases. However, it is obvious for karst researchers that dissolution void distribution is not random, but defined by parameters controlling the speleogenesis. We developed a method to analyse the 3D geometry of cave systems in order to demonstrate from a statistical point of view that karst conduits position is not random. The analysis of several among the largest cave systems in the World (more than 1500 km of analysed cave conduits) confirmed for the first time quantitatively that the development and position of karst conduits under phreatic conditions is strongly related to a restricted number of so called inception horizons. An inception horizon – a concept introduced by Lowe (1992) – is a part of a lithostratigraphic succession that is particularly susceptible to the effects of the earliest cave forming processes by virtue of physical, lithological or chemical deviation from the predominant carbonate facies within the formation. We demonstrate that probably less than 10 % of the existing bedding partings of a limestone sequence are inception horizons and guide more than 70 % of the phreatic conduits. Our analysis clearly confirms that the influence of these horizons onto the 3D geometry of cave systems is high. Based on the 3D analysis of cave systems as well as on field verifications, 18 inception horizons in six cave systems have been selected for field characterisation and sampling in order to identify the properties and processes that makes these particular lithostratigraphic horizons favourable to karstification. Around 200 rock samples from the horizons and the surrounding rock mass have been analysed. The results evidence that inception horizons have a thickness of some centimetres to decimetres and that it is possible to distinguish between 3 types of inception horizons: Inception horizons where the cave inception took place within the inception horizon (type 1); characterized by a slightly higher primary permeability, pyrite and quartz contents and lower matrix contents than the surrounding rock mass. Usually, fractures propagate through or occur within these horizons. Inception horizons where the cave inception took place at the contact with the inception horizon (type 2); characterized by a lower primary permeability and carbonate contents, but higher pyrite contents than the surrounding rock mass. Fractures usually are ending at these horizons. Inception horizons where the cave inception took place along bedding plane fractures (type 3); already a slippage of just a few millimetres, striation, brecciation and surface irregularities enhance openings along the sliding plane and cause a significant increase in permeability. Furthermore it can be assumed that for the inception of horizons of type 1 the primary permeability will be the relevant factor at beginning of karstification, whereas both the matrix and the pyrite contents are the key factors during the later phases of cave inception and gestation. Whereas for further cave development the total carbonate content will be crucial. For inception horizons of type 2, we can assume that the low primary permeability, the clogging of the pores by the clay minerals, the high content of pyrite (production of aggressive solutions within the horizon that concentrates the dissolution along the contact to the surrounding rock) and the ending of the fractures at the horizon are responsible for the enhanced karstification at the contact during cave inception and gestation phases. Using simple hydrogeological numerical modelling we show that an epigenic karstic rock massif can be subdivided into four speleogenetic zones: 1) vadose cave development zone above the water table, 2) the phreatic cave development and 3) gestation zone within the first tens of metres of the phreatic zone and 4) below them the inception zone. Each of these zones is characterized by typical speleogenetic processes as well as dissolution void distribution. Further, it was possible to explain and reproduce schematically the 3D pattern of different cave systems by using the position and orientation of the inception horizons and the history of the landscape evolution (i.e. the re- and discharge area). This forward analysis provides a first idea of the geometry of the conduits as well as a better understanding of the development of a karst system in time and space (vertical section). Finally, we evaluated the feasibility to combine the improved inception horizon hypothesis, to predict inception horizons, with other current applied methods to improve the prediction of dissolution voids. Furthermore, we proposed a scientific based risk assessment for underground engineering proposes. Essentially, it is now evidenced that it is possible to quantify the probability of karst occurrences inside a karst massif by reconstructing the hydrogeolgical history and identifying the few inception horizons that guide the karstification at a regional scale.

EPFL2009