Davit Varron

A multi-dimensional statistical rainfall threshold for deep landslides based on groundwater recharge and support vector machines

The rainfall threshold determination is widely used for estimating the minimum critical rainfall amount which may trigger slope failure. The aim of this study was to develop an objective approach for the determination of a statistical rainfall threshold of a deep-seated landslide. The determination is based on recharge estimation and a multi-dimensional rainfall threshold. This new method is compared with precipitation and with a conventional ‘two-dimensional’ rainfall threshold. The method is designed to be semiautomatic, enabling an eventual integration into a landslide warning system. The method consists in two independent parts: (i) unstable event identification based on displacement time series and (ii) multi-dimensional rainfall threshold determination based on support vector machines. The method produces very good results and constitutes an appropriate tool to define an objective and optimal rainfall threshold. In addition to shortened computation times, the non-necessity of pre-requisite hypotheses and a fully automatic implementation, the newly introduced multi-dimensional approach shows performances similar to the classical two-dimensional approach. This shows its relevance and its suitability to define a rainfall threshold. Lastly, this study shows that the recharge is a relevant parameter to be taken into account for deep-seated rainfall-induced landslides. Using the recharge rather than the precipitation significantly improves the delineation of a rainfall threshold separating stable and unstable events. The performance and accuracy of the multi-dimensional rainfall threshold developed for the Séchilienne landslide make it an appropriate method for integration into the present-day landslide warning system.

Hydrogeological Threshold Using Support Vector Machines and Effective Rainfall Applied to a Deep Seated Unstable Slope (Séchilienne, French Alps)

Rainfall threshold is a widely used method for estimating minimum critical rainfall amount which can yield a slope failure. Literature reviews show that most of the threshold studies are subjective and not optimal. For this study, effective rainfall was considered for threshold definition. Support vector machines (SVM) and automatic event identification were used in order to establish an optimal and objective threshold for the Sechilienne landslide. Effective rainfall does significantly improve threshold performance (misclassification rate of 7.08 % instead of 13.27 % for gross rainfall) and is a relevant parameter for threshold definition in deep-seated landslide studies. In addition, the accuracy of the Sechilienne SVM threshold makes it appropriate to be integrated into a landslide warning system. Finally, the ability to make predictions at a daily time step opens up an opportunity for destabilisation stage predictions, through the use of weather forecasting.

Non standard functional limit laws for the increments of the compound empirical distribution function

Let (Y i , Z i) i≥1 be a sequence of independent, identically distributed (i.i.d.) random vectors taking values in R k × R d , for some integers k and d. Given z ∈ R d , we provide a nonstandard functional limit law for the sequence of functional increments of the compound empirical process, namely ∆n,c(hn, z, ·) := 1 nhn n i=1 1 [0,·) Z i − z hn 1/d Y i. Provided that nhn ∼ c log n as n → ∞, we obtain, under some natural conditions on the conditional exponential moments of Y | Z = z, that ∆n,c(hn, z, ·) Γ almost surely, where denotes the clustering process under the sup norm on [0, 1) d. Here, Γ is a compact set that is related to the large deviations of certain compound Poisson processes.

Functional limit laws for local empirical processes in a spatial setting

We establish a functional limit theorem for local abstract empirical processes based on an independent, identically distributed sequence , with , and , where is a class of functions. Given two bandwidth sequences and fulfilling and given a probability measure on , we prove that almost surely, where is the random set of locations for which the sequence of subsets of admits Strassen-type sets as inner and outer topological limits. That result is proved under some standard structural conditions on and some regularity conditions on the law of .

Empirical likelihood confidence tubes for functional parameters in plug-in estimation

We consider the infinite-dimensional inference problem in which the parameter of interest is a multivariate trajectory that can be written as an explicit functional T of a number of probability distributions. We propose an empirical likelihood procedure to build simultaneous confidence regions for these trajectories. Our main assumption is the Hadamard differentiability of T under norms adapted to empirical measures, i.e., supremum norms indexed by Donsker classes of functions. In order to handle practical computational issues, the proposed method, which we prove to be consistent, is based on a first order expansion of T. We also prove a general result of independent interest in empirical likelihood theory. Three applications are provided.

A note on weak convergence, large deviations, and the bounded approximation property

Given a Banach space

A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process

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Some uniform in bandwidth functional results for the tail uniform empirical and quantile processes

having the bounded approximation property, we provide a paradigm that can be used to establish large deviation principles for sequences of random elements

Donsker and Glivenko-Cantelli theorems for a class of processes generalizing the empirical process

(X_n)\suite

Uniform in bandwidth exact rates for a class of kernel estimators

taking values in