# Temporal, spatial and extreme event analysis

6 ECTS

M2 MSIAM (DS)

## Instructors

Julien Chevallier (part I), Jean-François Coeurjolly (II) and Stéphane Girard (part III)

## Description

Modelling extreme temperatures, extreme river flows, earthquakes intensities, neuronal activity, map diseases, lightning strikes, forest fires, for example is a risk modelling and assessment task, which is tackled in statistics using point processes and extreme value theory.

On the one hand, point processes are a class of stochastic processes modelling random events in interaction. By event we can think of the time a neuron activates, an earthquake occurs, the time a tweet has been retweeted, etc or the location of a tree in a forest, the impact of a lightning strike, etc. The first two parts provide an introduction to stochastic models and statistical inference which could cover such applications. Main characteristics of such processes, standard models (properties, simulation) and statistical procedures to infer them will be presented.

On the other hand, taking into account extreme events such as heavy rainfalls, floods, extreme temperatures is often crucial in the statistical approach to risk modeling. In this context, the behavior of the distribution tail is then more important than the shape of the central part of the distribution. Extreme-value theory offers a wide range of tools for modeling and estimating the probability of extreme events.

## Course Outline

### Part I (9 hours) - Temporal point processes

• Definition and simulation of one-dimensional point processes (conditional/stochastic intensity);
• Likelihood and goodness-of-fit tests (illustration on the Poisson point process);
• Hawkes processes (estimation, goodness-of-fit, stationarity, ergodicity).

### Part II (12 hours) - Spatial point processes

• Definition and characterization of a spatial point process, intensity functions and conditional intensity functions; Poisson point process;
• Intensity estimation and summary statistics;
• Models for spatial point processes (Cox, determinantal and Gibbs point processes): characterization, simulation, statistical inference and validation.

### Part III (15 hours) - Extreme-value analysis

• Asymptotic behavior of the largest value of a sample. Extreme-value Distribution (EVD). Maximum domains of attraction (Fréchet, Weibull and Gumbel). Asymptotic behavior of excesses over a threshold. Generalized Pareto Distribution (GPD). Regularly varying functions.
• Estimation of the parameters of the EVD and GPD. Hill estimator. Application to the estimation of extreme quantiles. Illustration on simulated and real data.

## Prerequisites

Background on statistics and probability (master 1 level)

## Keywords

stochastic processes; dependence modelling; simulation and statistical inference; Poisson point process; quantiles; excess process.