RINA - Démonstrateur d’utilisation de l’intelligence artificielle pour la gestion opérationnelle des risques naturels gravitaires géologiques | FEREC 2020

2.1. REGULAR/IRREGULAR SAMPLING OF EVENTS 19 is the expected number of events occurring between times t and t + h Example. The most basic example of a temporal point process is a homogeneous Poisson process , which assumes that the events are independent of each other and events occurs at a constant rate i.e λ ( t ) = λ . In that case, times between events ∆ T i = T i − T i − 1 follow an exponential law. Example. Another popular example is the Hawkes process , which is a simple model of a self- exciting point process. The intensity of the Hawkes process is given by λ ( t | H t ) = µ + P t i <t g ( t − ti ) , where g ( . ) is a kernel function ( g ( s ) = 0 if s < 0 ) that represents the triggering effect from the past event Application to rockfall: As inter-event times distribution doesn’t follow an exponential law (previous chapter), Rockfall events is not a homogeneous Poisson Process. As previous basic statistics seem to show, rockfall event is at temporal level a point process with an inhomogeneous seasonal intensity rate λ ( t ) (probability that a rockfall occur in time interval [ t ; t + dt [ ). This intensity is correlated with rainfall point process which have a strong annual seasonality. We could model a point process with spatio-temporal intensity function λ ( t, x ) where variation among x could be explained by protection, geological zone... As general for these point process, we can define N ( t, x ) is the number of event between time interval [0 , t [ between location [0 , x [ . N ( t, x ) can be expressed from an 2D-intensity λ ( t, x ) which can be seen as instant rate of rockfall at time t and at location x . 2.1.2 Regular sampling: Times Series In this framework, the quantity of interest X t = ( X i ) i ∈ Z is evenly sampled each ∆ t , generally in time domain, so that i-th value is X i = X t [ i ∆ t ] . Definition 2. X t is said second-order stationary if : • E [ X i ] is constant over time and named mean of process µ = E [ X i ] • E [ X i X i + h ] doesn’t depend of considered time i but only of time delay h ∈ Z . We call auto- covariance: γ ( h ) = E [( X i − µ )( X i + h − µ )] Remark. • Usually we prefer its normalised version named auto-correlation ρ ( h ) = γ ( h ) γ (0) . γ ( h ) and ρ ( h ) are symmetric i.e γ ( − h ) = γ ( h ) for h ∈ Z . • Its spectral domain counterpart is the Power Spectral Density Γ( ν ) through Fourier Trans- form .