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

18 CHAPTER 2. TOWARD FORMALISATION • Regression models where we only focus on the dependencies between rockfall and rainfall. Those models don’t take into account time but we can introduce lagged time variable as exogenous one. If target values are binary, then we call it logistic regression. • Neural Networks with plenty of interconnected neurons and specifical functions. Due to sparsity of dataset, we investigate only Supervised Learning. In that case, networks ability to map inputs to target is learnt from labelled data. 2.1 Regular/Irregular sampling of events 2.1.1 Irregular sampling: Point process Sometimes, a physical phenomenon is just seen at irregular points (for d -dimensional space) or times, and can be modelled by a point process. In temporal dimension, a sample from point process is a set of increasing times ( T i ) i =1 ,...,N , each T i ∈ R + . This sample can be encode in a counting process N ( . ) : N : R + −→ N t 7 −→ N ( t ) = P i 1 T i <t N ( t ) represent the number of events occurring between time interval [0 , t [ . So, N [ a,b [ = N ( b ) − N ( a ) is number of points in [ a, b [ . Let λ ( . ) a integrable function defined on R + , named intensity . Mean of the random number of events can be computed from this intensity: by E [ N ( t )] = Λ( t ) = Z t 0 λ ( s ) ds Definition 1. Point process is call Inhomogeneous Poisson Process with rate λ ( t ) if : 1. N (0) = 0 and N ( t ) has independent increments, i.e N [ a,b [ is independent from N [ c,d [ as soon as [ a, b [ and [ c, d [ are disjoint intervals. 2. P [ N ( t + h ) − N ( t )] = λ ( t ) h + o ( h ) . Events occurs at time t with a the rate λ ( t ) . Moreover it should be is a simple process (no more than one point at each time, almost surely) Compared to general case, Poisson Process has independent increments! Such process are called "Poisson" because they have the following property. Property. The number of events in any interval [ t, t + h ] is a Poisson random variable P ( µ ) with mean parameter µ depending on the location t and the width h of the interval: µ = E [ N ( t + h ) − N ( t )] = Z t + h t λ ( s ) ds