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

46 APPENDIX B. BIBLIOGRAPHY ABOUT POINT PROCESS RELATED TO DEEP NEURAL NETWOR Figure B.1: NARX model illustration from paper [1] the RNN. A simple form of the input is the inter-event interval as x i = ( t i − t i − 1 ) or its logarithm as x i = ( log ( t i − t i − 1 )) . A hidden state h i of the RNN is updated as follows: h i = f ( W h h i − 1 + W x x i + b h ) where W h , W x , and b h denote the recurrent weight matrix, input weight matrix, and bias term, respectively, and f is an activation function. We here treat the hidden state of the RNN as a compact vector representation of the event history.The conditional intensity function is then formulated as a function of the elapsed τ time from the most recent event and the hidden state of the RNN, given as follows: λ ( t | H t ) = φ ( t − t i | h i ) where φ ( . ) is a non-negative function referred to as hazard function Du et al. (2016) assumed the following form for the hazard function φ ( τ | h i ) = exp( w t τ + v φ × h i + b φ ) The exponential function in the above equation is used to ensure the non-negativity of the intensity. In this model, the conditional intensity function exponentially decreases or increases with the elapsed time τ from the most recent event until the next event " Since the occurrence of an event may be triggered by what happened in the past, we can essentially specify models for the timing of the next event given what we have already known so far. More formally, a marked temporal point process is a random process of which the realization consists of a list of discrete events localized in time, t j , y j with the timing t j ∈ R + , the marker y j and j ∈ Z + " See Figure B.2 for architecture precision on RMTPP model. " Our key idea is to let the RNN (or its modern variant LSTM [23], GRU [5], etc.) model the nonlinear dependency over both of the markers and the timings from past events As shown in Figure 2, for the event occurring at the time t j of type y j , the pair ( t j , y j ) is fed as the input into a RNN unfolded up to the j+1 th event. The embedding h j − 1 represents the memory of the influence from the timings and the markers of past events. The neural network updates h j − 1 to h j by taking into account the effect of the current event ( t j , y j ) .The advantage of this formulation is that we explicitly embed the event history into a latent vector space, and by the elegant relation (4), we are now able to capture a general form of the