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

20 CHAPTER 2. TOWARD FORMALISATION • For our application, times series which values X i lie in N , they are generally named count times series. If we have 2 times series X t and Y t , like rainfall an rockfall daily times series for us, on can measure (linear) dependencies between them through: Definition 3. The cross-correlation between two times series X t and Y t , for a given time lag h ∈ N is : C h ( X t , Y t ) = P ( X t − X t )( Y t − h − Y t ) σ X σ Y where σ X = q P ( X t − X t ) 2 is the standard deviation of variable X 2.2 Regression models These models regress a target variable Y against explicative variables X = [ X 1 , X 2 , ..., X p ] without taking explicitly time in consideration (like a succession of samples without order). General Linear Model can be formulated as : E [ Y | X ] = g ( X β ) where β are coefficients vector, g ( . ) is the link function adapted to the probability law of Y . For instance, if Y ∈ { 0 , 1 , 2 ... } has Poisson distribution, g is logit ; if Y ∈ { 0 , 1 } is a Bernouilli variable, g is... Example. In next chapter, we first try regression with previous rain and number of rockfall as regressors. If NB t stands for the number of rockfall and R t for rainfall quantity at day t (with lag time or cumulative specified before), then regression vector is X = [ R t , R t − 1 , ..., R t − p , NB t − 1 , ...NB t − k ] . Number of lags p and k should be determined by different statistical test. 2.3 Neural Networks (NNs) From a statistical point of view, NNs are just non-linear function estimated f θ ( . ) from data, that attempt to map inputs to target ˆ Y = f ˆ θ ( X ) . Contrary to classical estimators, the number parameters card ( θ ) is often around million. They can be tuned thanks to their structure and effective parallel computing. 2.3.1 Learning process We say that a neural network learns from data because of its operating steps. First, all trainable parameters θ of the model, like weights of connection and biases are randomly initialised. After, the model improves itself to minimize a loss function L ( Y, ˆ Y ) = L θ that measures the error between predictions and true labels. This is achieved by iterating over each sample of the