Hawkes Process Kernel Structure Parametric Search with Renormalization Factors

Hawkes Processes are a type of point process for modeling self-excitation, i.e., when the occurrence of an event makes future events more likely to occur. The corresponding self-triggering function of this type of process may be inferred through an Unconstrained Optimization-based method for maximization of its corresponding Loglikelihood function. Unfortunately, the non-convexity of this procedure, along with the ill-conditioning of the initialization of the self- triggering function parameters, may lead to a consequent instability of this method. Here, we introduce Renormalization Factors, over four types of parametric kernels, as a solution to this instability. These factors are derived for each of the self-triggering function parameters, and also for more than one parameter considered jointly. Experimental results show that the Maximum Likelihood Estimation method shows improved performance with Renormalization Factors over sets of sequences of several different lengths.