Multiclass classification for Hawkes processes

We investigate the multiclass classification prob- lem where the features are event sequences. More precisely, the data are assumed to be generated by a mixture of simple linear Hawkes processes. In this new setting, the classes are discriminated by various triggering kernels. A challenge is then to build an efficient classification procedure. We de- rive the optimal Bayes rule and provide a two-step estimation procedure of the Bayes classifier. In the first step, the weights of the mixture are estimated; in the second step, an empirical risk minimization procedure is performed to estimate the parameters of the Hawkes processes. We establish the consis- tency of the resulting procedure and derive rates of convergence. Finally, the numerical properties of the data-driven algorithm are illustrated through a simulation study where the triggering kernels are assumed to belong to the popular parametric expo- nential family. It highlights the accuracy and the robustness of the proposed algorithm. In particular, even if the underlying kernels are misspecified, the procedure exhibits good performance.