Multiclass Classification for Hawkes Processes.

Summary We investigate the multiclass classification problem 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 derive the optimal Bayes rule and provide a twostep 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 consistency 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 exponential 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.