Multivariate Arrival Rate Estimation by Sum-of-squares Polynomial Splines and Decomposition

A novel method for the arrival rate estimation of multi-dimensional non-homogeneous Poisson processes is presented. The method provides a smooth, piecewise polynomial spline estimator of any prescribed order of differentiability. At the heart of the algorithm is a maximum-likelihood optimization model with sum of squares polynomial constraints, which characterize a proper subset of smooth arrival rate functions. It is investigated when the underlying spline space is dense in the space of continuous arrival rate functions. The method is easily parallelized, exploiting the sparsity of the neighborhood structure of the underlying splines, using an augmented Lagrangian decomposition approach. As a numerical illustration with real-world data, the (spatio-temporal) arrival rate of accidents on the New Jersey Turnpike is estimated.