Arrival Rate Approximation by Nonnegative Cubic Splines

We describe an optimization method to approximate the arrival-rate function of a nonhomogeneous Poisson process based on observed arrival data. We estimate the function by cubic splines, using an optimization model based on the maximum-likelihood principle. A critical feature of the model is that the splines are constrained to be nonnegative everywhere. We enforce these constraints by using a characterization of nonnegative polynomials by positive semidefinite matrices. We also describe versions of our model that allow for periodic arrival-rate functions and input data of limited time precision. We formulate the estimation problem as a convex nonlinear program, and solve it with standard nonlinear optimization packages. We present numerical results using both an actual record of e-mail arrivals over a period of 60 weeks, and artificially generated data sets. We also present a cross-validation procedure for determining an appropriate number of spline knots to model a set of arrival observations.