Arrival rate approximation by nonnegative cubic splines

We describe an optimization method to approximate the arrival rate function of a non-homogeneous 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 everywhere nonnegative. 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 sixty 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.