Arrival rate approximation by nonnegative cubic splines

We describe an optimization method to approximate the arrival rate of data such as e-mail messages, website visits, changes to databases, and changes to websites mirrored by other servers. We model these arrival rates as non-homogeneous Poisson process based on observed arrival data. We estimate the arrival function by cubic splines using the maximum likelihood principle. A critical feature of the model is that the splines are constrained to be everywhere nonnegative. We formulate this constraint 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 precision. We formulate the estimation problem as a convex program related to semidefinite programming and solve it with a standard nonlinear optimization package called KNITRO. 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.