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Arrival Rate Approximation by Nonnegative Cubic Splines

Department of Management Science and Information Systems, Business School and RUTCOR, Rutgers University, Piscataway, New Jersey 08854
Department of Management Science and Information Systems, Business School and RUTCOR, Rutgers University, Piscataway, New Jersey 08854
Faculty of Engineering and Natural Sciences, Sabanci University, Orhanli, Tuzla, 34956 Istanbul, Turkey
RUTCOR, Rutgers University, Piscataway, New Jersey 08854
alizadeh{at}rutcor.rutgers.edu
jeckstei{at}rci.rutgers.edu
nnoyan{at}sabanciuniv.edu
grudolf{at}rutcor.rutgers.edu

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.

Subject classifications: nonlinear programming; applications; probability; statistics; nonparametric.
History: Received May 2005; revision received February 2007; accepted March 2007.