Piecewise-quadratic rate smoothing: The cyclic context

Even when they are known to be continuous, Poisson-process rate functions are sometimes specified as piecewise constant. To better approximate the unknown continuous rate function, we fit a piecewise-quadratic function. In addition to maintaining the rate´s integral over each time interval, at each interval´s end point we match the rates and their first derivatives. For every interval with negative rates, we force non-negativity by taking the maximum of zero and the quadratic-function value, modifying the quadratic to maintain the integral value. These rate functions can be used alone or applied after one or more iterations of I-SMOOTH, our existing algorithm designed for the same problem. We provide examples. Finally, we discuss random-process generation from piecewise-quadratic rate functions.