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.