I-SMOOTH: Iteratively Smoothing Mean-Constrained and Nonnegative Piecewise-Constant Functions

Continuous nonnegative functions, such as Poisson rate functions, are sometimes approximated as piecewise-constant functions. We consider the problem of automatically smoothing such functions while maintaining the integral of each piece and maintaining nonnegativity everywhere, without specifying a parametric function. We develop logic for SMOOTH (Smoothing via Mean-constrained Optimized-Objective Time Halving), a quadratic-optimization algorithm that yields a smoother nonnegative piecewise-constant rate function having twice as many time intervals, each of half the length. I-SMOOTH (Iterated SMOOTH) iterates the SMOOTH formulation to create a sequence of piecewise-constant rate functions that, in the limit, yields a nonparametric continuous function. We consider two contexts: finite-horizon and cyclic. We develop a sequence of computational simplifications for SMOOTH, moving from numerically minimizing the quadratic objective function, to numerically computing a matrix inverse, to a closed-form matrix inverse obtained as finite sums, to optimal decision-variable values that are linear combinations of the given rates, and to simple approximations.