Monotone smoothing splines with bounds

TL;DR

This paper formulates monotone smoothing splines with bounds as a constrained variational problem, proves solution properties, and introduces a branch and bound algorithm for practical computation, demonstrated on neuroscience and distribution fitting data.

Contribution

It presents a novel formulation, theoretical guarantees, and an efficient algorithm for monotone spline smoothing with bounds.

Findings

Existence and uniqueness of solutions are established.

A new branch and bound algorithm effectively computes the spline solutions.

Applications include neuroscience data and cumulative distribution function fitting.

Abstract

The problem of monotone smoothing splines with bounds is formulated as a constrained minimization problem of the calculus of variations. Existence and uniqueness of solutions of this problem is proved, as well as the equivalence of it to a finite dimensional but nonlinear optimization problem. A new algorithm for computing the solution which is a spline curve, using a branch and bound technique, is presented. The method is applied to examples in neuroscience and for fitting cumulative distribution functions from data.