Monotone Smoothing Splines with Bounds
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Monotone Smoothing Splines with Bounds Sara Maad Sasane1
Received: 5 November 2019 / Accepted: 24 January 2020 © The Author(s) 2020
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. Keywords Monotone spline · Curve fitting · KKT conditions · Algorithm · Cumulative distribution function
1 Introduction Splines and smoothing splines have a long history of application in many fields. The basic history is outlined in Egerstedt and Martin, [3] and in Wahba, [19]. See also [20] for an introduction to the use of smoothing splines in statistics. In this paper we return to the problem of monotone smoothing splines, which was previously studied in [3, 4, 8, 9]. A classical application is to determine the average growth curve of a population of juveniles. Suppose we have a population of perhaps 30 children whose heights are measured every 6 months from age 2 to 20. It is easy to fit a smoothing spline to the data set but there is no guarantee of monotonicity. Since people just do not get shorter and then regrow, the smoothing spline is not appropriate for this and other similar situations. Instead, the appropriate tool is the monotone smoothing spline. Charles, Sun and Martin [2] used monotone smoothing splines in calculating distribution functions. There a problem can arise in that the spline may become greater than 1 at some point and by monotonicity it can never decrease, and so this violates the condition that a cumulative distribution only takes values between 0 and 1, a problem that was not addressed in [2]. In this paper, we solve this problem by imposing a condition that the spline is bounded above by some number xmax (which would be 1 in the case of cumulative distribution functions). We also present an application of monotone
B S. Maad Sasane 1
Lund University, Lund, Sweden
S. Maad Sasane
smoothing splines in mathematical biology, where sigmoidally shaped function commonly occur. The main difficulty of the problem is the monotonicity constraint x˙ ≥ 0, which has to hold at every point t in the interval under consideration. This infinite dimensional constraint is handled by a vector space version of the Karush–Kuhn–Tucker theorem, and using this, it is possible to reduce the original infinite dimensional problem to a finite dimensional problem which can be solved numerically. The paper is organized as follows: In Sect. 2, we formulate the curve fitting problem as a constrained Calculus of Variations problem, and in Sect. 3, we show existence and uniqueness of the minimizer of this minimization problem, and hence existence and uniqueness
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