Unimodal regression using Bernstein–Schoenberg splines and penalties

Abstract

Research in the field of nonparametric shape constrained regression has been intensive. However, only few publications explicitly deal with unimodality although there is need for such methods in applications, for example, in dose-response analysis. In this article, we propose unimodal spline regression methods that make use of Bernstein-Schoenberg splines and their shape preservation property. To achieve unimodal and smooth solutions we use penalized splines, and extend the penalized spline approach toward penalizing against general parametric functions, instead of using just difference penalties. For tuning parameter selection under a unimodality constraint a restricted maximum likelihood and an alternative Bayesian approach for unimodal regression are developed. We compare the proposed methodologies to other common approaches in a simulation study and apply it to a dose-response data set. All results suggest that the unimodality constraint or the combination of unimodality and a penalty can substantially improve estimation of the functional relationship.

Adaptive risk bounds in unimodal regression

“…For classes of unimodal functions indexed by a smoothness parameter, the authors prove rates of convergence of the mode of the LSE and also show that the rates are minimax optimal up to logarithmic factors, for a given smoothness class. Köllmann et al (2014) propose the use of a penalized estimator based on splines to estimate the underlying unimodal function, but without studying the risk properties of their estimator. Hence, apparently little is known about the behavior of the LSE θ as an estimator of θ * .…”

Identifying optimal dosage regimes under safety constraints: An application to long term opioid treatment of chronic pain

“…To estimate the Q-functions at t = 1, 2 we use Random Forests [30] and to estimate the Q-function at baseline we use L 2 -penalized unimodal regression with Bernstein-Schoenberg Splines [31] with the amount of penalization chosen by cross-validated mean squared error. At baseline the state S 0 is a vector composed of unity, age, weight ( blweight ), baseline pain score ( Y 0 ), and the baseline safety event indicator ( Z 0 ).…”

Change-point estimation using shape-restricted regression splines

“…This function estimator tends to have a "spike" at the largest observation near the true mode, and hence the mode estimator tends to have a larger variance than that for a smoothed function estimator. The convergence rate ofm satisfies that lim n→∞ sup{n/(log(n)) 2γ } 1/(2s+1) |m − m| < ∞ a.s.. Köllmann, Bornkamp, and Ickstadt (2014) discuss penalized spline methods to achieve unimodal and smooth estimation of a functional relationship in a scenario of dose-response analysis. They use a restricted maximum likelihood approach to choose the tuning parameter and to estimate B-spline coefficients.…”

“…Unimodal Case. We compare our estimatorm S with the estimatorm P of Shoung and Zhang (2001) and the estimatorm U of Köllmann et al (2014). We choose two unimodal functions.…”