Bayesian Nonparametric Monotone Regression

TL;DR

This paper introduces a Bayesian nonparametric method for monotone regression using Bernstein polynomials and Dirichlet process priors, enabling inference on derivatives with uncertainty quantification, demonstrated on aerosol concentration data.

Contribution

It proposes a novel Bayesian nonparametric approach for monotone regression that allows closed-form derivative inference and effective clustering of basis functions.

Findings

Performs comparably to existing methods on wavy functions

Outperforms others on linear functions

Provides full uncertainty quantification for derivatives

Abstract

In many applications there is interest in estimating the relation between a predictor and an outcome when the relation is known to be monotone or otherwise constrained due to the physical processes involved. We consider one such application--inferring time-resolved aerosol concentration from a low-cost differential pressure sensor. The objective is to estimate a monotone function and make inference on the scaled first derivative of the function. We proposed Bayesian nonparametric monotone regression which uses a Bernstein polynomial basis to construct the regression function and puts a Dirichlet process prior on the regression coefficients. The base measure of the Dirichlet process is a finite mixture of a mass point at zero and a truncated normal. This construction imposes monotonicity while clustering the basis functions. Clustering the basis functions reduces the parameter space and allows the estimated regression function to be linear. With the proposed approach we can make closed-formed inference on the derivative of the estimated function including full quantification of uncertainty. In a simulation study the proposed method performs similar to other monotone regression approaches when the true function is wavy but performs better when the true function is linear. We apply the method to estimate time-resolved aerosol concentration with a newly-developed portable aerosol monitor. The R package bnmr is made available to implement the method.