Bayesian Nonparametric Monotone Regression

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.

Keywords: Aerosol monitors; Bernstein polynomials; Dirichlet process; Fine particulate matter; monotone regression.

Figure 1:

Various representations of the Bernstein polynomial (BP) basis functions. Panel 1a shows the 51 BP basis functions of order M = 50 (Ψ(x, M)). Panel 1b shows the transformed BP basis represented as Ψ(x, M)A⁻¹ as described in 2.1. This transformation is used for both BNMR and BISOREG. Panel 1c shows the posterior mode group of basis functions selected to be included into the model with BISOREG. This is a subset of the transformed basis functions shown in Panel 1a. Panel 1d shows the posterior model combination of basis functions included with BNMR. This includes the intercept and three basis functions which are each a linear combination of one to three of the basis functions shown in panel 1b and subsequently linear combinations of the basis functions shown in Panel 1a. Results from all 12 runs are shown in the supplemental material.

Figure 2:

Simulation results for the number of non-zero regression coefficients (dashed line) and the number of unique values of the non-zero regression coefficient (solid line) for BISOREG (triangle) and BNMR (×). The number of unique non-zero values is always equal to the total number of non-zero values in BISOREG.

Figure 3:

Estimated pressure drop from the MARS data for one run. Each panel shows the estimates and 95% intervals for each method separately. Results from all 12 runs are shown in the supplemental material.

Figure 4:

Estimated PM_2.5 concentration from the MARS data. Panel 4a shows the posterior mean and 95% interval from BNMR and Panel 4b shows the posterior mean and 95% interval from BISOREG. The dashed line in each panel is the PM_2.5 concentration measured with the TEOM. Results from all 12 runs are shown in the supplemental material. Results with other methods are also shown in the supplemental material.