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.

中文翻译：

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贝叶斯非参数单调回归


在许多应用中，当已知预测变量和结果之间的关系是单调的或由于所涉及的物理过程而受到其他约束时，人们有兴趣估计预测变量和结果之间的关系。我们考虑这样一种应用——通过低成本压差传感器推断时间分辨气溶胶浓度。目标是估计单调函数并对函数的缩放一阶导数进行推断。我们提出了贝叶斯非参数单调回归，它使用伯恩斯坦多项式基来构造回归函数，并将狄利克雷过程置于回归系数之上。狄利克雷过程的基本测度是零质点和截断法线的有限混合。这种构造在对基函数进行聚类的同时强加了单调性。对基函数进行聚类可以减少参数空间并允许估计的回归函数是线性的。通过所提出的方法，我们可以对估计函数的导数进行封闭式推断，包括不确定性的完全量化。在模拟研究中，当真实函数是波动的时，所提出的方法的表现与其他单调回归方法类似，但当真实函数是线性时，表现更好。我们应用该方法通过新开发的便携式气溶胶监测仪来估计时间分辨气溶胶浓度。 R 包 bnmr 可用于实现该方法。