Bayesian Nonparametric Monotone Regression.

Area of Science:

• Environmental Science
• Data Science
• Statistical Modeling

Background:

• Estimating constrained relationships, such as monotone functions, is crucial in scientific applications.
• Inferring time-resolved aerosol concentration from low-cost differential pressure sensors presents a practical challenge.
• Existing methods may lack accuracy or flexibility for specific physical constraints.

Purpose of the Study:

• To develop a Bayesian nonparametric monotone regression approach for estimating constrained relationships.
• To apply this method for inferring time-resolved aerosol concentration from differential pressure sensor data.
• To enable inference on the derivative of the estimated function, quantifying uncertainty.

Main Methods:

• Utilized Bernstein polynomial basis for regression function construction.
• Employed a Dirichlet process prior on regression coefficients with a mixture base measure (mass point at zero and truncated normal).
• This approach imposes monotonicity and clusters basis functions, allowing for linear function estimation and closed-form derivative inference.

Main Results:

• The proposed method demonstrates comparable performance to existing monotone regression techniques for wavy functions.
• It exhibits superior performance when the true underlying function is linear.
• Successful application in estimating time-resolved aerosol concentration using a portable aerosol monitor.

Conclusions:

• The Bayesian nonparametric monotone regression method effectively handles physical constraints in regression.
• It provides a robust tool for applications like aerosol concentration monitoring, with accurate uncertainty quantification.
• An R package, bnmr, is available for implementing the proposed methodology.