Reducing the Bias and Uncertainty of Free Energy Estimates by Using Regression to Fit Thermodynamic Integration Data

TL;DR

This paper introduces a polynomial regression method with optimized lambda point selection to improve the accuracy and reduce the bias and uncertainty in free energy estimates from thermodynamic integration data.

Contribution

The study demonstrates that polynomial regression combined with Chebyshev node sampling enhances free energy estimate accuracy over traditional quadrature methods.

Findings

High-degree polynomial regression yields more accurate free energy differences.

Using Chebyshev nodes for lambda values improves estimate precision.

Regression with non-equidistant points reduces bias and uncertainty.

Abstract

This report presents the application of polynomial regression for estimating free energy differences using thermodynamic integration. We employ linear regression to construct a polynomial that optimally fits the thermodynamic integration data, and thus reduces the bias and uncertainty of the resulting free energy estimate. Two test systems with analytical solutions were used to verify the accuracy and precision of the approach. Our results suggest that regression with a high degree of polynomials give the most accurate free energy difference estimates, but often with a slightly larger variance, compared to commonly used quadrature techniques. High degrees of polynomials possess the flexibility to closely fit the thermodynamic integration data but are often sensitive to small changes in data points. To further improve overall accuracy and reduce uncertainty, we also examine the use of Chebyshev nodes to guide the selection of non-equidistant lambda values for the thermodynamic integration scheme. We conclude that polynomial regression with non-equidistant lambda values delivers the most accurate and precise free energy estimates for thermodynamic integration data. Software and documentation is available at http://www.phys.uidaho.edu/ytreberg/software