Reducing the bias and uncertainty of free energy estimates by using regression to fit thermodynamic integration data

This report presents the application of polynomial regression for estimating free energy differences using thermodynamic integration data, i.e., slope of free energy with respect to the switching variable lambda. 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 use of regression with high degree of polynomials provides the most accurate free energy difference estimates, but often with slightly larger uncertainty, compared to commonly used quadrature techniques. High degree polynomials possess the flexibility to closely fit the thermodynamic integration data but are often sensitive to small changes in the data points. Thus, we also used Chebyshev nodes to guide in the selection of nonequidistant lambda values for use in thermodynamic integration. We conclude that polynomial regression with nonequidistant lambda values delivers the most accurate and precise free energy estimates for thermodynamic integration data for the systems considered here. Software and documentation is available at http://www.phys.uidaho.edu/ytreberg/software.