Monte Carlo errors with less errors
Ulli Wolff
TL;DR
The paper tackles the challenge of estimating means and statistical errors for nonlinear functions of primary observables in Monte Carlo simulations by explicitly estimating and summing autocorrelation functions, yielding a robust effective integrated autocorrelation time. It introduces the Gamma-method, extends to replica runs for bias control, and provides practical estimators for the variance and error of derived quantities, including an automatic window optimization for truncating autocorrelation sums. The contributions include explicit error and bias formulas, a consistent framework for both primary and derived quantities, and a MATLAB implementation that supports replica-based consistency checks. This approach enhances reliability and efficiency in analyzing expensive simulations by accurately accounting for autocorrelation effects rather than relying on binning alone.
Abstract
We explain in detail how to estimate mean values and assess statistical errors for arbitrary functions of elementary observables in Monte Carlo simulations. The method is to estimate and sum the relevant autocorrelation functions, which is argued to produce more certain error estimates than binning techniques and hence to help toward a better exploitation of expensive simulations. An effective integrated autocorrelation time is computed which is suitable to benchmark efficiencies of simulation algorithms with regard to specific observables of interest. A Matlab code is offered for download that implements the method. It can also combine independent runs (replica) allowing to judge their consistency.