Statistical Taylor Expansion: A New and Path-Independent Method for Uncertainty Analysis
This addresses uncertainty propagation in applied mathematics, offering a standardized approach for numerical computations, though it appears incremental as an extension of conventional Taylor expansion.
The paper tackles the problem of uncertainty analysis in numerical computations by introducing statistical Taylor expansion, a method that replaces precise inputs with random variables to compute mean, deviation, and reliable factors, resulting in path-independent analytic results and highlighting the impact of library function errors.
As a rigorous statistical approach, statistical Taylor expansion extends the conventional Taylor expansion by replacing precise input variables with random variables of known distributions and sample counts to compute the mean, the deviation, and the reliable factor of each result. It tracks the propagation of the input uncertainties through intermediate steps, so that the final analytic result becomes path independent. Therefore, it differs fundamentally from common approaches in applied mathematics that optimize computational path for each calculation. Statistical Taylor expansion may standardize numerical computations for analytic expressions. This study also introduces the implementation of statistical Taylor expansion termed variance arithmetic and presents corresponding test results across a wide range of mathematical applications. Another important conclusion of this study is that numerical errors in library functions can significantly affect results. It is desirable that each value from library functions be accomplished by an uncertainty deviation. The possible link between statistical Taylor expansion and quantum physics is discussed as well.