Statistical Taylor Expansion: A New and Path-Independent Method for Uncertainty Analysis

As a rigorous statistical approach, statistical Taylor expansion extends the conventional Taylor expansion by replacing precise input variables with random variables of known distributions to compute the means and standard deviations of the results. Statistical Taylor expansion tracks the propagation of the input uncertainties through intermediate steps, causing the variables in intermediate analytic expressions to become interdependent, while the final analytic result becomes path independent. This fundamentally distinguishes it from common approaches in applied mathematics that optimize computational path for each calculation. In essence, statistical Taylor expansion may standardize numerical computations for analytic expressions. Its statistical nature enables reliable testing of results when the sample size is sufficiently large, as well as includes sample count in result qualification. This study also introduces an 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. For instance, periodic numerical errors in trigonometric library functions may resonate with periodic signals, producing large numerical errors in the outcomes. AMS subject classifications: G.1.0