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 means and standard deviations of the results. Statistical Taylor expansion traces the dependency of the input uncertainties in the intermediate steps, so that the variables in the intermediate analytic expressions can no longer be regarded as independent of each other, and the result of the analytic expression is path independent. Thus, it differs fundamentally from the conventional common approaches in applied mathematics which optimize execution path for each calculation. In fact, statistical Taylor expansion may standardize numerical calculations for analytic expressions. Its statistical nature allows religious testing of its result when the sample size is large enough. This paper also introduces an implementation of statistical Taylor expansion called variance arithmetic and presents corresponding test results in a very wide range of mathematical applications. Another important conclusion of this paper is that the numerical errors in the library function can have significant effects on the result. For example, the periodic numerical errors in the trigonometric library functions can resonate with periodic signals, producing large numerical errors in the results.