用随机变量替代确定值，实现不确定性传播的精确路径无关分析
Statistical Taylor Expansion: A New and Path-Independent Method for Uncertainty Analysis
- 将输入变量替换为已知分布的随机变量，逐层追踪不确定性传播
- 结果与计算路径无关，且可量化追踪精度，避免传统方法偏差
- 适用于数学建模、回归分析等场景，适合需高可信度的工程应用
统计泰勒展开是传统泰勒展开的严格拓展，将每个确定性输入变量替换为具有已知分布和样本量的随机变量，进而计算输出的均值、方差及置信边界。通过追踪输入不确定性在所有中间步骤中的传播，使最终结果具有路径无关性，并能精确量化追踪质量。这一特性从根本上区别于依赖计算路径的传统数值方法。本文提出了名为方差算术的实现方式，并在多种数学应用中验证其性能。研究还揭示了库函数中数值误差的潜在显著影响、传统回归中将输入不确定性作为权重的缺陷，以及离散傅里叶变换的建模误差。同时引入了统计代数的概念。
原文摘要 · Abstract (English)
Statistical Taylor expansion is a rigorous extension of conventional Taylor expansion that replaces each precise input variable with a random variable of known distribution and sample count, then computes the mean, deviation, and a bounding reliability of every result. By tracking the propagation of input uncertainties through all intermediate steps, it renders the final result path-independent, with precise quantification of the tracking quality. This path-independence sets it fundamentally apart from conventional numerical approaches, which are path-dependent. This study presents an implementation called variance arithmetic and demonstrates its performance across diverse mathematical applications. This study also reveals the potentially substantial impact of numerical errors in library functions, the defect of applying input uncertainties as weights in conventional regression, and the modeling error of the discrete Fourier transformation. The concept of statistical algebra is also introduced.
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