A Conjugate Model for Dimensional Analysis

Dimensional analysis (DA) is a methodology widely used in physics and engineering. The main idea is to extract key variables based on physical dimensions. Its overlooked importance in statistics has been recognized recently. However, most literature treats DA as merely a preprocessing tool, leading to multiple statistical issues. In particular, there are three critical aspects: (a) the nonunique choice of basis quantities and dimensionless variables; (b) the statistical representation and testing of DA constraints; (c) the spurious correlations between post-DA variables. There is an immediate need for an appropriate statistical methodology that integrates DA and the quantitative modeling. In this article, we propose a power-law type of “DA conjugate” model that is useful for incorporating dimensional information and analyzing post-DA variables. Adapting the similar idea of “conjugacy” in Bayesian analysis, we show that the proposed modeling technique not only produces flexible and effective results, but also provides good solutions to the above three issues. A modified projection pursuit regression analysis is implemented to fit the additive power-law model. A numerical study on ocean wave speed is discussed in detail to illustrate and evaluate the advantages of the proposed procedure. Supplementary materials for this article are available online.

量纲分析（Dimensional Analysis，DA）是物理学与工程领域广泛应用的方法论。其核心思路为基于物理量纲提取关键变量。该方法在统计学中长期被忽视的重要性，近来已得到学界的广泛认知。然而，绝大多数现有文献仅将量纲分析视作单一的预处理工具，由此引发了诸多统计学层面的问题。具体而言，存在三类关键问题：(a) 基量与无量纲变量的选择不具备唯一性；(b) 量纲分析约束的统计学表征与检验；(c) 经量纲分析处理后的变量间存在虚假相关。当前亟需一套能够整合量纲分析与量化建模的适配性统计学方法论。本文提出一种幂律型“量纲共轭（DA conjugate）”模型，可有效融入量纲信息并用于分析经量纲分析处理后的变量。借鉴贝叶斯分析中“共轭性”的相似思路，本文证明所提出的建模方法不仅可生成灵活且有效的分析结果，还能为上述三类问题提供优良的解决方案。本文采用改进的投影寻踪回归分析方法，以拟合可加性幂律模型。此外，本文通过针对海浪速度的数值实验，详细阐述并验证了所提方法的优势。本文的补充材料可在线获取。