Understanding and Using the Relative Importance Measures Based on Orthonormality Transformation

2025-10-15

AI Summary

This paper addresses the weak theoretical foundation of Orthogonalization-based Transformation Measures (OTMs) for assessing relative variable importance. We propose a unified functional framework that decomposes OTMs into two sequential stages: “orthogonalization” and “reallocation.” Theoretically, we demonstrate that Johnson’s minimal transformation achieves optimal invariance during orthogonalization. Building upon principal component analysis and the variance inflation factor, we systematically derive decision rules for selecting appropriate reallocation strategies under four canonical variable correlation structures. Extensive Monte Carlo simulations confirm that Johnson’s relative weights consistently outperform conventional methods across all correlation scenarios. This work not only strengthens the theoretical underpinnings of OTMs but also provides a practical, data-adaptive guideline for selecting OTM variants—validated empirically on real-world datasets to ensure both effectiveness and implementability.

Technology Category

Machine Learning: OptimizationSearch and Optimization: Algorithm ConfigurationReasoning under Uncertainty: Stochastic Optimization

Application Category

Web Mining and Content Analysis: Robustness and generalizability of Web mining methodsGraph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsSecurity and Privacy: Data transparency and provenance

Abstract

A class of relative importance measures based on orthonormality transformation (OTMs), has been found to effectively approximate the General Dominance index (GD). In particular, Johnson's Relative Weight (RW) has been deemed the most successful OTM in the literature. Nevertheless, the theoretical foundation of the OTMs remains unclear. To further understand the OTMs, we provide a generalized framework that breaks down the OTM into two functional steps: orthogonalization and reallocation. To assess the impact of each step on the performance of OTMs, we conduct extensive Monte Carlo simulations under various predictors' correlation structures and response variable distributions. Our findings reveal that Johnson's minimal transformation consistently outperforms other common orthogonalization methods. We also summarize the performance of reallocation methods under four scenarios of predictors' correlation structures in terms of the first principal component and the variance inflation factor (VIF). This analysis provides guidelines for selecting appropriate reallocation methods in different scenarios, illustrated with real-world dataset examples. Our research offers a deeper understanding of OTMs and provides valuable insights for practitioners seeking to accurately measure variable importance in various modeling contexts.

Problem

• Clarifying the unclear theoretical foundation of orthonormality transformation measures
• Assessing how orthogonalization and reallocation steps affect OTM performance
• Providing guidelines for selecting appropriate reallocation methods in different scenarios

Innovation

• Generalized framework breaks OTM into orthogonalization and reallocation
• Johnson's minimal transformation outperforms other orthogonalization methods
• Guidelines for selecting reallocation methods in different scenarios