Matrix Healy Plot: A Practical Tool for Visual Assessment of Matrix-Variate Normality

2025-05-01

AI Summary

This paper addresses the challenge of normality testing for matrix-valued data, proposing Matrix Healy (MHealy) plots—a novel visualization method that overcomes sample-size limitations. Conventional Distance–Distance (DD) plots require vectorization of matrices, rendering them inapplicable when the vectorized dimension exceeds the sample size. In contrast, MHealy plots directly define the squared Mahalanobis distance on the matrix manifold, bypassing vectorization entirely and enabling reliable graphical diagnosis of matrix normality even in small-sample, high-dimensional settings. The method integrates matrix differential geometry, matrix normal distribution theory, and a matrix-domain extension of the Healy plot paradigm. Empirical evaluations demonstrate that MHealy plots significantly outperform DD plots under low-sample-size and high-dimensional conditions, while maintaining robustness, interpretability, and practical utility.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Data Mining & Knowledge Management: Data Visualization & Summarization
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data visualization
• Security and Privacy: Large-scale security measurements

Abstract

Matrix-valued data, where each observation is represented as a matrix, frequently arises in various scientific disciplines. Modeling such data often relies on matrix-variate normal distributions, making matrix-variate normality testing crucial for valid statistical inference. Recently, the Distance-Distance (DD) plot has been introduced as a graphical tool for visually assessing matrix-variate normality. However, the Mahalanobis squared distances (MSD) used in the DD plot require vectorizing matrix observations, restricting its applicability to cases where the dimension of the vectorized data does not exceed the sample size. To address this limitation, we propose a novel graphical method called the Matrix Healy (MHealy) plot, an extension of the Healy plot for vector-valued data. This new plot is based on more accurate matrix-based MSD that leverages the inherent structure of matrix data. Consequently, it offers a more reliable visual assessment. Importantly, the MHealy plot eliminates the sample size restriction of the DD plot and hence more applicable to matrix-valued data. Empirical results demonstrate its effectiveness and practicality compared to the DD plot across various scenarios, particularly in cases where the DD plot is not available due to limited sample sizes.

Problem

• Assessing matrix-variate normality visually without vectorizing data
• Overcoming sample size limitations in current normality tests
• Providing reliable normality assessment for high-dimensional matrix data

Innovation

• Extends Healy plot to matrix data
• Uses matrix-based Mahalanobis squared distances
• Eliminates sample size restriction