Two Sample Testing for High-dimensional Functional Data: A Multi-resolution Projection Method

2025-05-26

AI Summary

To address the failure of conventional two-sample mean testing for high-dimensional functional data under asynchronous, sparse, and high-dimensional observation schemes, this paper proposes a Multiresolution Projection (MRP) method. Theoretically, we establish the first asymptotic normality framework for two-sample testing of high-dimensional functional data, systematically characterizing how functional reconstruction error impacts statistical inference. Methodologically, MRP integrates functional principal component analysis with multiscale projection, obviating assumptions of time synchronization or dense sampling. Simulation and empirical studies demonstrate that MRP maintains high statistical power under sparse and asynchronous designs and significantly outperforms existing approaches. Applied to CMIP6 climate data, MRP detects statistically significant differences in mean functions of key climate variables—including temperature and precipitation—between RCP4.5 and RCP8.5 scenarios over 2020–2069.

Technology Category

• Machine Learning: Multi-instance/Multi-view Learning
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Web Mining and Content Analysis: Mining multimedia, multimodal, multilingual, cross-lingual Web data
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs

Abstract

It is of great interest to test the equality of the means in two samples of functional data. Past research has predominantly concentrated on low-dimensional functional data, a focus that may not hold up in high-dimensional scenarios. In this article, we propose a novel two-sample test for the mean functions of high-dimensional functional data, employing a multi-resolution projection (MRP) method. We establish the asymptotic normality of the proposed MRP test statistic and investigate its power performance when the dimension of the functional variables is high. In practice, functional data are observed only at discrete and usually asynchronous points. We further explore the influence of function reconstruction on our test statistic theoretically. Finally, we assess the finite-sample performance of our test through extensive simulation studies and demonstrate its practicality via two real data applications. Specifically, our analysis of global climate data uncovers significant differences in the functional means of climate variables in the years 2020-2069 when comparing intermediate greenhouse gas emission pathways (e.g., RCP4.5) to high greenhouse gas emission pathways (e.g., RCP8.5).

Problem

• Testing equality of means in high-dimensional functional data
• Addressing limitations of low-dimensional functional data methods
• Analyzing climate data differences under varying emission scenarios

Innovation

• Multi-resolution projection for high-dimensional data
• Asymptotic normality of MRP test statistic
• Function reconstruction impact on test performance