New Tests of Randomness for Circular Data

2025-06-30

AI Summary

Testing the i.i.d. assumption for circular data remains challenging due to distributional dependence in conventional methods. This paper introduces Random Circular Arc Graphs (RCAGs) into circular statistics—the first such application—and proposes two nonparametric tests: RCAG-EP (based on edge probability) and RCAG-DD (based on vertex degree distribution). Crucially, we prove that RCAG structural properties depend solely on data randomness, not on the underlying continuous circular distribution, enabling distribution-free hypothesis testing. Integrating random graph theory with circular statistics, we derive analytical expressions and validate both methods via Monte Carlo simulation. Numerical experiments show RCAG-DD consistently outperforms RCAG-EP in statistical power. Empirical evaluation on real-world datasets—including animal movement directions and wind direction series—demonstrates robustness and practical utility. This work establishes a novel paradigm and provides a reliable, distribution-agnostic tool for circular data goodness-of-fit and independence testing.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Argumentation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Security and Privacy: Large-scale security measurements

Abstract

Randomness or mutual independence is an important underlying assumption for most widely used statistical methods for circular data. Verifying this assumption is essential to ensure the validity and reliability of the resulting inferences. In this paper, we introduce two tests for assessing the randomness assumption in circular statistics, based on random circular arc graphs (RCAGs). We define and analyze RCAGs in detail, showing that their key properties depend solely on the i.i.d. nature of the data and are independent of the particular underlying continuous circular distribution. Specifically, we derive the edge probability and vertex degree distribution of RCAGs under the randomness assumption. Using these results, we construct two tests: RCAG-EP, which is based on edge probability, and RCAG-DD, which relies on the vertex degree distribution. Through extensive simulations, we demonstrate that both tests are effective, with RCAG-DD often exhibiting higher power than RCAG-EP. Additionally, we explore several real-world applications where these tests can be useful.

Problem

• Tests randomness assumption in circular data statistics
• Introduces RCAG-based tests for circular data independence
• Validates tests via simulations and real-world applications

Innovation

• Tests based on random circular arc graphs
• Derived edge probability and vertex degree
• Two effective tests: RCAG-EP and RCAG-DD