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Hausdorff correlation for interval-valued random objects
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DOI：10.1007/s11222-025-10743-2.png)
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Abstract
En 中文
Analyzing the correlation between interval-valued data presents an essential yet challenging problem in modern statistical research due to the lack of basic geometric and algebraic structures. Existing methods are often limited by their reliance on algebraic formulations or assumptions about the underlying distribution of true values within intervals. Moreover, they primarily focus on simple midpoint-range interval representations, restricting their applicability to more complex interval structures, e.g., when the interval contains multiple segments. To address these limitations, we introduce the Fr & eacute;chet framework into the interval metric space equipped with the Hausdorff distance, extending the notions of Fr & eacute;chet mean and proposing a more general and straightforward interval dependency measure, called Hausdorff correlation. The proposed method offers a strong geometric interpretation, revealing the relationship between random intervals and their Hausdorff mean, while also accommodating a broader range of interval forms. From a theoretical perspective, we establish the foundational properties of the proposed framework, proving the existence and uniqueness of the Hausdorff mean. Empirical evaluations on both synthetic and real-world datasets demonstrate the distinctiveness and effectiveness of Hausdorff correlation and its superior performance in feature selection compared to existing methods. In particular, a real-world Wearable Watch Dataset analysis shows the Hausdorff correlation successfully captures the relationship between multi-segment sleep intervals and physiological indicators, where existing methods fail to provide meaningful estimates.
Keywords:
Hausdorff distance
Metric space
Interval-valued data
Multi-segment intervals
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