Partial distance correlation with methods for dissimilarities
Open accessAbstract
Distance covariance and distance correlation are scalar coefficients that characterize independence of random vectors in arbitrary dimension. Properties, extensions and applications of distance correlation have been discussed in the recent literature, but the problem of defining the partial distance correlation has remained an open question of considerable interest. The problem of partial distance correlation is more complex than partial correlation partly because the squared distance covariance is not an inner product in the usual linear space. For the definition of partial distance correlation, we introduce a new Hilbert space where the squared distance covariance is the inner product. We define the partial distance correlation statistics with the help of this Hilbert space, and develop and implement a test for zero partial distance correlation. Our intermediate results provide an unbiased estimator of squared distance covariance, and a neat solution to the problem of distance correlation for dissimilarities rather than distances.
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Adding a constant to off-diagonal elements of each distance matrix prior to computation of cMDS is the recommended solution for non-Euclidean distances.
“The recommended solution is to add a constant to off-diagonal elements of each distance matrix prior to computation of cMDS, which has a solution derived in Cailliez (1983) and recently applied in the formulation of partial distance correlation (Székely and Rizzo 2014).”
From Similarity-based multimodal regression (2024) · cites this paper as Székely and Rizzo 2014
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- Authors
- Székely, G. J., & Rizzo, M. L.
- Affiliations
- Bowling Green State University (US)
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 2014-12-01
- Volume
- 42
- Issue
- 6
- Pages
- 2382-2412
- ISSN
- 0090-5364 (ISSN-L)
- 2168-8966
- OpenAlex
- W2077398228
- Cited by
- 34
- References
- 32
- Citations by year
- 2025: 2 · 2024: 3 · 2023: 9 · 2022: 9 · 2021: 3 · 2020: 1 · 2019: 1 · 2018: 1 · 2016: 3 · 2014: 1 · 2013: 1
- FWCI
- 1.41
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- The Annals of Statistics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- Project Euclid (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Keywords
- distance covariance, distance correlation, independence testing, Hilbert spaces, dissimilarity, unbiased estimators, multivariate dependence
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