Geometric Estimation of Multivariate Dependency
This paper proposes a geometric estimator of dependency between a pair of multivariate random variables. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-08-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/21/8/787 |
| _version_ | 1811212928202833920 |
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| author | Salimeh Yasaei Sekeh Alfred O. Hero |
| author_facet | Salimeh Yasaei Sekeh Alfred O. Hero |
| author_sort | Salimeh Yasaei Sekeh |
| collection | DOAJ |
| description | This paper proposes a geometric estimator of dependency between a pair of multivariate random variables. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we call the geometric mutual information (GMI), which is equivalent to the Henze−Penrose divergence. between the joint distribution of the multivariate samples and the product of the marginals. The GMI has many of the same properties as standard MI but can be estimated from empirical data without density estimation; making it scalable to large datasets. The proposed empirical estimator of GMI is simple to implement, involving the construction of an minimal spanning tree (MST) spanning over both the original data and a randomly permuted version of this data. We establish asymptotic convergence of the estimator and convergence rates of the bias and variance for smooth multivariate density functions belonging to a Hölder class. We demonstrate the advantages of our proposed geometric dependency estimator in a series of experiments. |
| first_indexed | 2024-04-12T05:37:42Z |
| format | Article |
| id | doaj.art-52b26ab5920d400990c5203cef3a63c6 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-12T05:37:42Z |
| publishDate | 2019-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-52b26ab5920d400990c5203cef3a63c62022-12-22T03:45:46ZengMDPI AGEntropy1099-43002019-08-0121878710.3390/e21080787e21080787Geometric Estimation of Multivariate DependencySalimeh Yasaei Sekeh0Alfred O. Hero1Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USADepartment of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USAThis paper proposes a geometric estimator of dependency between a pair of multivariate random variables. The proposed estimator of dependency is based on a randomly permuted geometric graph (the minimal spanning tree) over the two multivariate samples. This estimator converges to a quantity that we call the geometric mutual information (GMI), which is equivalent to the Henze−Penrose divergence. between the joint distribution of the multivariate samples and the product of the marginals. The GMI has many of the same properties as standard MI but can be estimated from empirical data without density estimation; making it scalable to large datasets. The proposed empirical estimator of GMI is simple to implement, involving the construction of an minimal spanning tree (MST) spanning over both the original data and a randomly permuted version of this data. We establish asymptotic convergence of the estimator and convergence rates of the bias and variance for smooth multivariate density functions belonging to a Hölder class. We demonstrate the advantages of our proposed geometric dependency estimator in a series of experiments.https://www.mdpi.com/1099-4300/21/8/787Henze–Penrose mutual informationFriedman–Rafsky test statisticgeometric mutual informationconvergence ratesbias and variance tradeoffoptimizationminimal spanning trees |
| spellingShingle | Salimeh Yasaei Sekeh Alfred O. Hero Geometric Estimation of Multivariate Dependency Entropy Henze–Penrose mutual information Friedman–Rafsky test statistic geometric mutual information convergence rates bias and variance tradeoff optimization minimal spanning trees |
| title | Geometric Estimation of Multivariate Dependency |
| title_full | Geometric Estimation of Multivariate Dependency |
| title_fullStr | Geometric Estimation of Multivariate Dependency |
| title_full_unstemmed | Geometric Estimation of Multivariate Dependency |
| title_short | Geometric Estimation of Multivariate Dependency |
| title_sort | geometric estimation of multivariate dependency |
| topic | Henze–Penrose mutual information Friedman–Rafsky test statistic geometric mutual information convergence rates bias and variance tradeoff optimization minimal spanning trees |
| url | https://www.mdpi.com/1099-4300/21/8/787 |
| work_keys_str_mv | AT salimehyasaeisekeh geometricestimationofmultivariatedependency AT alfredohero geometricestimationofmultivariatedependency |