Four simple axioms of dependence measures

Móri, Tamás F.; Székely, Gábor

doi:10.1007/s00184-018-0670-3

Cited by 24 publications

(23 citation statements)

References 34 publications

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“…transformations consist of all Euclidean transformations and all (nonzero) scaling (Móri and Székely (2019)). The following properties hold:…”

Section: Multivariate Independence Testing Proceduresmentioning

confidence: 99%

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The Binary Expansion Randomized Ensemble Test

Lee¹,

Zhang²,

Kosorok³

2024

STAT SINICA

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Recently, the binary expansion testing framework was introduced to test the independence of two continuous random variables by utilizing symmetry statistics that are complete sufficient statistics for dependence. We develop a new test based on an ensemble approach that uses the sum of squared symmetry statistics and distance correlation. Simulation studies suggest that this method improves the power while preserving the clear interpretation of the binary expansion testing. We extend this method to tests of independence of random vectors in arbitrary dimension. Through random projections, the proposed binary expansion randomized ensemble test transforms the multivariate independence testing problem into a univariate problem. Simulation studies and data example analyses show that the proposed method provides relatively robust performance compared with existing methods.

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“…transformations consist of all Euclidean transformations and all (nonzero) scaling (Móri and Székely (2019)). The following properties hold:…”

Section: Multivariate Independence Testing Proceduresmentioning

confidence: 99%

“…The third benefit of our method is "invariance". Móri and Székely (2019) introduced axioms for a measure to be a dependence measure. If a measure ∆ satisfies ∆(f (X), g(Y )) = ∆(X, Y ) where f, g are similarity transformations, it is called invariant with respect to similarity transformations.…”

Section: Multivariate Independence Testing Proceduresmentioning

confidence: 99%

The Binary Expansion Randomized Ensemble Test

Lee¹,

Zhang²,

Kosorok³

2024

STAT SINICA

View full text Add to dashboard Cite

show abstract

“…Measures of variable dependence can be roughly divided into the following four categories: grid-based method, mutual information estimation, distance/kernel-based statistics and correlation-based methods [9]. These measures are summarized in Table 1.…”

Section: Literature Reviewmentioning

confidence: 99%

A Matrix Iteration Algorithm With Pruning for Pinpointing Multivariate Correlations From High Dimensional Data Sets

et al. 2021

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There are a few dependent multivariate relationships among high dimensional data sets. Then how to identify these dependent variables from high dimensional data sets is an important issue for data analysis. Now, the most frequently used method is the enumeration method, that is all multivariate relationships in the high dimensional data sets are examined. However, the time complexity of the enumeration method is exponential (2 n ) and the calculation load is very heavy when the dimension is high. Aiming at solving this problem, the matrix iteration algorithm with pruning (MIP) is proposed for pinpointing multivariate dependent relationships in high dimensional data sets without examining all multivariate relationships. Some not dependent relationships are ignored without examined by the pruning process of the proposed MIP and the computing burden is reduced. The maximal information coefficient (MIC) is adopted as the measure of correlations in the proposed MIP algorithm due to the excellent properties, generality and equitability, of MIC. In the case of the data set with 5 variables, more than 50% multivariate relationships are pruned without examining. Numerical experiments also show that the calculating burden is greatly reduced. Compared to the enumeration method, 82.5% calculating time and 98.5% calculating times of multivariate relationships are saved for the data set with two dependent multivariate relationships among 30 variables in the experiment. The proposed MIP algorithm is effective for pinpointing multivariate dependent relationships from data sets with high dimensions.INDEX TERMS correlation, high dimensions, maximal information coefficient, pruning algorithm

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“…A prerequisite for the continuity (A4) is the finiteness of the measure d, cf. [31]. Thus all considerations for (normalized) multivariance are under the moment condition (5) and for the multicorrelation we have to assume (18).…”

Section: Axiomatic Classification Of Dependence Measuresmentioning

confidence: 99%

“…A similarity transform is any combination of translations, rotations, and reflections and non zero scalings (using the same scaling factor for all components of a vector), cf [31]…”

mentioning

confidence: 99%

Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance

Böttcher

2019

Open Statistics

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Distance multivariance is a multivariate dependence measure, which can detect dependencies between an arbitrary number of random vectors each of which can have a distinct dimension. Here we discuss several new aspects, present a concise overview and use it as the basis for several new results and concepts: In particular, we show that distance multivariance unifies (and extends) distance covariance and the Hilbert-Schmidt independence criterion HSIC, moreover also the classical linear dependence measures: covariance, Pearson's correlation and the RV coefficient appear as limiting cases. Based on distance multivariance several new measures are defined: a multicorrelation which satisfies a natural set of multivariate dependence measure axioms and m-multivariance which is a dependence measure yielding tests for pairwise independence and independence of higher order. These tests are computationally feasible and under very mild moment conditions they are consistent against all alternatives. Moreover, a general visualization scheme for higher order dependencies is proposed, including consistent estimators (based on distance multivariance) for the dependence structure.Many illustrative examples are provided. All functions for the use of distance multivariance in applications are published in the R-package multivariance.

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Four simple axioms of dependence measures

Cited by 24 publications

References 34 publications

The Binary Expansion Randomized Ensemble Test

The Binary Expansion Randomized Ensemble Test

A Matrix Iteration Algorithm With Pruning for Pinpointing Multivariate Correlations From High Dimensional Data Sets

Dependence and dependence structures: estimation and visualization using the unifying concept of distance multivariance

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