HoeffdingD

HoeffdingD[v1,v2]

gives Hoeffding's dependence measure for the vectors v1 and v2.

HoeffdingD[m]

gives Hoeffding's dependence measure for the matrix m.

HoeffdingD[m1,m2]

gives Hoeffding's dependence measure for the matrices m1 and m2.

HoeffdingD[dist]

gives Hoeffding's matrix for the multivariate symbolic distribution dist.

HoeffdingD[dist,i,j]

gives the ^(th) element of for the multivariate symbolic distribution dist.

Details

  • HoeffdingD[v1,v2] gives Hoeffding's dependence measure between v1 and v2.
  • Hoeffding's is a measure of dependence based on the relative order of elements in the two lists.
  • Hoeffding's between v1 and v2 is given by 30 (-2 (n-2) R+(n-3) (n-2) S+Q)/TemplateBox[{{n, -, 4}, 5}, Pochhammer], where is the number of observations in v1, , , , for , is the rank of v1i, is the rank of v2i, and is equal to Boole[a<b].
  • The arguments v1 and v2 can be any real‐valued vectors of equal length greater than 5.
  • For a matrix m with columns, HoeffdingD[m] is a × matrix of the dependence measures between columns of m.
  • For an × matrix m1 and an × matrix m2, HoeffdingD[m1,m2] is a × matrix of the dependence measures between columns of m1 and columns of m2.
  • HoeffdingD[dist,i,j] is given by 30 Expectation[(F[x,y]-G[x]H[y])^2,{x,y}disti,j], where F[x,y], G[x], and H[y] are the CDFs of the ^(th), ^(th), and ^(th) marginals of dist respectively.
  • HoeffdingD[dist] gives a matrix where the ^(th) entry is given by HoeffdingD[dist,i,j].

Examples

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Basic Examples  (4)

Hoeffding's for two vectors:

Hoeffding's for a matrix:

Hoeffding's for two matrices:

Compute Hoeffding's for a bivariate distribution:

Compare to a simulated value:

Scope  (7)

Data  (4)

Exact input yields exact output:

Approximate input yields approximate output:

Works with large arrays:

SparseArray data can be used:

Distributions and Processes  (3)

Hoeffding's matrix for a continuous multivariate distribution:

Hoeffding's matrix for derived distributions:

Data distribution:

Hoeffding's matrix for a random process at times and :

Applications  (3)

Hoeffding's is typically used to detect non-monotonic dependency structures:

Hoeffding's tends to be larger for dependent vectors:

The value tends to 0 for independent vectors:

Hoeffding's can detect linear dependence:

SpearmanRho and KendallTau are more sensitive to linear dependence:

Hoeffding's can also detect many types of nonlinear dependence:

Use HoeffdingDTest to determine if the value is statistically significant:

Properties & Relations  (4)

Larger values of Hoeffding's indicate increasing dependence:

Hoeffding's matrix is symmetric:

The diagonal elements of Hoeffding's matrix are 1:

Hoeffding's for a continuous bivariate distribution:

Wolfram Research (2012), HoeffdingD, Wolfram Language function, https://reference.wolfram.com/language/ref/HoeffdingD.html.

Text

Wolfram Research (2012), HoeffdingD, Wolfram Language function, https://reference.wolfram.com/language/ref/HoeffdingD.html.

CMS

Wolfram Language. 2012. "HoeffdingD." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HoeffdingD.html.

APA

Wolfram Language. (2012). HoeffdingD. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HoeffdingD.html

BibTeX

@misc{reference.wolfram_2025_hoeffdingd, author="Wolfram Research", title="{HoeffdingD}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/HoeffdingD.html}", note=[Accessed: 22-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_hoeffdingd, organization={Wolfram Research}, title={HoeffdingD}, year={2012}, url={https://reference.wolfram.com/language/ref/HoeffdingD.html}, note=[Accessed: 22-April-2025 ]}