# 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  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 , 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 realvalued 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  ,  , and  marginals of dist respectively.
• HoeffdingD[dist] gives a matrix where the  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: