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HoeffdingD

HoeffdingD[v₁,v₂]

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

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 v_1i, is the rank of v_2i, 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}dist_i,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:

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